Constructive set theory

Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory.
The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach.
On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

In addition to rejecting the principle of excluded middle (), constructive set theories often require some logical quantifiers in their axioms to be set bounded, motivated by results tied to impredicativity.

Introduction

Constructive outlook

Preliminary on the use of intuitionistic logic 
The logic of the set theories discussed here is constructive in that it rejects the principle of excluded middle , i.e. that the disjunction  automatically holds for all propositions . As a rule, to prove the excluded middle for a proposition , i.e. to prove the particular disjunction , either  or  needs to be explicitly proven. When either such proof is established, one says the proposition is decidable, and this then logically implies the disjunction holds. Similarly, a predicate  for  in a domain  is said to be decidable when the more intricate statement  is provable. Non-constructive axioms may enable proofs that formally claim decidability of such  (and/or ) in the sense that they prove excluded middle for  (resp. the statement using the quantifier above) without demonstrating the truth of either side of the disjunction(s). This is often the case in classical logic. In contrast, axiomatic theories deemed constructive tend to not permit many classical proofs of statements involving properties that are provenly computationally undecidable.

The law of noncontradiction is a special case of the propositional form of modus ponens. Using the former with any negated statement , one valid De Morgan's law thus implies  already in the more conservative minimal logic. In words, intuitionistic logic still posits: It is impossible to rule out a proposition and rule out its negation both at once, and thus the rejection of any instantiated excluded middle statement for an individual proposition is inconsistent. Here the double-negation captures that the disjunction statement now provenly can never be ruled out or rejected, even in cases where the disjunction may not be provable (for example, by demonstrating one of the disjuncts, thus deciding ) from the assumed axioms.

More generally, constructive mathematical theories tend to prove classically equivalent reformulations of classical theorems. For example, in constructive analysis, one cannot prove the intermediate value theorem in its textbook formulation, but one can prove theorems with algorithmic content that, as soon as double negation elimination and its consequences are assumed legal, are at once classically equivalent to the classical statement. The difference is that the constructive proofs are harder to find.

The intuitionistic logic underlying the set theories discussed here, unlike minimal logic, still permits double negation elimination for individual propositions  for which excluded middle holds. In turn the theorem formulations regarding finite objects tends to not differ from their classical counterparts. Given a model of all natural numbers, the equivalent for predicates, namely Markov's principle, does not automatically hold, but may be considered as an additional principle.

In an inhabited domain and using explosion, the disjunction  implies the existence claim , which in turn implies . Classically, these implications are always reversible. If one of the former is classically valid, it can be worth trying to establish it in the latter form. For the special case where  is rejected, one deals with a counter-example existence claim , which is generally constructively stronger than a rejection claim : Exemplifying a  such that  is contradictory of course means it is not the case that  holds for all possible . But one may also demonstrate that  holding for all  would logically lead to a contradiction without the aid of a specific counter-example, and even while not being able to construct one. In the latter case, constructively, here one does not stipulate an existence claim.

Imposed restrictions on a set theory 
Compared to the classical counterpart, one is generally less likely to prove the existence of relations that cannot be realized. A restriction to the constructive reading of existence apriori leads to stricter requirements regarding which characterizations of a set  involving unbounded collections constitute a (mathematical, and so always meaning total) function. This is often because the predicate in a case-wise would-be definition may not be decidable. Adopting the standard definition of set equality via extensionality, the full Axiom of Choice is such a non-constructive principle that implies  for the formulas permitted in one's adopted Separation schema, by Diaconescu's theorem. Similar results hold for the Axiom of Regularity existence claim, as shown below. The latter has a classically equivalent inductive substitute.
So a genuinely intuitionistic development of set theory requires the rewording of some standard axioms to classically equivalent ones. Apart from demands for computability and reservations regrading of impredicativity, technical question regarding which non-logical axioms effectively extend the underlying logic of a theory is also a research subject in its own right.

Metalogic 
With computably undecidable propositions already arising in Robinson arithmetic, even just Predicative separation lets one define elusive subsets easily. In stark contrast to the classical framework, constructive set theories may be closed under the rule that any property that is decidable for all sets is already equivalent to one of the two trivial ones,  or . Also the real line may be taken to be indecomposable in this sense. Undecidability of disjunctions also affects the claims about total orders such as that of all ordinal numbers, expressed by the provability and rejection of the clauses in the order defining disjunction . This determines whether the relation is trichotomous. A weakened theory of ordinals in turn affects the proof theoretic strength defined in ordinal analysis.

In exchange, constructive set theories can exhibit attractive disjunction and existence properties, as is familiar from the study of constructive arithmetic theories. These are features of a fixed theory which metalogically relate judgements of propositions provable in the theory. Particularly well-studied are those such features that can be expressed in Heyting arithmetic, with quantifiers over numbers and which can often be realized by numbers, as formalized in proof theory. In particular, those are the numerical existence property and the closely related disjunctive property, as well as being closed under Church's rule, witnessing any given function to be computable.

A set theory does not only express theorems about numbers, and so one may consider a more general so-called strong existence property that is harder to come by, as will be discussed. A theory has this property if the following can be established: For any property , if the theory proves that a set exist that has that property, i.e. if the theory claims the existence statement, then there is also a property  that uniquely describes such a set instance. More formally, for any predicate  there is a predicate  so that

The role analogous to that of realized numbers in arithmetic is played here by defined sets proven to exist according to the theory, and so this is a subtle question concerning term construction and the theories strength. While many theories discussed tend have all the various numerical properties, the existence property can easily be spoiled, as will be discussed. Weaker forms of existence properties have been formulated.

Some classical theories can in fact also be constrained so as to exhibit the strong existence property. Zermelo–Fraenkel set theory  with the constructible universe postulate, , or  with sets all taken to be ordinal-definable, , do have the existence property. For contrast, consider the theory  given by  plus the full axiom of choice existence postulate. Recall that this collection of axioms proves the well-ordering theorem, implying well-orderings exists for any set. In particular, this means that relations  formally exist that establish the well-ordering of  (i.e. the theory claims the existence of a least element for all subsets of  with respect to those relations). This is despite the fact that definability of such an ordering is known to be independent of . The latter implies that for no particular formula  in the language of the theory does the theory prove that the corresponding set is a well-ordering relation of the reals. So  formally proves the existence of a subset  with the property of being a well-ordering relation, but at the same time no particular set  for which the property could be validated can possibly be defined.

Anti-classical principles 
As mentioned above, a constructive theory  may exhibit the numerical existence property, , for some number  and where  denotes the corresponding numeral in the formal theory. Here one must carefully distinguish between provable implications between two propositions, , and metalogical properties of the form . When adopting an established proof theoretical schema of the latter type as an inference rule and nothing new can be proven, one says the theory  is closed under that rule.

One may instead consider adjoining the rule corresponding to the meta-theoretical property as an implication (in the sense of "") to , as a schema or in quantified form. A situation commonly studied is that of a fixed  exhibiting the meta-theoretical property of the following type: For an instance from some collection of formulas of a particular form, here captured via  and , one established the existence of a number  so that . Here one may then postulate , where the bound  is a number variable in language of the theory. For example, Church's rule is an admissible rule in first-order Heyting arithmetic  and, furthermore, the corresponding Church's thesis principle  may consistently be adopted as an axiom. The new theory with the principle added is anti-classical, in that it may not be consistent anymore to also adopt . Similarly, adjoining the excluded middle principle  to some theory , the theory thus obtained may prove new, strictly classical statements, and this may spoil some of the meta-theoretical properties that were previously established for . In such a fashion,  may not be adopted in , also known as Peano arithmetic .

The focus in this subsection shall be on set theories with quantification over a fully formal notion of an infinite sequences space, i.e. function space, as it will be introduced further below.
A translation of Church's rule into the language of the theory itself may here read

Kleene's T predicate together with the result extraction expresses that any input number  being mapped to the number  is, through , witnessed to be a computable mapping. Here  now denotes a set theory model of the standard natural numbers and  is an index with respect to a fixed program enumeration. Stronger variants have been used, which extend this principle to functions  defined on domains  of low complexity. The principle rejects decidability for the predicate  defined as , expressing that  is the index of a computable function halting on its own index. Weaker, double negated forms of the principle may be considered too, which do not require the existence of a recursive implementation for every , but which still make principles inconsistent that claim the existence of functions which provenly have no recursive realization. Some forms of a Church's thesis as principle are even consistent with the classical, weak so called second-order arithmetic theory , a subsystem of the two-sorted first-order theory .

The collection of computable functions is classically subcountable, which classically is the same as being countable. But classical set theories will generally claim that  holds also other functions than the computable ones. For example there is a proof in  that total functions (in the set theory sense) do exist that cannot be captured by a Turing machine.
Taking the computable world seriously as ontology, a prime example of an anti-classical conception related the Markovian school is the permitted subcountability of various uncountable collections. Adopting the subcountability of the collection of all unending sequences of natural numbers () as an axiom in a constructive theory, the "smallness" (in classical terms) of this collection, in some set theoretical realizations, is then already captured by the theory itself.
A constructive theory may also adopt neither classical nor anti-classical axioms and so stay agnostic towards either possibility.

Constructive principles already prove  for any . And so for any given element  of , the corresponding excluded middle statement for the proposition cannot be negated. Indeed, for any given , by noncontradiction it is impossible to rule out  and rule out its negation both at once, and the relevant De Morgan's rule applies as above. But a theory may in some instances also permit the rejection claim . Adopting this does not necessitate providing a particular  witnessing the failure of excluded middle for the particular proposition , i.e. witnessing the inconsistent . Predicates  on an infinite domain  correspond to decision problems. Motivated by provenly computably undecidable problems, one may reject the possibility of decidability of a predicate without also making any existence claim in .
As another example, such a situation is enforced in Brouwerian intuitionistic analysis, in a case where the quantifier ranges over infinitely many unending binary sequences and  states that a sequence  is everywhere zero. Concerning this property, of being conclusively identified as the sequence which is forever constant, adopting Brouwer's continuity principle strictly rules out that this could be proven decidable for all the sequences.

So in a constructive context with a so-called non-classical logic as used here, one may consistently adopt axioms which are both in contradiction to quantified forms of excluded middle, but also non-constructive in the computable sense or as gauged by meta-logical existence properties discussed previously. In that way, a constructive set theory can also provide the framework to study non-classical theories.

History and overview 
Historically, the subject of constructive set theory (often also "") begun with John Myhill's work on the theories also called  and .
In 1973, he had proposed the former as a first-order set theory based on intuitionistic logic, taking the most common foundation  and throwing out the Axiom of choice as well as the principle of the excluded middle, initially leaving everything else as is. However, different forms of some of the  axioms which are equivalent in the classical setting are inequivalent in the constructive setting, and some forms imply , as will be demonstrated. In those cases, the intuitionistically weaker formulations were consequently adopted. The far more conservative system  is also a first-order theory, but of several sorts and bounded quantification, aiming to provide a formal foundation for Errett Bishop's program of constructive mathematics.

The main discussion presents sequence of theories in the same language as , leading up to Peter Aczel's well studied , and beyond. Many modern results trace back to Rathjen and his students.
 is also characterized by the two features present also in Myhill's theory:
On the one hand, it is using the Predicative Separation instead of the full, unbounded Separation schema, see also Lévy hierarchy.
Boundedness can be handled as a syntactic property or, alternatively, the theories can be conservatively extended with a higher boundedness predicate and its axioms. Secondly, the impredicative Powerset axiom is discarded, generally in favor of related but weaker axioms. The strong form is very casually used in classical general topology.
Adding  to a theory even weaker than  recovers , as detailed below.
The system, which has come to be known as Intuitionistic Zermelo–Fraenkel set theory (), is a strong set theory without . It is similar to , but less conservative or predicative.
The theory denoted  is the constructive version of , the classical Kripke–Platek set theory without a form of Powerset and where even the Axiom of Collection is bounded.

Models 
Many theories studied in constructive set theory are mere restrictions of Zermelo–Fraenkel set theory () with respect to their axiom as well as their underlying logic. Such theories can then also be interpreted in any model of .

For a set theory context without infinite sets, constructive first-order arithmetic can also be taken as an apology for most axioms adopted in : The arithmetic theory  is bi-interpretable with a weak constructive set theory, as also described in the article on Heyting arithmetic. One may arithmetically characterize a membership relation "" and with it prove - instead of the existence of a set of natural numbers  - that all sets in its theory are in bijection with a (finite) von Neumann natural, a principle denoted . This context further validates Extensionality, Pairing, Union, Binary Intersection (which is related to the Axiom schema of predicative separation) and the Set Induction schema. Taken as axioms, the aforementioned principles constitute a set theory that is already identical with the theory given by  minus the existence of  but plus  as axiom. All those axioms are discussed in detail below.
Relatedly,  also proves that the hereditarily finite sets fulfill all the previous axioms. This is a result which persists when passing on to  and  minus Infinity.

As far as constructive realizations go there is a relevant realizability theory. Relatedly, Aczel's theory constructive Zermelo-Fraenkel  has been interpreted in a Martin-Löf type theories, as sketched in the section on . In this way, theorems provable in this and weaker set theories are candidates for a computer realization.

More recently, presheaf models for constructive set theories have been introduced. These are analogous to presheaf models for intuitionistic set theory developed by Dana Scott in the 1980s. Realizability models of  within the effective topos have been identified, which, say, at once validate full Separation, relativized dependent choice , independence of premise  for sets, but also the subcountability of all sets, Markov's principle  and Church's thesis  in the formulation for all predicates.

Subtheories of ZF

Notation

Language 
The propositional connective symbols used to form syntactic formulas are standard. The axioms of set theory give a means to prove equality "" of sets and that symbol may, by abuse of notation, be used for classes. Negation "" of elementhood "" is often written "", and usually the same goes for non-equality "". However, in a context with apartness relations, for example when dealing with sequences, the latter symbol is also used for something different.

Variables 
Below the Greek  denotes a proposition or predicate variable in axiom schemas and  or  is used for particular such predicates. The word "predicate" is sometimes used interchangeably with "formulas" as well, even in the unary case.

Quantifiers range over sets and those are denoted by lower case letters.
As is common, one may use argument brackets to express predicates, for the sake of highlighting particular free variables in their syntactic expression, as in "".

One abbreviates  by  and  by . The syntactic notion of bounded quantification in this sense can play a role in the formulation of axiom schemas, as seen below.

Two ways to express disjointness capture many of the intuitionistically valid negation rules: . Using the above notation, this is a purely logical equivalence and below the proposition will furthermore be expressible as .

Express the subclass claim , i.e. , by .
The similar notion of subset-bounded quantifiers, as in , has been used in set theoretical investigation as well, but will not be further highlighted here.

Unique existence  here means .

If there provenly exists a set inside a class, meaning , then one calls it inhabited. One may also use quantification in  to express this as . The class  is then provenly not the empty set, introduced below. While classically equivalent, constructively non-empty is a weaker notion with two negations and ought to be called not uninhabited. Unfortunately, the word for the more useful notion of 'inhabited' is rarely used in classical mathematics.

Classes 
As is also common in the study of set theories, one makes use set builder notation for classes, which, in most contexts, are not part of the object language but used for concise discussion. In particular, one may introduce notation declarations of the corresponding class via "", for the purpose of expressing  as . Logically equivalent predicates can be used to introduce the same class.
One also writes  as shorthand for . For example, one may consider  and this is also denoted .
Further extensions of such notation are in common used in set theory, giving meaning to statements like "", and so on.

For a predicate , trivially . And so follows that .

Through the use of a 2-ary predicate , a set  may be characterized as  holding for all , where the right hand side may depend on the actual variable , and possibly even on membership in  itself.
The convenient notational relation between  and  functions akin to a simpler, special case of this. Axioms of the form  postulate that the class of all sets for which  holds actually forms a set, and this is sometimes less formally expressed as .

Equality
Denote by  the statement expressing that two classes have exactly the same elements, i.e. , or equivalently . This is not to be conflated with the concept of equinumerosity also used below.

The following axiom gives a means to prove equality "" of two sets, so that through substitution, any predicate about  translates to one of .

By the logical properties of equality, the converse direction holds automatically.

In a constructive interpretation, the elements of a subclass  of  may come equipped with more information than those of , in the sense that being able to judge  is being able to judge . And (unless the whole disjunction follows from axioms) in the Brouwer–Heyting–Kolmogorov interpretation, this means to have proven  or having rejected it.
As  may be not decidable for all elements in , the two classes must a priori be distinguished.

Consider a predicate  that provenly holds for all elements of a set , so that , and assume that the class on the left hand side is established to be a set. Note that, even if this set on the left informally also ties to proof-relevant information about the validity of  for all the elements, the Extensionality axiom postulates that, in our set theory, the set on the left hand side is judged equal to the one on the right hand side.

The above analysis also shows that a statement of the form , which in informal class notation may be expressed as , is then equivalently expressed as . In particular, this means that such -theorems (e.g. the ones provable from full mathematical induction) enable treating the class defined on the right hand side of the equality to be used like the set on the left.

Alternative approaches
While often adopted, this axiom has been criticized in constructive thought, as it effectively collapses differently defined properties, or at least the sets viewed as the extension of these properties, a Fregian notion.

Modern type theories may instead aim at defining the demanded equivalence "" in terms of functions, see e.g. type equivalence. The related concept of function extensionality is often not adopted in type theory.

Other frameworks for constructive mathematics might instead demand a particular rule for equality or apartness come for the elements  of each and every set  discussed. Even then, the above definition can be used to characterize equality of subsets  and .
Note that adopting "" as a symbol in a predicate logic theory makes equality of two terms a quantifier-free expression.

Merging sets
Define class notation for the pairing of a few given elements via disjunctions. E.g.  is the quantifier-free statement , and likewise  says , and so on.

Two other basic existence postulates given some other sets are as follows. Firstly,

Given the definitions above,  expands to , so this is making use of equality and a disjunction. The axiom says that for any two sets  and , there is at least one set , which hold at least those two sets.

With bounded Separation below, also the class  exists as a set. Denote by  the standard ordered pair model , so that e.g.  denotes another bounded formula in the formal language of the theory.

And then, using existential quantification and a conjunction,

saying that for any set , there is at least one set , which holds all the members , of 's members .
The minimal such set is the union. 

The two axioms are commonly formulated stronger, in terms of "" instead of just "", although this is technically redundant in the context of : As the Separation axiom below is formulated with "", for statements  the equivalence can be derived, given the theory allows for separation using . In cases where  is an existential statement, like here in the union axiom, there is also another formulation using a universal quantifier.

Also using bounded Separation, the two axioms just stated together imply the existence of a binary union of two classes  and , when they have been established to be sets, denoted by  or . For a fixed set , to validate membership  in the union of two given sets  and , one needs to validate the  part of the axiom, which can be done by validating the disjunction of the predicates defining the sets  and , for . In terms of the associated sets, it is done by validating the disjunction .

The union and other set forming notations are also used for classes. For instance, the proposition  is written . Let now . Given , the decidability of membership in , i.e. the potentially independent statement , can also be expressed as . But, as for any excluded middle statement, the double-negation of the latter holds: That union isn't not inhabited by . This goes to show that partitioning is also a more involved notion, constructively.

Set existence
The property that is false for any set corresponds to the empty class, which is denoted by  or zero, . That the empty class is a set readily follows from other axioms, such as the Axiom of Infinity below. But if, e.g., one is explicitly interested in excluding infinite sets in one's study, one may at this point adopt the

Introduction of the symbol  (as abbreviating notation for expressions in involving characterizing properties) is justified as uniqueness for this set can be proven. As  is false for any , the axiom then reads .

For a set , define the successor set  as  and write  for . Its interplay with the membership relation has a recursive clause, in the sense that . By reflexivity of equality, , and in particular  is always inhabited.
A sort of blend between pairing and union, an axiom more readily related to the successor is the Axiom of adjunction. This is all relevant for the standard modeling of individual Neumann ordinals.

A simple and provenly false proposition then is, for example, , corresponding to  in the standard arithmetic model.
Again, here symbols such as  are treated as convenient notation and any proposition really translates to an expression using only "" and logical symbols, including quantifiers. Accompanied by a metamathematical analysis that the capabilities of the new theories are equivalent in an effective manner, formal extensions by symbols such as  may also be considered.

BCST 
The following makes use of axiom schemas, i.e. axioms for some collection of predicates.
Some of the stated axiom schemas shall allow for any collection of set parameters as well (meaning any particular named variables ). That is, instantiations of the schema are permitted in which the predicate (some particular ) also depends on a number of further set variables and the statement of the axiom is understood with corresponding extra outer universal closures (as in ).

Separation 
Basic constructive set theory  consists of several axioms also part of standard set theory, except the Separation axiom is weakened.
Beyond the four axioms above, it postulates Predicative Separation as well as the Replacement schema.

This axiom amounts to postulating the existence of a set  obtained by the intersection of any set  and any predicatively described class .
For any  proven to be a set, when the predicate is taken as , one obtains the binary intersection of sets and writes . Intersection corresponds to conjunction in an analog way to how union corresponds to disjunction.

When the predicate is taken as the negation , one obtains the difference principle, granting existence of any set . Note that sets like  or  are always empty. So, as noted, from Separation and the existence of at least one set (e.g. Infinity below) will follow the existence of the empty set  (also denoted ).
Within this conservative context of , the Bounded Separation schema is actually equivalent to Empty Set plus the existence of the binary intersection for any two sets. The latter variant of axiomatization does not make use of a formula schema.

For a proposition , a recurring trope in the constructive analysis of set theory is to view the predicate  as the subclass  of the second ordinal . 
If it is provable that  holds, or , or , then  is inhabited, or empty (uninhabited), or non-empty (not uninhabited), respectively.
Clearly,  is equivalent to the proposition  (in the model of the naturals this means  being smaller than  because, equivalently, ), while  is equivalent to  (so that, equivalently, ). The union that is part of the successor operation definition above may be used to express the excluded middle statement  as . In words,  is decidable if and only if the successor of  is larger than the smallest ordinal . The proposition   is decided either way through establishing how   is smaller: By  already being smaller than , or by  being 's direct predecessor. Another way to express excluded middle for  is as the existence of a least number member of the inhabited class .
If ones separation axiom allows for separation with , then  is a subset, which may be called the truth value associated with . Two truth values can be proven equal, as sets, by proving an equivalence. In terms of this terminology, the collection of proof values can a priori be understood to be rich. Unsurprisingly, decidable propositions have one of a binary set of truth values. 

The axiom schema of Predicative Separation is also called Bounded Separation, as in Separation for set-bounded quantifiers only. The scope of specified subsets that can be proven to exist is enriched with further set existence postulate. Bounded Separation is a schema that takes into account syntactic aspects of set defining predicates, up to provable equivalence. The permitted formulas are denoted by  in the set theoretical Lévy hierarchy, in analogy to  in the arithmetical hierarchy. (Note however that the arithmetic classification is sometimes expressed not syntactically but in terms of subclasses of the naturals. Also, the bottom level of the arithmetical hierarchy has several common definitions, some not allowing the use of some total functions. The distinction is not relevant on the level  or higher. Finally note that a  classification of a formula may be expressed up to equivalence in the theory.)

The schema is also the way in which Mac Lane weakens a system close to Zermelo set theory , for mathematical foundations related to topos theory. See also Kripke-Platek set theory.

No universal set 
By a remark in the section on merging sets, a set cannot consistently ruled out to be a member of a class of the form . A constructive proof that it is in that class contains information. Now if  is a set, then the class on the right is not a set. The following demonstrates this in the special case when  is empty, i.e. when the right side is the universal class.

The following holds for any relation , giving a purely logical condition by which two terms  and  non-relatable

The expression  is a bounded one and thus allowed in separation. By virtue of the rejection of the final disjunct above, , Russel's construction shows that . So for any set , Predicative Separation alone implies that there exists a set which is not a member of . In particular, no universal set can exist in this theory.

In a theory with the axiom of regularity, like , of course that subset  can be proven to be equal to  itself. As an aside, in a theory with stratification like Intuitionistic New Foundations, a universal set may exist because use of the syntactic expression  may be disallowed in proofs of existence by, essentially, separation.

Already the special case  implies that the subclass  of the universal class is proper as well.

Predicativity 
The restriction in the axiom is also gatekeeping impredicative definitions: Existence should at best not be claimed for objects that are not explicitly describable, or whose definition involves themselves or reference to a proper class, such as when a property to be checked involves a universal quantifier. So in a constructive theory without Axiom of power set, when  denotes some 2-ary predicate, one should not generally expect a subclass  of  to be a set, in case that it is defined, for example, as in
,
or via a similar definitions involving any quantification over the sets . Note that if this subclass  of  is provenly a set, then this subset itself is also in the unbounded scope of set variable . In other words, as the subclass property  is fulfilled, this exact set , defined using the expression , would play a role in its own characterization.

While predicative Separation leads to fewer given class definitions being sets, it must be emphasized that many class definitions that are classically equivalent are not so when restricting oneself to constructive logic. So in this way, one gets a broader theory, constructively.
Due to the potential undecidability of general predicates, the notion of subset and subclass is more elaborate in constructive set theories than in classical ones. This remains true if full Separation is adopted, as in the theory , which however spoils the existence property as well as the standard type theoretical interpretations, and in this way spoils a bottom-up view of constructive sets. As an aside, as subtyping is not a necessary feature of constructive type theory, constructive set theory can be said to quite differ from that framework.

Replacement 
Next consider the

It is granting existence, as sets, of the range of function-like predicates, obtained via their domains. In the above formulation, the predicate is not restricted akin to the Separation schema, but this axiom already involves an existential quantifier in the antecedent. Of course, weaker schemas could be considered as well.

With the Replacement schema, this theory proves that the equivalence classes or indexed sums are sets. In particular, the Cartesian product, holding all pairs of elements of two sets, is a set. In turn, for any fixed number, the corresponding product expression, say , can be constructed as a set. The axiomatic requirements for sets recursively defined in the language are discussed further below. Equality of elements inside a set  is decidable if the corresponding relation as a subset of  is decidable, in which case  is sometimes called discrete.

Replacement is not necessary in the design of a weak constructive set theory that is bi-interpretable with Heyting arithmetic . However, some form of induction is. Replacement together with Set Induction (introduced below) also suffices to axiomize hereditarily finite sets constructively and that theory is also studied without Infinity. For comparison, consider the very weak classical theory called General set theory that interprets the class of natural numbers and their arithmetic via just Extensionality, Adjunction and full Separation.

Replacement is relevant for function comprehension can be seen as a form of comprehension more generally. Only when assuming  does Replacement already imply full Separation. In , Replacement is mostly important to prove the existence of sets of high rank, namely via instances of the axiom schema where  relates relatively small set  to bigger ones, .

Constructive set theories commonly have Axiom schema of Replacement, sometimes restricted to bounded formulas. However, when other axioms are dropped, this schema is actually often strengthened - not beyond , but instead merely to gain back some provability strength. Such stronger axioms exist that do not spoil the strong existence properties of a theory, as discussed further below.

The discussion now proceeds with axioms granting existence of objects which, in different but related form, are also found in dependent type theories, namely products and the collection of natural numbers as a completed set. Infinite sets are particularly handy to reason about operations applied to sequences defined on unbounded domains, say the formal differentiation of a generating function or the addition of two Cauchy sequences.

ECST 
For some fixed predicate  and a set , the statement  expresses that  is the smallest (in the sense of "") among all sets  for which  holds true, and that it is always a subset of such . The aim of the axiom of infinity is to eventually obtain unique smallest inductive set.

In the context of common set theory axioms, one statement of infinitude is to state that a class is inhabited and also includes a chain of membership (or alternatively a chain of supersets). That is,
.
More concretely, denote by  the inductive property,
.
In terms of a predicate  underlying the class so that , the latter translates to .

Write  for the general intersection . (A variant of this definition may be considered which requires , but we only use this notion for the following auxiliary definition.)

One commonly defines a class , the intersection of all inductive sets. (Variants of this treatment may work in terms of a formula that depends on a set parameter  so that .) The class  exactly holds all  fulfilling the unbounded property . The intention is that if inductive sets exist at all, then the class  shares each common natural number with them, and then the proposition , by definition of "", implies that  holds for each of these naturals. While bounded separation does not suffice to prove  to be the desired set, the language here forms the basis for the following axiom, granting natural number induction for predicates that constitute a set.

The elementary constructive Set Theory  has the axiom of  as well as the postulate

Going on, one takes  to denote the now unique smallest inductive set, an unbounded von Neumann ordinal. It contains the empty set and, for each set in , another set in  that contains one element more.

Symbols called zero and successor are in the signature of the theory of Peano. In , the above defined successor of any number also being in the class  follow directly from the characterization of the natural naturals by our von Neumann model. Since the successor of such a set contains itself, one also finds that no successor equals zero. So two of the Peano axioms regarding the symbols zero and the one regarding closedness of  come easily. Fourthly, in , where  is a set,  can be proven to be an injective operation.

For some predicate of sets , the statement  claims it for all subsets of the set of naturals, and the axiom now proves such sets do exist. Such quantification is also possible in second-order arithmetic.

The pairwise order "" on the naturals is captured by their membership relation "". It is important to note that the theory proves the order as well as the equality relation on this set to be decidable. Not only is no number smaller than , but induction implies that among subsets of , it is exactly the empty set which has no least member. The contrapositive of this proves the double-negated least number existence for all non-empty subsets of . Another valid principle also classically equivalent to it is least number existence for all inhabited decidable subsets. That said, the bare existence claim for the inhabited subset  of  is equivalent to excluded middle for , and a constructive theory will therefore not prove  to be well-ordered.

Weaker formulations of infinity 
Should it need motivation, the handiness of postulating an unbounded set of numbers in relation to other inductive properties becomes clear in the discussion of arithmetic in set theory further below. But as is familiar from classical set theory, also weak forms of Infinity can be formulated. For example, one may just postulate the existence of some inductive set,  - such an existence postulate suffices when full Separation may then be used to carve out the inductive subset  of natural numbers, the shared subset of all inductive classes. Alternatively, more specific mere existence postulates may be adopted. Either which way, the inductive set then fulfills the following  predecessor existence property in the sense of the von Neumann model:

Without making use of the notation for the previously defined successor notation, the extensional equality to a successor  is captured by . This expresses that all elements  are either equal to  or themselves hold a predecessor set  which shares all other members with .

Observe that through the expression "" on the right hand side, the property characterizing  by its members  here syntactically again contains the symbol  itself. Due to the bottom-up nature of the natural numbers, this is tame here. Assuming -set induction, no two sets have this property.
Also note that there are also longer formulations of this property, avoiding "" in favor unbounded quantifiers.

Number bounds 
Adopting an Axiom of Infinity, the set-bounded quantification legal in predicates used in -Separation then explicitly permits numerically unbounded quantifiers - the two meanings of "bounded" should not be confused.
With  at hand, call a class of numbers  bounded if the following existence statement holds

This or the contrapositive variant

are statements of finiteness. For decidable properties, these are -statements in arithmetic, but with the Axiom of Infinity, the two quantifiers are set-bound. Denote an initial segment of the natural numbers, i.e.

for any  and including the empty set, by
.
This set equals  and so at this point "" is mere notation for its predecessor (i.e. not involving subtraction function).
To reflect more closely the discussion of functions below, write the above condition as
.
For a class , the logically positive unboundedness statement

is now also one of infinitude. It is  in the decidable arithmetic case. To validate infinitude of a set, this property even works if the set holds other elements besides infinitely many of members of .

Moderate induction in ECST 
Is instructive to recall the way in which a theory with set comprehension and extensionality ends up encoding predicate logic. Like any class in set theory, a set can be read as corresponding to predicates on sets. For example, an integer is even if it is a member of the set of even integers, or a natural number has a successor if it is a member of the set of natural numbers that have a successor. 
For a less primitive example, fix some set  and let  denote the existential statement that the function space on the finite ordinal into  exist. The predicate will be denoted  below, and here the existential quantifier is not merely one over natural numbers, nor is it bounded by any other set. Now a proposition like the finite exponentiation principle  and, less formally, the equality  are just two ways of formulating the same desired statement, namely an -indexed conjunction of existential propositions where  ranges over the set of all naturals. Via extensional identification, the second form expresses the claim using notation for subclass comprehension and the bracketed object on the right hand side may not even constitute a set. If that subclass is not provably a set, it may not actually be used in many set theory principles in proofs, and establishing the universal closure  as a theorem may not be possible. The set theory can thus be strengthened by more set existence axioms, to be used with predicative bounded Separation, but also by just postulating stronger -statements.

The second universally quantified conjunct in the strong axiom of Infinity expresses mathematical induction for all  in the universe of discourse, i.e. for sets. This is because the consequent of this clause, , states that all  fulfill the associated predicate. Being able to use predicative separation to define subsets of , the theory proves induction for all predicates  involving only set-bounded quantifiers. This role of set-bounded quantifiers also means that more set existence axioms impact the strength of this induction principle, further motivating the function space and collection axioms that will be a focus of the rest of the article. Notably,  already validates induction with quantifiers over the naturals, and hence induction as in the first-order arithmetic theory . 
The so called axiom of full mathematical induction for any predicate (i.e. class) expressed through set theory language is far stronger than the bounded induction principle valid in . The former induction principle could be directly adopted, closer mirroring second-order arithmetic. In set theory it also follows from full (i.e. unbounded) Separation, which says that all predicates on  are sets. Mathematical induction is also superseded by the (full) Set induction axiom.

Warning note: In naming induction statements, one must take care not to conflate terminology with arithmetic theories. The first-order induction schema of natural number arithmetic theory claims induction for all predicates definable in the language of first-order arithmetic, namely predicates of just numbers. So to interpret the axiom schema of , one interprets these arithmetical formulas. In that context, the bounded quantification specifically means quantification over a finite range of numbers.
One may also speak about the induction in the two-sorted, first order theory of second-order arithmetic  in a form explicitly expressed for subsets of the naturals. That class of subsets can be taken to correspond to a richer collection of formulas than the first-order arithmetic definable ones. In the program of reverse mathematics, all mathematical objects discussed are encoded as naturals or subsets of naturals. Subsystems of  with very low complexity comprehension studied in that framework have a language that does not merely express arithmetical sets, while all sets of naturals particular such theories prove to exist are just computable sets. Theorems therein can be a relevant reference point for weak set theories with a set of naturals, predicative separation and only some further restricted form of induction. Constructive reverse mathematics exists as a field but is less developed than its classical counterpart.  shall moreover not be confused with the second-order formulation of Peano arithmetic . Typical set theories like the one discussed here are also first-order, but those theories are not arithmetics and so formulas may also quantify over the subsets of the naturals. When discussing the strength of axioms concerning numbers, it is also important to keep in mind that the arithmetical and the set theoretical framework do not share a common signature. Likewise, care must always be taken with insights about totality of functions. In computability theory, the μ operator enables all partial general recursive functions (or programs, in the sense that they are Turing computable), including ones e.g. non-primitive recursive but -total, such as the Ackermann function. The definition of the operator involves predicates over the naturals and so the theoretical analysis of functions and their totality depends on the formal framework and proof calculus at hand.

Functions

General note on programs and functions 
Naturally, the meaning of existence claims is a topic of interest in constructivism, be it for a theory of sets or any other framework.
Let  express a property such that a mathematical framework validates what amounts to the statement

A constructive proof calculus may validate such a judgement in terms of programs on represented domains and some object representing a concrete assignment , providing a particular choice of value in  (a unique one), for each input from . Expressed through the rewriting , this function object maybe be understood as witnessing the proposition. Consider for example the notions of proof in through realizability theory or function terms in a type theory with a notion of quantifiers. The latter captures proof of logical proposition through programs via the Curry–Howard correspondence.

Depending on the context, the word "function" may be used in association with a particular model of computation, and this is a priori  narrower than what is discussed in the present set theory context. One notion of program is formalized by partial recursive "functions" in computability theory. But beware that here the word "function" is used in a way that also comprises partial functions, and not just "total functions". The scare quotes are used for clarity here, as in a set theory context there is technically no need to speak of total functions, because this requirement is part of the definition of a set theoretical function and partial function spaces can be modeled via unions. At the same time, when combined with a formal arithmetic, partial function programs provides one particularly sharp notion of totality for functions. By Kleene's normal form theorem, each partial recursive function on the naturals computes, for the values where it terminates, the same as , for some partial function program index , and any index will constitute some partial function. A program can be associated with a  and may be said to be -total whenever a theory proves , where  amounts to a primitive recursive program and  is related to the execution of . Kreisel proved that the class of partial recursive functions proven -total by  is not enriched when  is added. As a predicate in , this totality constitutes an undecidable subset of indices, highlighting that the recursive world of functions between the naturals is already captured by a set dominated by . As a third warning, note that this notion is really about programs and several indices will in fact constitute the same function, in the extensional sense.

A theory in first-order logic, such as the axiomatic set theories discussed here, comes with a joint notion of total and functional for a binary predicate , namely . Such theories relate to programs only indirectly. If  denotes the successor operation in a formal language of a theory being studied, then any number, e.g.  (the number three), may metalogically be related to the standard numeral, e.g. . Similarly, programs in the partial recursive sense may be unrolled to predicates and weak assumptions suffice so that such a translation respects equality of their return values. Among finitely axiomizable sub-theories of , classical Robinson arithmetic  exactly fulfills this. Its existence claims are intended to only concern natural numbers and instead of using the full mathematical induction schema for arithmetic formulas, the theories' axioms postulate that every number is either zero or that there exists a predecessor number to it. Focusing on -total recursive functions here, it is a meta-theorem that the language of arithmetic expresses them by -predicates  encoding their graph such that  represents them, in the sense that it correctly proves or rejects  for any input-output pair of numbers  and  in the meta-theory. Now given a correctly representing , the predicate  defined by  represents the recursive function just as well, and as this only validates the smallest return value, the theory also proves functionality for all inputs  in the sense of . Given a representing predicate, then at the cost of making use of , one can always also systematically (i.e. with a ) prove the graph to be total functional.

Which predicates are provably functional for various inputs, or even total functional on their domain, generally depends on the adopted axioms of a theory and proof calculus. For example, for the diagonal halting problem, which cannot have a -total index, it is -independent whether the corresponding graph predicate on  (a decision problem) is total functional, but  implies that it is. Proof theoretical function hierarchies provide examples of predicates proven total functional in systems going beyond .
Which sets proven to exist do constitute a total function, in the sense introduced next, also always depends on the axioms and the proof calculus. Finally, note that the soundness of halting claims is a metalogical property beyond consistency, i.e. a theory may be consistent and from it one may prove that some program will eventually halt, despite this never actually occurring when said program is ran. More formally, assuming consistency of a theory does not imply it is also arithmetically -sound.

Total functional relations 
In set theory language here, speak of a function class when  and provenly
.
Notably, this definition involves quantifier explicitly asking for existence - an aspect which is particularly important in the constructive context. In words: For every , it demands the unique existence of a  so that . 
In the case that this holds one may use function application bracket notation and write . The above property may then be stated as . This notation may be extended to equality of function values. Some notational conveniences involving function application will only work when a set has indeed been established to be a function.
Let  (also written ) denote the class of sets that fulfill the function property. This is the class of functions from  to  in a pure set theory. Below the notation  is also used for , for the sake of distinguishing it from ordinal exponentiation. When functions are understood as just function graphs as here, the membership proposition  is also written . The boolean-valued  are among the classes discussed in the next section.

By construction, any such function respects equality in the sense that , for any inputs from . This is worth mentioning since also more broader concepts of "operations" exist in the literature, which may not respect this. Variants of the functional predicate definition using apartness relations on setoids have been defined as well. 
A subset of a function is still a function and the function predicate may also be proven for enlarged chosen codomain sets. As noted, care must be taken with nomenclature "function", a word which sees use in most mathematical frameworks. When a function set itself is not tied to a particular codomain, then this set of pairs is also member of a function space with larger codomain. This do not happen when by the word one denotes the subset of pairs paired with a codomain set, i.e. a formalization in terms of . This is mostly a matter of bookkeeping, but affects how other predicates are defined, question of size. This choice is also just enforced by some mathematical frameworks.

If both the domain  and considered codomain  are sets, then the above predicate only involves bounded quantifiers. Common notions such as injectivity and surjectivity can be expressed in a bounded fashion as well, and thus so is bijectivity. Both of these tie in to notions of size. Importantly, injection existence between any two sets provides a preorder. A power class does not inject into its underlying set and the latter does not map onto the former. Surjectivity is formally a more complex definition. Note that injectivity shall be defined positively, not by its contrapositive, which is common practice in classical mathematics. The version without negations is sometimes called weakly injective. The existence of value collisions is a strong notion of non-injectivity. And regarding surjectivity, similar considerations exist for outlier-production in the codomain.

Whether a subclass (or predicate for that matter) can be judged to be a function set, or even total functional to begin with, will depend on the strength of the theory, which is to say the axioms one adopts. And notably, a general class could also fulfill the above defining predicate without being a subclass of the product , i.e. the property is expressing not more or less than functionality w.r.t. the inputs from .
Now if the domain is a set, the function comprehension principle, also called axiom of unique choice  or non-choice, says that a function as a set, with some codomain, exists well. (And this principle is valid in a theory like . Also compare with the Replacement axiom.) That is, the mapping information exists as set and it has a pair for each element in the domain. Of course, for any set from some class, one may always associate unique element of the singleton , which shows that merely a chosen range being a set does not suffice to be granted a function set.
It is a metatheorem for theories containing  that adding a function symbol for a provenly total class function is a conservative extension, despite this formally changing the scope of bounded Separation.
In summary, in the set theory context the focus is on capturing particular total relations that are functional. To delineate the notion of function in the theories of the previous subsection (a 2-ary logical predicate defined to express a functions graph, together with a proposition that it is total and functional) from the "material" set theoretical notion here, one may explicitly call the latter graph of a function, anafunction or set function. The axiom schema of Replacement can also be formulated in terms of the ranges of such set functions.

Finitude 
One defines three distinct notions involving surjections. For a general set to be (Bishop-)finite shall mean there is a bijective function to a natural. If the existence of such a bijection is proven impossible, the set is called non-finite. Secondly, for a notion weaker than finite, to be finitely indexed (or Kuratowski-finite) shall mean that there is a surjection from a von Neumann natural number onto it. Call a set subfinite if it is the subset of a finite set, which thus injects into that finite set. Thirdly, for a notion even weaker than finitely indexed, to be subfinitely indexed means to be in the surjective image of a subfinite set, and in  this just means to be the subset of a finitely indexed set, meaning the subset can also be taken on the image side instead of the domain side. A set exhibiting either of those three notions can be understood to be majorized by a finite set, but in the second case the relation between the sets members is not necessarily fully understood, and in the third not even membership with respect to some superset it is necessary fully understood.
The claim that being finite is equivalent to being subfinite, for all sets, is equivalent to .
More finiteness properties for a set  can be defined, e.g. expressing the existence of some large enough natural such that a certain class of functions on the naturals always fail to map to distinct elements in . One definition considers some notion of non-injectivity into . Other definitions consider functions to a fixed superset of  with more elements.

Terminology for conditions of finiteness and infinitude may vary. Notably, subfinitely indexed sets (a notion necessarily involving surjections) are sometimes called subfinite (which can be defined without functions). The property of being finitely indexed could also be denoted "finitely countable", to fit the naming logic, but is by some authors also called finitely enumerable (which might be confusing as this suggest an injection in the other direction). Relatedly, the existence of a bijection with a finite set has not established, one may say a set is not finite, but this use of language is then weaker than to claim the set to be non-finite. The same issue applies to countable sets (not proven countable vs. proven non-countable), et cetera. A surjective map may also be called an enumeration.

Infinitude 
The set  itself is clearly unbounded. In fact, for any surjection from a finite range onto , one may construct an element that is different from any element in the functions range. Where needed, this notion of infinitude can also be expressed in terms of an apartness relation on the set in question. Being not Kuratowski-finite implies being non-finite and indeed the naturals shall not be finite in any sense. Commonly, the word infinite is used for the negative notion of being non-finite. Further, observe that , unlike any of its members, can be put in bijection with some of its proper unbounded subsets, e.g. those of the form  for any . This validates the formulations of Dedekind-infinite. So more generally than the property of infinitude in the previous section on number bounds, one may call a set infinite in the logically positive sense if one can inject  into it. A set that is even in bijection with  may be called countably infinite. A set is Tarski-infinite if there is some chain of subsets such that each has some new elements compared to its predecessor. There are indeed plenty of properties characterizing infinitude even in classical  and that theory does not prove all non-finite sets to be infinite in the injection existence sense, albeit it there holds when further assuming countable choice.  without any choice even permits cardinals aside the aleph numbers, and there can then be sets that negate both of the above properties, i.e. they are both non-Dedekind-infinite and non-finite (also called Dedekind-finite infinite sets).

Call an inhabited set countable if there exists a surjection from  onto it and subcountable if this can be done from some subset of . Call a set enumerable if there exists an injection to , which renders the set discrete. Notably, all of these are function existence claims. The empty set is not inhabited but generally deemed countable too, and note that the successor set of any countable set is countable. The set  is trivially infinite, countable and enumerable, as witnessed by the identity function. Also here, in strong classical theories many of these notions coincide in general and, as a result, the naming conventions in the literature are inconsistent. An infinite, countable set is equinumeros to . 

There are also various ways to characterize logically negative notion. The notion of uncountability, in the sense of being not countable, is also discussed in conjunction with the exponentiation axiom further below. Another notion of uncountability of  is given when one can produce a member in the compliment of any of 's countable subsets. More properties of finiteness may be defined as negations of such properties, et cetera.

Characteristic functions 
Separation lets us cut out subsets of products , at least when they are described in a bounded fashion. Given any , one is now led to reason about classes such as

Since , one has 

and so 
.
But be aware that in absence of any non-constructive axioms  may in generally not be decidable, since one requires an explicit proof of either disjunct. Constructively, when  cannot be witnessed for all the , or uniqueness of the terms  associated with each  cannot be proven, then one cannot judge the comprehended collection to be total functional.
Case in point: The classical derivation of Schröder–Bernstein relies on case analysis - but to constitute a function, particular cases shall actually be specifiable, given any input from the domain. It has been established that Schröder–Bernstein cannot have a proof on the base of  plus constructive principles. So to the extent that intuitionistic inference does not go beyond what is formalized here, there is no generic construction of a bijection from two injections in opposing directions.

But being compatible with , the development in this section still always permits "function on " to be interpreted as a completed object that is also not necessarily given as lawlike sequence. Applications may be found in the common models for claims about probability, e.g. statements involving the notion of "being given" an unending random sequence of coin flips, even if many predictions can also be expressed in terms of spreads.

If indeed one is given a function , it is the characteristic function actually deciding membership in some set  so that

and then by convention  and  as well as any equivalent of the formulas  and  with  free may be referred to as a decidable property or set on . One may call a collection  searchable for  if existence is actually decidable,

Now consider the case . If , say, then the range  of  is an inhabited, counted set, by Replacement. However, the  need not be again a decidable set itself, since the claim  is equivalent to the rather strong . Moreover,  is also equivalent to  and so one can state undecidable propositions about  also when membership in  is decidable.
This also plays out like this classically in the sense that statements about  may be independent, but any classical theory then nonetheless claims the joint proposition . Consider the set  of all indices of proofs of an inconsistency of the theory at hand, in which case the universally closed statement  is a consistency claim. In terms of arithmetic principles, assuming decidability of this would be - or arithmetic -. This and the stronger related , or arithmetic -, is discussed below.

Computable sets 
Going back to more generality, given a general predicate  on the numbers (say one defined from Kleene's T predicate), let again

Given any natural , then

In classical set theory,  by  and so excluded middle also holds for subclass membership. If the class  has no numerical bound, then successively going through the natural numbers , and thus "listing" all numbers in  by simply skipping those with , classically always constitutes an increasing surjective sequence . There, one can obtain a bijective function. In this way, the class of functions in typical classical set theories is provenly rich, as it also contains objects that are beyond what we know to be effectively computable, or programmatically listable in praxis.

In computability theory, the computable sets are ranges of non-decreasing total functions in the recursive sense, at the level  of the arithmetical hierarchy, and not higher. Deciding a predicate at that level amounts to solving the task of eventually finding a certificate that either validates or rejects membership.
As not every predicate  is computably decidable, also the theory  alone will not claim (prove) that all unbounded  are the range of some bijective function with domain . See also Kripke's schema.
Note that bounded Separation nonetheless proves the more complicated arithmetical predicates to still constitute sets, the next level being the computably enumerable ones at .

There is a large corpus of computability theory notions regarding how general subsets of naturals relate to one another. For example, one way to establish a bijection of two such sets is by relating them through a computable isomorphism, which is a computable permutation of all the naturals. The latter may in turn be established by a pair of particular injections in opposing directions.

Boundedness criteria 
Any subset  injects into . If  is decidable and inhabited with , the sequence

i.e.

is surjective onto , making it a counted set. That function also has the property .

Now consider a countable, bounded set . With  as above, it follows that

A set  such that this loose bounding statement holds for all sequences taking values in  (or an equivalent formulation of this property) is called pseudo-bounded. The intention of this property would be to still capture that  is eventually exhausted, albeit now this is expressed in terms of the function space  (which is bigger than  in the sense that  always injects into ). The related notion familiar from topological vector space theory is formulated in terms of ratios going to zero for all sequences ( in the above notation). For a decidable, inhabited set, validity of pseudo-boundedness, together with the counting sequence defined above, grants a bound for all the elements of .

The principle that any inhabited, pseudo-bounded subset of  that is just countable (but not necessarily decidable) is always also bounded is called -. The principle also holds generally in many constructive frameworks, such as the Markovian base theory , which is a theory postulating exclusively lawlike sequences with nice number search termination properties. However, - is independent of .

Choice functions 
Not even classical  proves each union of a countable set of two-element sets to be countable again. Indeed, models of  have been defined that negate the countability of such a countable union of pairs. Assuming countable choice rules out that model as an interpretation of the resulting theory. This principle is independent of  - A naive proof strategy for that statement fails at the accounting of infinitely many existential instantiations. 

A choice principle postulates that certain selections can always be made in a joint fashion so that they are also manifested as a single set function in the theory. As with any independent axiom, this raises the proving capabilities while restricting the scope of possible (model-theoretic) interpretations of the (syntactic) theory. A function existence claim can often be translated to the existence of inverses, orderings, and so on. Choice moreover implies statements about cardinalities of different sets, e.g. they imply or rule out countability of sets. Adding full choice to  does not prove any new -theorems, but it is strictly non-constructive, as shown below. The development here proceeds in a fashion agnostic to any of the variants described next.
 Axiom of countable choice  (or ): If , one can form the one-to-many relation-set . The axiom of countable choice would grant that whenever , one can form a function mapping each number to a unique value. The existence of such sequences is not generally provable on the base of  and countable choice is not -conservative over that theory. Countable choice into general sets can also be weakened further. One common consideration is to restrict the possible cardinalities of the range of , giving the weak countable choice into countable, finite or even just binary sets (). One may consider the version of countable choice for functions into  (called  or ), as is implied by the constructive Church's thesis principle, i.e. by postulating that all total arithmetical relations are recursive.  in arithmetic may be understood as a form of choice axiom. Another means of weakening countable choice is by restricting the involved definitions w.r.t. their place in the syntactic hierarchies (say -). The weak Kőnig's lemma , which breaks strictly recursive mathematics as further discussed below, is stronger than - and is itself sometimes viewed as capturing a form of countable choice. In the presence of a weak form of countable choice, the lemma becomes equivalent to the non-constructive principle of more logical flavor, . Constructively, weak form of choice is required for well-behaved Cauchy reals. Countable choice is not valid in the internal logic of a general topos, which can be seen as models of constructive set theories.
 Axiom of dependent choice : Countable choice is implied by the more general axiom of dependent choice, extracting a sequence in an inhabited , given any entire relation . In set theory this is again an infinite set of pairs, a subset of  and one is granted to pass from several existence statements to a unique existence statement for every natural. An appropriate formulation of dependent choice is adopted in several constructive frameworks, e.g., by some schools that understand unending sequences as ongoing constructions instead of completed objects. At least those cases seem benign where, for any , next value existence  can be validated in a computable fashion. The corresponding recursive function , if it exists, is then conceptualized as being able to return a value at infinitely many potential inputs , but these do not have to be evaluated all together at once. It also holds in many realizability models. In the condition of the formally similar recursion theorem, one is already given a unique choice at each step, and that theorem lets one combine them to a function on . So also with  one may consider forms of the axiom with restrictions on . Via the bounded separation axiom in , the principle also is equivalent to a schema in two bounded predicate variables: Keeping all quantifiers ranging over , one may further narrow this set domain using a unary -predicate variable, while also using any 2-ary -predicate instead of the relation set .
 Relativized dependent choice : This is the schema just using two general classes, instead of requiring  and  be sets. The domain of the choice function granted to exist is still just . Over , it implies full mathematical induction, which, in turn allows for function definition on  through the recursion schema. When  is restricted to -definitions, it still implies mathematical induction for -predicates (with an existential quantifier over sets) as well as . In , the schema  is equivalent to .
 -: A family of sets is better controllable if it comes indexed by a function. A set  is a base if all indexed families of sets  over it, have a choice function , i.e. . A collection of sets holding  and its elements and which is closed by taking indexed sums and products (see dependent type) is called -closed. While the axiom that all sets in the smallest -closed class are a base does need some work to formulate, it is the strongest choice principle over  that holds in the type theoretical interpretation .
 Axiom of choice : This is the "full" choice function postulate concerning domains that are general sets  containing inhabited sets, with the codomain given as their general union. Given a collection of sets such that the logic allows to make a choice in each, the axiom grants that there exists a set function that jointly captures a choice in all. It is typically formulated for all sets but has also been studied in classical formulations for sets only up to any particular cardinality. A standard example is choice in all inhabited subsets of the reals, which classically equals the domain . For this collection there can be no uniform element selection prescription that provably consistutes a choice function on the base of . Also, when restricted to the Borel algebra of the reals,  alone does not prove the existence of a function selecting a member from each non-empty such Lebesgue-measurable subset. (The set  is the σ-algebra generated by the finite intervals . It strictly includes those intervals, in the sense of , but in  also only has the cardinality of the reals itself.) Striking existence claims implied by the axiom are abound.  proves  exists and then the axiom of choice also implies dependent choice. Critically in the present context, it moreover also implies instances of  via Diaconescu's theorem. For  or theories extending it, this means full choice at the very least proves  for all -formulas, a non-constructive consequence not acceptable, for example, from a computability standpoint.

Diaconescu's theorem
To highlight the strength of full Choice and its relation to matters of intentionality, one should consider the classes

from the proof of Diaconescu's theorem. They are as contingent as the proposition  involved in their definition and they are not proven finite. Nonetheless, the setup entails several consequences. Referring back to the introductory elaboration on the meaning of such convenient class notation, as well as to the principle of distributivity, . So unconditionally,  as well as , and in particular they are inhabited. As  in any model of Heyting arithmetic, using the disjunctive syllogism both  and  each imply . The two statements are indeed equivalent to the proposition, as clearly . The latter also says that validity of  means  and  share all members, and there are two of these. As  are  are then sets, also  by extensionality. Conversely, assuming they are equal means  for any , validating all membership statements. So both the membership statements as well as the equalities are found to be equivalent to . Using the contrapositive results in the weaker equivalence of disjuncts . Of course, explicitly  and so one actually finds in which way the sets can end up being different. As functions preserves equality by definition,  indeed holds for any  with domain .

In the following assume a context in which  are indeed established to be sets, and thus subfinite sets. The general axiom of choice claims existence of a function  with . It is important that the elements  of the function's domain are different than the natural numbers  in the sense that a priori less is known about the former. When forming then union of the two classes,  is a necessary but then also sufficient condition. Thus  and one is dealing with functions  into a set of two distinguishable values. With choice come the conjunction  in the codomain of the function, but the possible function return values are known to be just  or . Using the distributivity, there arises a list of conditions, another disjunction. Expanding what is then established, one finds that either both  as well as the sets equality holds, or that the return values are different and  can be rejected. The conclusion is that the choice postulate actually implies  whenever a Separation axiom allows for set comprehension using undecidable proposition .

Analysis of Diaconescu's theorem
So full choice is non-constructive in set theory as defined here. The issue is that when propositions are part of set comprehension (like when  is used to separate, and thereby define, the classes  and  from ), the notion of their truth values are ramified into set terms of the theory. Equality defined by the set theoretical axiom of extensionality, which itself is not related to functions, in turn couples knowledge about the proposition to information about function values. To recapitulate the final step in terms function values: On the one hand, witnessing  implies  and  and this conclusion independently also applies to witnessing . On the other hand, witnessing  implies the two function arguments are not equal and this rules out . There are really only three combinations, as the axiom of extensionality in the given setup makes  inconsistent. So if the constructive reading of existence is to be preserved, full choice may be not adopted in the set theory, because the mere claim of function existence does not realize a particular function.

To better understand why one cannot expect to be granted a definitive (total) choice function with domain , consider naive function candidates. Firstly, an analysis of the domain is in order. The surjection  witnesses that  is finitely indexed. It was noted that its members are subfinite and also inhabited, since regardless of  it is the case that  and . So naively, this would seem to make  a contender for a choice function. When  can be rejected, then this is indeed the only option. But in the case of provability of , when , there is extensionally only one possible function input to a choice function. So in that situation, a choice function would explicitly have type , for example  and this would rule out the initial contender. For general , the domain of a would-be choice function is not concrete but contingent on  and not proven finite. When considering the above functional assignment , then neither unconditionally declaring  nor  is necessarily consistent. Having identified  with , the two candidates described above can be represented simultaneously via  (which is not proven finite either) with the subfinite "truth value of " given as . As , postulating , or , or the classical principle  here would indeed imply that  is a natural, so that the latter set  constitutes a choice function into . And as in the constructive case, given a particular choice function - a set holding either exactly one or exactly two pairs - one could actually infer whether  or whether  does hold. Vice versa, the third and last candidate  can be captured as part of , where . Such a  had already been considered in the early section on the axiom of separation. Again, the latter  here is a classical choice function either way also, where  functions as a (potentially undecidable) "if-clause". Constructively, the domain and values of such -dependent would-be functions are not understood enough to prove them to be a total functional relation into .

For computable semantics, set theory axioms postulating (total) function existence lead to the requirement for halting recursive functions. From their function graph in individual interpretations, one can infer the branches taken by the "if-clauses" that were undecided in the interpreted theory. 
But on the level of the synthetic frameworks, when they broadly become classical from adopting full choice, these extensional set theories theories contradict the constructive Church's rule.

Regularity implies PEM
The axiom of regularity states that for every inhabited set , there exists an element  in , which shares no elements with . As opposed to the axiom of choice, this existence claim reaches for an element of every inhabited set, i.e. the "domain" are all inhabited sets. Its formulation does not involve a unique existence claim but instead guarantees a set with a specific property.
As the axiom correlates membership claims at different rank, the axiom also ends up implying :

The proof from choice above had used a particular set . The proof in this paragraph also assumes Separation applies to  and uses , for which  by definition. It was already explained that  and so one wants to prove excluded middle for the latter. This  is defined using the clause , so that any given  fulfills the disjunction . Now let  be the postulated member with the empty intersection property. If , then , while otherwise . This establishes excluded middle for .

Demanding that the set of naturals is well-ordered with respect to it standard order relation imposes the same condition on the inhabited set . So the least number principle has the same non-constructive implication.

Arithmetic 
The four Peano axioms for  and , characterizing  as a model of the natural numbers in the constructive set theory , have been discussed. The order "" of natural numbers is captured by membership "" in this von Neumann model and this set is discrete, i.e. also  is decidable. But the theory does actually not enable primitive recursion in  for function definitions of what would be  (where "" here denotes the Cartesian product of set, not to be confused with multiplication above). Indeed, despite having the Replacement axiom, the theory does not prove the addition function  to be a set function. 

The first-order theory of Heyting arithmetic  has the same signature and non-logical axioms as Peano arithmetic . In contrast, the signature of set theory does not contain addition "" or multiplication "". Next, it is clarified which set theory axiom may be asserted to prove existence of the latter as a function set, together with their desired relation to zero and successor. Far beyond just the equality predicate, the obtained model of arithmetic then validates

for any quantifier-free formula. Indeed,  is -conservative over  and double-negation elimination is possible for any Harrop formula.

Arithmetic functions from recursion 
Going a step beyond , the axiom granting definition of set functions via iteration-step set functions must be added: For any set , set  and , there must also exist a function  attained by making use of the former, namely such that  and . This iteration- or recursion principle is akin to the transfinite recursion theorem, except it is restricted to set functions and finite ordinal arguments, i.e. there is no clause about limit ordinals. It functions as the set theoretical equivalent of a natural numbers object in category theory.
This then enables a full interpretation of Heyting arithmetic  in our set theory, including addition and multiplication functions.

With this,  and  are well-founded, in the sense of the inductive subsets formulation. Further, arithmetic of rational numbers  can then also be defined and its properties, like uniqueness and countability, be proven.

Recursion from set theory axioms 
Recall that  is short for , where  is short for the total function predicate, a proposition in terms of uses bounded quantifiers. If both sides are sets, then by extensionality this is also equivalent to . (Although by slight abuse of formal notation, as with the symbol "", the symbol "" is also commonly used with classes anyhow.) 

A set theory with the -model enabling recursion principle, spelled out above, will also prove that, for any naturals  and , the function spaces

are sets. Indeed, bounded recursion suffices, i.e. the principle for -defined classes. Conversely, the recursion principle can be proven from a definition involving the union of recursive functions on finite domains. Relevant for this is the class of partial functions on  such that all of its members have a return values only up to some natural number bound, which may be expressed by . Existence of this as a set becomes provable assuming that the individual function spaces  all form sets themselves. To this end

With this axiom, any such space is now a set of subsets of  and this is strictly weaker than full Separation. Notably, adoption of this principle has genuine set theoretical flavor, in contrast to a direct embedding of arithmetic principles into our theory. And it is a modest principle insofar as when further assuming full induction or full exponentiation, taking  to function spaces , or to n-fold Cartesian products, does preserve countability.

In  plus finite exponentiation, the recursion principle is a theorem. Moreover, enumerable forms of the pigeon hole principle can now also be proven, e.g. that on a finitely indexed set, every auto-injection is also a surjection. As a consequence, the cardinality of finite sets, i.e. the finite von Neumann ordinal, is provably unique. Powersets of finite sets exist. The finitely indexed discrete sets are just the finite sets. In particular, finitely indexed subsets of  are finite. Taking quotients or taking the binary union or Cartesian product of two sets preserve finiteness, sub-finiteness and being finitely indexed.

The set theory axioms listed so far incorporates first-order arithemtic and suffices as formalized framework for a good portion of common mathematics. The restriction to finite domains is lifted in the strictly stronger exponentiation axiom below. However, also that axiom does not entail the full induction schema for formulas with unbound quantifiers over the domain of sets, nor a dependent choice principle. Likewise, there are Collection principles that are constructively not implied by Replacement, as discussed further below. A consequence of this is that for some statements of higher complexity or indirection, even if concrete instances of interest may well be provable, the theory may not prove the universal closure.
Stronger than this theory with finite exponentiation is  plus full induction. It implies the recursion principle even for classes and such that  is unique. Already that recursion principle when restricted to  does prove finite exponentiation, and also the existence of a transitive closure for every set with respect to  (since union formation is ). With it more common constructions preserve countability. 
General unions over a finitely indexed set of finitely indexed sets are again finitely indexed, when at least assuming induction for -predicates (with respect to the set theory language, and this then holds regardless of the decidability of their equality relations.)

Induction without infinite sets 
Before discussing even classically uncountable sets, this last section takes a step back to a context more akin to . The addition of numbers, considered as relation on triples, is an infinite collection, just like collection of natural numbers themselves. But note that induction schemas may be adopted (for sets, ordinals or in conjunction with a natural number sort), without ever postulating that the collection of naturals exists as a set. As noted, Heyting arithmetic  is bi-interpretable with such a constructive set theory, in which all sets are postulated to be in bijection with an ordinal. The BIT predicate is a common means to encode sets in arithmetic.

This paragraph lists a few weak natural number induction principles studied in the proof theory of arithmetic theories with addition and multiplication in their signature. This is the framework where these principles are most well understood. The theories may be defined via bounded formulations or variations on induction schemas that may furthermore only allow for predicates of restricted complexity. On the classical first-order side, this leads to theories between the Robinson arithmetic  and Peano arithmetic : The theory  does not have any induction.  has full mathematical induction for arithmetical formulas and has ordinal , meaning the theory lets one encode ordinals of weaker theories as recursive relation on just the naturals. Theories may also include additional symbols for particular functions.
Many of the well studied arithmetic theories are weak regarding proof of totality for some more fast growing functions. Some of the most basic examples of arithmetics include elementary function arithmetic , which includes induction for just bounded arithmetical formulas, here meaning with quantifiers over finite number ranges. The theory has a proof theoretic ordinal (the least not provenly recursive well-ordering) of .
The -induction schema for arithmetical existential formulas allows for induction for those properties of naturals a validation of which is computable via a finite search with unbound (any, but finite) runtime. The schema is also classically equivalent to the -induction schema.
The relatively weak classical first-order arithmetic which adopts that schema is denoted  and proves the primitive recursive functions total. The theory  is -conservative over primitive recursive arithmetic .
Note that the -induction is also part of the second-order reverse mathematics base system , its other axioms being  plus -comprehension of subsets of naturals. The theory  is -conservative over . Those last mentioned arithmetic theories all have ordinal .

Let us mention one more step beyond the -induction schema. Lack of stronger induction schemas means, for example, that some unbounded versions of the pigeon hole principle are unprovable. One relatively weak one being the Ramsey theorem type claim here expressed as follows: For any  and coding of a coloring map , associating each  with a color , it is not the case that for every color  there exists a threshold input number  beyond which  is not ever the mappings return value anymore. (In the classical context and in terms of sets, this claim about coloring may be phrased positively, as saying that there always exists at least one return value  such that, in effect, for some unbounded domain  it holds that . In words, when  provides infinite enumerated assignments, each being of one of  different possible colors, it is claimed that a particular  coloring infinitely many numbers always exists and that the set can thus be specified, without even having to inspect properties of . When read constructively, one would want  to be concretely specifiable and so that formulation is a stronger claim.) Higher indirection, than in induction for mere existential statements, is needed to formally reformulate such a negation (the Ramsey theorem type claim in the original formulation above) and prove it. Namely to restate the problem in terms of the negation of the existence of one joint threshold number, depending on all the hypothetical 's, beyond which the function would still have to attain some color value. More specifically, the strength of the required bounding principle is strictly between the induction schema in  and .
For properties in terms of return values of functions on finite domains, brute force verification through checking all possible inputs has computational overhead which is larger the larger the size of the domain, but always finite. Acceptance of an induction schema as in  validates the former so called infinite pigeon hole principle, which concerns unbounded domains, and so is about mappings with infinitely many inputs.

It is worth noting that in the program of predicative arithmetic, even the mathematical induction schema has been criticized as possibly being impredicative, when natural numbers are defined as the object which fulfill this schema, which itself is defined in terms of all naturals.

Exponentiation 
Classical  without the Powerset axiom is still consistent with all existing sets of reals being subcountable, and there even countable. Such a theory essentially amounts to second-order arithmetic.

Possible choice principles were discussed, a weakened form of the Separation schema was already adopted, and more of the standard  axioms shall be weakened for a more predicative and constructive theory. The first one of those is the Powerset axiom, which is adopted in the form of the space of characteristic functions, itself tied to exactly the decidable subclasses. The following axiom  is strictly stronger than its pendant for finite domains discussed above:

The formulation here again uses the convenient notation for function spaces, as discussed above. In words, the axiom says that given two sets , the class  of all functions is, in fact, also a set.
This is certainly required, for example, to formalize the object map of an internal hom-functor like 

Adopting existence statements like Exponentiation, i.e. function spaces being sets, quantification  over the elements of certain classes of functions now only range over sets. The axiom directly enriches the domain of sets, but with it also enables the derivation of yet more, via bounded Separation, which in turn also strengthens other axioms. Notably, these bounded quantifiers now range over function spaces that are provably uncountable, and hence even classically uncountable. E.g. the collection of all functions  where , i.e. the set  of points underlying the Cantor space, is uncountable, by Cantor's diagonal argument, and can at best be taken to be a subcountable set. In this theory one may now also quantify over subspaces of spaces like , which may be understood as collections of decidable predicates, and which is a third order notion on the naturals. (In this section and beyond, the symbol for the semiring of natural numbers in expressions like  is used, or written , just to avoid conflation of cardinal- with ordinal exponentiation.) Without further axioms, intuitionistic mathematics has models in recursive functions but also forms of hypercomputation. Such a theory does not prove a function space like  to be not enumerable, in the sense of injections out of it.

Roughly, classically uncountable sets tend to not have computably decidable equality.

On countable sets 
With Exponentiation, the union of all finite sequences over a countable set is now a countable set. For any countable family of counting functions together with their ranges, the theory proves the union of those ranges to be countable. In contrast, not assuming countable choice, even  is consistent with the uncountable set  being the union of a countable set of countable sets.

At last, any finitely indexed union of a family of subfinitely indexed resp. subcountable sets is itself subfinitely indexed resp. subcountable as well. Of course, many other theorems about the various function existence predicates hold, especially when assuming countable choice.

The set theory now proves the existence of any primitive recursive function in , and in particular in the uncountable function spaces out of . Indeed, with function spaces and the finite von Neumann ordinals as domains, we can model  as discussed, and thus encode ordinals in the arithmetic. One then furthermore obtains the ordinal-exponentiated number  as a set, which may be characterized as , the counted set of words over an infinite alphabet.

As far as comprehension goes, the dependent or indexed products  are now sets. The theory now proves the collection of all the countable subsets of any set (the collection is a subclass of the powerclass) to be a set.

The class of subsets of a set 
The characterization of the class  of all subsets of a set  involves unbounded universal quantification, namely . Here  has been defined in terms of the membership predicate  above. So in a mathematical set theory framework, the power class is defined not in a bottom-up construction from its constituents (like an algorithm on a list, that e.g. maps ) but via a comprehension over all sets. If  is a set, that defining quantification even ranges over , which makes the axiom of powerset impredicative. The statement  itself is .

The richness of the powerclass in a theory without excluded middle can best be understood by considering small classically finite sets. For any proposition , consider the subclass  of  (i.e.  or ). It equals  when  can be rejected and it equals  (i.e. ), when  can be proven. But  may also not be decidable at all. Consider three different undecidable proposition, none of which provenly imply another. They can ben used to define three subclasses of the singleton , none of which are provenly the same. In this view, the powerclass  of the singleton, usually denoted by , is called the truth value algebra and does not necessarily provenly have only two elements.

With Exponentiation,  being a set already implies Powerset for sets in general. The proof is via replacement for the association of  to , and an argument why all subsets are covered.

The set , of functions taking values in  (i.e. ) on a set , injects into the function space  and corresponds to just the decidable subsets of . 
If the theory proves  above a set (as for example  unconditionally does), then the subset  of  is a function  with . To claim that  is to claim that excluded middle holds for .

It has been pointed out that the empty set  and the set  itself are of course two subsets of . Whether also  is true in a theory is contingent on a simple disjunction:
.
So assuming  for just bounded formulas, predicative Separation then lets one demonstrate that the powerclass  is a set. And so in this context, also full Choice proves Powerset.
Moreover, with bounded excluded middle,  is in bijection with . See also the discussion of  below.
Note that cardinal relations involving uncountable sets are also elusive in , where the characterization of uncountability simplifies to . For example, it is independent whether all such  have , nor does it prove that , see continuum hypothesis and the related Easton's theorem.

Full Separation is equivalent to just assuming that each individual subclass of  is a set. Assuming full Separation, both full Choice and Regularity prove .

Assuming  in this theory, Set induction becomes equivalent to Regularity and Replacement becomes capable of proving full Separation.

Metalogic
While the theory  does not exceed the consistency strength of Heyting arithmetic, adding Excluded Middle gives a theory proving the same theorems as classical  minus Regularity!
Thus, adding Regularity as well as either  or full Separation to  gives full classical .
Adding full Choice and full Separation gives  minus Regularity.

So this would thus lead to a theory beyond the strength of typical type theory.

Category and type theoretic notions 
So in this context with Exponentiation, first-order arithmetic has a model and all function spaces between sets exist. The latter are more accessible than the classes containing all subsets of a set, as is the case with exponential objects resp. subobjects in category theory.
In category theoretical terms, the theory  essentially corresponds to constructively well-pointed Cartesian closed Heyting pretoposes with (whenever Infinity is adopted) a natural numbers object. Existence of powerset is what would turn a Heyting pretopos into an elementary topos.
Every such topos that interprets  is of course a model of these weaker theories, but locally Cartesian closed pretoposes have been defined that e.g. interpret theories with Exponentiation but reject full Separation and Powerset.
A form of  corresponds to any subobject having a complement, in which case we call the topos Boolean. Diaconescu's theorem in its original topos form says that this hold iff any coequalizer of two nonintersecting monomorphisms has a section. The latter is a formulation of choice. Barr's theorem states that any topos admits a surjection from a Boolean topos onto it, relating to classical statements being provable intuitionistically.

In type theory, the expression "" exists on its own and denotes function spaces, a primitive notion. These types (or, in set theory, classes or sets) naturally appear, for example, as the type of the currying bijection between  and , an adjunction.
A typical type theory with general programming capability - and certainly those that can model , which is considered a constructive set theory - will have a type of integers and function spaces representing , and as such also include types that are not countable. This is just to say, or implies, that among the function terms , none have the property of being a surjection.

Constructive set theories are also studied in the context of applicative axioms.

Analysis 
In this section the strength of  is elaborated on.
For context, possible further principles are mentioned, which are not necessarily classical and also not generally considered constructive.
Here a general warning is in order: When reading proposition equivalence claims in the computable context, one shall always be aware which choice, induction and comprehension principles are silently assumed.
See also the related constructive analysis, feasible analysis and computable analysis.

The theory proves uniqueness of Archimedean, Dedekind complete (pseudo-)ordered fields, with equivalence by a unique isomorphism. The prefix "pseudo" here highlights that the order will, in any case, constructively not always be decidable. This result is relevant assuming complete such models exist as sets.

Cauchy sequences 
Exponentiation implies recursion principles and so in , one can comfortably reason about sequences , their regularity properties such as , or about shrinking intervals in . So this enables speaking of Cauchy sequences and their arithmetic. This is also the approach to analysis taken in .

Cauchy reals 
Any Cauchy real number is a collection of such sequences, i.e. subset of a set of functions on  constructed with respect to an equivalence relation. Exponentiation together with bounded separation prove the collection of Cauchy reals to be a set, thus somewhat simplifying the logically treatment of the reals.

Even in the strong theory  with a strengthened form of Collection, the Cauchy reals are poorly behaved when not assuming a form of countable choice, and  suffices for most results. This concerns completeness of equivalence classes of such sequences, equivalence of the whole set to the Dedekind reals, existence of a modulus of convergence for all Cauchy sequences and the preservation of such a modulus when taking limits. An alternative approach that is slightly better behaved is to work a collection of Cauchy reals together a choice of modulus, i.e. not with just the real numbers but with a set of pairs, or even with a fixed modulus shared by all real numbers.

Towards the Dedekind reals 
As in the classical theory, Dedekind cuts are characterized using subsets of algebraic structures such as : The properties of being inhabited, numerically bounded above, "closed downwards" and "open upwards" are all bounded formulas with respect to the given set underlying the algebraic structure. A standard example of a cut, the first component indeed exhibiting these properties, is the representation of  given by

(Depending on the convention for cuts, either of the two parts or neither, like here, may makes use of the sign .)

The theory given by the axioms so far validates that a Pseudo-order|pseudo-ordered field that is also Archimedean and Dedekind complete, if it exists at all, is in this way characterized uniquely, up to isomorphism.
However, the existence of just function spaces such as  does not grant  to be a set, and so neither is the class of all subsets of  that do fulfill the named properties.
What is required for the class of Dedekind reals to be a set is an axiom regarding existence of a set of subsets and this is discussed further below in the section on Binary refinement.
In a context without  or Powerset, countable choice into finite sets is assumed to prove the uncountability of the set of all Dedekind reals.

Whether Cauchy or Dedekind reals, fewer statements about the arithmetic of the reals are decidable, compared to the classical theory.

Constructive schools 
Most schools for constructive analysis validate some choice and also -, as defined in the second section on number bounds. Here are some other propositions employed in theories of constructive analysis that are not provable using just base intuitionistic logic:
 On the recursive mathematics side (the "Russian" or "Markovian" constructive framework with many abbreviations, e.g. ), first one has Markov's principle , which is a form of proof by contradiction motivated by (unbound memory capacity) computable search. This has notable impact on statements about real numbers, as touched upon below. In this school one further even has the anti-classical constructive Church's thesis principle , generally adopted for number-theoretic functions. Church's thesis principle expressed in the language of set theory and formulated for set functions postulates that these all correspond to computable programs that eventually halt on any argument. In computability theory, the natural numbers corresponding to indices of codes of the computable functions which are total are  in the arithmetical hierarchy, meaning membership of any index is affirmed by validating a  proposition. This is to say that such a collection of functions is still a mere subclass of the naturals and so is, when put in relation to some classical function spaces, conceptually small. In this sense, adopting  postulate makes  into a "sparse" set, as viewed from classical set theory. Subcountability of sets can also be postulated independently. 
 So on another end, on the Brouwerian intuitionist side (), there are bar induction, the decidable fan theorem  saying decidable bars are uniform, which are amongst the weakest often discussed principles, Kripke's schema (with countable choice turning all subclasses of  countable), or even Brouwer's anti-classical continuity principle, determining return values of what is established a function on unending sequences already through just finite initial segments. 

Certain laws in both of those schools contradict , so that choosing to adopt all principles from either school disproves theorems from classical analysis.  is still consistent with some choice, but contradicts the classical  and , explained next. The independence of premise rule with set existence premises is not fully understood, but as a number theoretic principle it is in conflict with the Russian school axioms in some frameworks. Notably,  also contradicts , meaning the constructive schools also cannot be combined in full. Indeed, some of the principles cannot be combined constructively simply because they together imply forms of , for example  plus the countability of all subsets of the naturals. These combinations are then naturally also not consistent with further anti-classical principles.

Indecomposability
Denote the class of all sets by . Decidability of membership in a class  can be expressed as membership in . 
We also note that, by definition, the two extremal classes  and  are trivially decidable. Membership in those two is equivalent to the trivial propositions  resp. .

Call a class  indecomposable or cohesive if, for any predicate ,

This expresses that the only properties that are decidable on  are the trivial properties. It is well studied in intuitionistic analysis.

The so called indecomposability schema  (Unzerlegbarkeit) for set theory is a possible principle which states that the whole class  is indecomposable. Extensionally speaking,  postulates that the two trivial classes are the only classes that are decidable with respect to the class of all sets. 
For a simple motivating predicate, consider membership  in the first non-trivial class, which is to say the property  of being empty. This property is non-trivial to the extent that it separates some sets: The empty set is a member of , by definition, while a plethora of sets are provenly not members of . But, using Separation, one may of course also define various sets for which emptiness is not decidable in a constructive theory at all, i.e. membership in  is not provable for all sets. So here the property of emptiness does not partition the set theoretical domain of discourse into two decidable parts. For any such non-trivial property, the contrapositive of  says that it cannot be decidable over all sets.

 is implied by the uniformity principle , which is consistent with  and discussed below.

Non-constructive principles 
Of course  and many principles defining intermediate logics are non-constructive.  and , which is  for just negated propositions, can be presented as De Morgan's rules. 
More specifically, this section shall be concerned with statements in terms of predicates - especially weaker ones, expressed in terms of a few quantifiers over sets, on top of decidable predicates on numbers. Referring back to the section on characteristic functions, one may call a collection  searchable if it is searchable for all its decidable subsets . This is a form of - for . Note that in the context of Exponentiation, such proposition on sets are now set-bound.

Particularly valuable in the study of constructive analysis are non-constructive claims commonly formulated in terms of the collection of all binary sequences and the characteristic functions  on the arithmetic domain  are well studied. Here  is a decidable proposition at each numeral , but, as demonstrated previously, quantified statements in terms of  may not be. As is known from the incompleteness theorem and its variations, already in first-order arithmetic, example functions on  can be characterized such that if  is consistent, the competing - disjuncts, of low complexity, are each -unprovable (even if  proves the disjunction of the two axiomatically.)

More generally, the arithmetic -, a most prominent non-constructive, essentially logical statement goes by the name limited principle of omniscience . In the constructive set theory  introduced below, it implies -, , the -version of the fan theorem, but also  discussed below. Recall examples of famous sentences that can be written down in a -fashion, i.e. of Goldbach-type: Goldbach conjecture, Fermat's last theorem but also the Riemann hypothesis are among them. Assuming relativized dependent choice  and the classical  over  does not enable proofs of more -statements.
 postulates a disjunctive property, as does the weaker decidability statement for functions being constant (-sentences) , the arithmetic -. The two are related in a similar way as is  versus  and they essentially differ by .
 in turn implies the so-called "lesser" version . This is the (arithmetic) -version of the non-constructive De Morgan's rule for a negated conjunction. There are, for example, models of the strong set theory  which separate such statements, in the sense that they may validate  but reject .

Disjunctive principles about -sentences generally hint at equivalent formulations deciding apartness in analysis in a context with mild choice or . 
The claim expressed by  translated to real numbers is equivalent to the claim that either equality or apartness of any two reals is decidable (it in fact decides the trichotomy). It is then also equivalent to the statement that every real is either rational or irrational - without the requirement for or construction of a witness for either disjunct. Likewise, the claim expressed by  for real numbers is equivalent that the ordering  of any two reals is decidable (dichotomy). It is then also equivalent to the statement that if the product of two reals is zero, then either of the reals is zero - again without a witness. Indeed, formulations of the three omniscience principles are then each equivalent to theorems of the apartness, equality or order of two reals in this way. 
Yet more can be said about the Cauchy sequences that are augmented with a modulus of convergence.

A famous source of computable undecidability - and in turn also of a broad range of undecidable propositions - is the predicate expressing a computer program to be total.

Infinite trees 
Through the relation between computability and the arithmetical hierarchy, insights in this classical study are also revealing for constructive considerations. A basic insight of reverse mathematics concerns computable infinite finitely branching binary trees. Such a tree may e.g. be encoded as an infinite set of finite sets
,
with decidable membership, and those trees then provenly contain elements of arbitrary big finite size. The so called Weak Kőnig's lemma  states: For such , there always exists an infinite path in , i.e. an infinite sequence such that all its initial segments are part of the tree. In reverse mathematics, the second-order arithmetic subsystem  does not prove . To understand this, note that there are computable trees  for which no computable such path through it exists. To prove this, one enumerates the partial computable sequences and then diagonalizes all total computable sequences in one partial computable sequences . One can then roll out a certain tree , one exactly compatible with the still possible values of  everywhere, which by construction is incompatible with any total computable path.

In , the principle  implies  and -, a very modest form of countable choice introduced above. The former two are equivalent assuming that choice principle already in the more conservative arithmetic context.  is also equivalent to the Brouwer fixed point theorem and other theorems regarding values of continuous functions on the reals. The fixed point theorem in turn implies the intermediate value theorem, but always be aware that these claims may depend on the formulation, as the classical theorems about encoded reals can translate to different variants when expressed in a constructive context.

The , and some variants thereof, concerns infinite graphs and so its contrapositives gives a condition for finiteness. Again to connect to analysis, over the classical arithmetic theory , the claim of  is for example equivalent to the Borel compactness regarding finite subcovers of the real unit interval.  is a closely related existence claim involving finite sequences in an infinite context. Over , they are actually equivalent. In  those are distinct, but, after again assuming some choice, here then  implies .

Induction

Mathematical induction 
It was observed that in set language, induction principles can read , with the antecedent  defined as further above, and with  meaning  where the set  always denotes the standard model of natural numbers. Via the strong axiom of Infinity and predicative Separation, the validity of induction for set-bounded or -definitions was already established and thoroughly discussed. For those predicates involving only quantifiers over , it validates induction in the sense of the first-order arithmetic theory. In a set theory context where  is a set, this induction principle can be used to prove predicatively defined subclasses of  to be the set  itself. The so called full induction schema now postulates set equality of  to all its inductive subclasses. As in the classical theory, it is also implied when passing to the impredicative full Separation schema. As stated in the section on Choice, induction principles such as this are also implied by various forms of choice principles. 

The recursion principle for set functions mentioned in the section dedicated to arithmetic is also implied by the full induction schema over one's structure modeling the naturals (e.g. ). So for that theorem, granting a model of Heyting arithmetic, it represents an alternative to exponentiation principles.

Predicate formulas used with the schema are to be understood as formulas in first-order set theory. The zero  denotes the set  as above, and the set  denotes the successor set of , with . By Axiom of Infinity above, it is again a member of . Beware that unlike in an arithmetic theory, the naturals here are not the abstract elements in the domain of discourse, but elements of a model. As has been observed in previous discussions, when accepting , not even for all predicatively defined sets is the equality to such a finite von Neumann ordinal necessarily decidable.

Set Induction 
Like mathematical induction above, the strictly stronger full set induction the axiom is usually formulated as a schema in terms of predicates, and thus has a different character than induction principles proven from predicative set theory axioms. A variant of the axiom just for bounded formulas is also studied independently and may be derived from other axioms.

Here  holds trivially and so this covers to the "bottom case"  in the standard framework.

In , the axiom proves induction in transitive sets and so in particular also for transitive sets of transitive sets. The latter then is an adequate definition of the ordinals, and even a -formulation. Induction in turn enables ordinal arithmetic in this sense. Set induction allows definitions of class functions by transfinite recursion. The study of the various principles that grant set definitions by induction, i.e. inductive definitions, is a main topic in the context of constructive set theory and their comparatively weak strengths. This also holds for their counterparts in type theory. Replacement is not required to prove induction over the set of naturals from set induction, but that axiom is necessary for their arithmetic modeled within the set theory.

The axiom of regularity is a single statement with universal quantifier over sets and not a schema. As show, it implies , and so is non-constructive. Now for  taken to be the negation of some predicate  and writing  for the class , induction reads

Via the contrapositive, set induction implies all instances of regularity but only with double-negated existence in the conclusion. In the other direction, given enough transitive sets, regularity implies each instance of set induction.

Metalogic
The theory formulated above can be expressed as  with its collection axioms discarded in favour of the weaker Replacement and Exponentiation axioms. It proves the Cauchy reals to be a set, but not the class of Dedekind reals.

Call an ordinal itself trichotomous if the irreflexive membership relation "" among its members is trichotomous. Like the axiom of regularity, set induction restricts the possible models of "" and thus that of a set theory, as was the motivation for the principle in the 20's. But the constructive theory here does not prove a trichotomy for all ordinals, while the trichotomous ordinals are not well behaved with respect to the notion of successor and rank.

The added proof-theoretical strength attained with Induction in the constructive context is significant, even if dropping Regularity in the context of  does not reduce the proof-theoretical strength.
Even without Exponentiation, the present theory with set induction has the same proof theoretic strength as  and proves the same functions recursive. Specifically, its proof-theoretic large countable ordinal is the Bachmann–Howard ordinal. This is also the ordinal of classical or intuitionistic Kripke–Platek set theory.
It is consistent even with  to assume that the class of  trichotomous ordinals form a set. The current theory augmented with this ordinal set existence postulate proves the consistency of .

Aczel was also one of the main developers or Non-well-founded set theory, which rejects set induction.

Relation to ZF
The theory also constitutes a presentation of Zermelo–Fraenkel set theory  in the sense that variants of all its eight axioms are present. Extensionality, Pairing, Union and Replacement are indeed identical. Separation is adopted in a predicative form and with Infinity is stated in a strong formulation, as in the classical case, they already imply the Empty Set axiom. Exponentiation for finite domains and full mathematical induction are also implied by their stronger adopted variants. Without the principle of excluded middle, the theory here is lacking, in its classical form, full Separation, Powerset as well as Regularity. Accepting  now exactly leads into the classical theory.

The following highlights the different readings of a formal theory. Let  denote the continuum hypothesis and  so that . Then  is inhabited by  and any set that is established to be a member of  either equals  or . Induction on  implies that it cannot consistently be negated that  has some least natural number member. The value of such a member can be shown to be independent of theories such as . Nonetheless, any classical set theory like  proves there exists such a number.

Strong Collection 
Having discussed all the weakened forms of the classical set theory axioms, Replacement and Exponentiation can be further strengthened without losing a type theoretical interpretation, and in a way that is not going beyond .

So firstly, one may reflect upon the strength of the axiom of replacement, also in the context of the classical set theory. For any set  and any natural , there exists the product  recursively given by , which have ever deeper rank. Induction for unbound predicates proves that these sets exist for all of the infinitely many naturals. Replacement "for " now moreover states that this infinite class of products can be turned into the infinite set, . This is also not a subset of any previously established set.

Going beyond those axioms also seen in Myhill's typed approach, consider the discussed constructive theory with Exponentiation and Induction, but now strengthened by the collection schema. In  it is equivalent to Replacement, unless the powerset axiom is dropped. In the current context the strong axiom presented supersedes Replacement, due to not requiring the binary relation definition to be functional, but possibly multi-valued.

In words, for every total relation, there exists an image set  such that the relation is total in both directions. Expressing this via a raw first-order formulation leads to a somewhat repetitive format. The antecedent states that one considers relation  between sets  and  that are total over a certain domain set , that is,  has at least one "image value"  for every element  in the domain. This is more general than an inhabitance condition  in a set theoretical choice axiom, but also more general than the condition of Replacement, which demands unique existence . In the consequent, firstly, the axioms states that then there exists a set  which contains at least one "image" value  under , for every element of the domain. Secondly, in this axioms formulation it then moreover states that only such images  are elements of that new codomain set . It is guaranteeing that  does not overshoot the codomain of  and thus the axiom is also expressing some power akin to a Separation procedure. The principle may be used in the constructive study of larger sets beyond the everyday need of analysis.

Metalogic
This theory without , without unbounded separation and without "naive" Power set enjoys various nice properties. For example, as opposed to  with its subset collection schema below, it has the existence property.

Constructive Zermelo–Fraenkel

Binary refinement
The so called binary refinement axiom says that for any  there exists a set  such that for any covering , the set  holds two subsets  and  that also do this covering job, . It is a weakest form of the powerset axiom and at the core of some important mathematical proofs.
Fullness below, for relations between the set  and the finite , implies that this is indeed possible.

Taking another step back,  plus Recursion and plus Binary refinement already proves that there exists an Archimedean, Dedekind complete pseudo-ordered field. That set theory also proves that the class of left Dedekind cuts is a set, not requiring Induction or Collection. And it moreover proves that function spaces into discrete sets are sets (there e.g. ), without assuming . Already over the weak theory  (which is to say without Infinity) does binary refinement prove that function spaces into discrete sets are sets, and therefore e.g. the existence of all characteristic function spaces .

Subset Collection
The theory known as  adopts the axioms of the previous sections plus a stronger form of Exponentiation. It is by adopting the following alternative to Exponentiation, which can again be seen as a constructive version of the Power set axiom:

An alternative that is not a schema is elaborated on below.

Fullness
For given  and , let  be the class of all total relations between  and . This class is given as

As opposed to the function definition, there is no unique existence quantifier in . The class  represents the space of "non-unique-valued functions" or "multivalued functions" from  to , but as set of individual pairs with right projection in . The second clause says that one is concerned with only these relations, not those which are total on  but also extend their domain beyond .

One does not postulate  to be a set, since with Replacement one can use this collection of relations between a set  and the finite , i.e. the "bi-valued functions on ", to extract the set  of all its subsets. In other words  being a set would imply the Powerset axiom.

Over , there is a single, somewhat clearer alternative axiom to the Subset Collection schema. It postulates the existence of a sufficiently large set  of total relations between  and .

This says that for any two sets  and , there exists a set  which among its members inhabits a still total relation  for any given total relation .

On a given domain , the functions are exactly the sparsest total relations, namely the unique valued ones. Therefore, the axiom implies that there is a set such that all functions are in it. In this way, Fullness implies Exponentiation. It further implies binary refinement, already over .

The Fullness axiom, as well as dependent choice, is in turn also implied by the so-called Presentation Axiom about sections, which can also be formulated category theoretically.

Metalogic of CZF
 has the numerical existence property and the disjunctive property, but there are concessions:  lacks the existence property due to the Subset Schema or Fullness axiom. The existence property is not lacking when the weaker Exponentiation or the stronger but impredicative Powerset axiom axiom is adopted instead. The latter is in general lacking a constructive interpretation.

Unprovable claims
The theory is consistent with some anti-classical assertions, but on its own proves nothing not provable in . Some prominent statements not proven by the theory (nor by , for that matter) are part of the principles listed above, in the sections on constructive schools in analysis, on the Cauchy construction and on non-constructive principles. What follows concerns set theoretical concepts:

The bounded notion of a transitive sets of transitive sets is a good way to define ordinals and enables induction on ordinals. But notably, this definition includes some -subsets in . So assuming that the membership of  is decidable in all successor ordinals  proves  for bounded formulas in . Also, neither linearity of ordinals, nor existence of power sets of finite sets are derivable in this theory, as assuming either implies Power set. The circumstance that ordinals are better behaved in the classical than in the constructive context manifests in a different theory of large set existence postulates.

Consider the functions the domain of which is  or some . These are sequences and their ranges are counted sets. Denote by  the class characterized as the smallest codomain such that the ranges of the aforementioned functions into  are also itself members of . In , this is the set of hereditarily countable set , which is uncountable as it also contains all countable ordinals.  does not prove such a set exists, even when countable choice is assumed.

Finally, the theory does not prove that all function spaces formed from sets in the constructible universe  are sets inside , and this holds even when assuming Powerset instead of the weaker Exponentiation axiom. So this is a particular statement preventing  from proving the class  to be a model of .

Ordinal analysis
Taking  and dropping set induction gives a theory that is conservative over  for arithmetic statements, in that sense that it proves the same arithmetical statements for its -model . Adding back just mathematical induction gives a theory with proof theoretic ordinal , which is the first common fixed point of the Veblen functions  for . This is the same ordinal as for  and is below the Feferman–Schütte ordinal .
Exhibiting a type theoretical model, the full theory  goes beyond , its ordinal still being the modest Bachmann–Howard ordinal.
Assuming the class of trichotomous ordinals is a set raises the proof theoretical strength of  (but not of ).

Being related to inductive definitions or bar induction, the regular extension axiom  raises the proof theoretical strength of . This large set axiom, granting the existence of certain nice supersets for every set, is proven by .

Models
The category of sets and functions of  is a -pretopos. Without diverging into topos theory, certain extended such -pretopoi contain models of . The effective topos contains a model of this  based on maps characterized by certain good subcountability properties.

Separation, stated redundantly in a classical context, is constructively not implied by Replacement. The discussion so far only committed to the predicatively justified bounded Separation. Note that full Separation (together with ,  and also  for sets) is validated in some effective topos models, meaning the axiom does not spoil cornerstones of the restrictive recursive school.

Related are type theoretical interpretations.
In 1977 Aczel showed that  can still be interpreted in Martin-Löf type theory, using the propositions-as-types approach. More specifically, this uses one universe and -types, providing what is now seen a standard model of  in .
This is done in terms of the images of its functions and has a fairly direct constructive and predicative justification, while retaining the language of set theory. Roughly, there are two "big" types , the sets are all given through any  on some , and membership of a  in the set is defined to hold when .
Conversely,  interprets .
All statements validated in the subcountable model of the set theory can be proven exactly via  plus the choice principle -, stated further above.
As noted, theories like , and also together with choice, have the existence property for a broad class of sets in common mathematics.
Martin-Löf type theories with additional induction principles validate corresponding set theoretical axioms.

Soundness and Completeness theorems of , with respect to realizability, have been established.

Breaking with ZF
As a rule of inference,  is closed under Troelstra's general uniformity for both  and . One may adopt it as an anti-classical axiom schema, the uniformity principle denoted ,

This is also often formulated for . Now for a binary set of labels ,  implies the indecomposability schema , as noted.

One may postulate the subcountability of all sets. This already holds true in the type theoretical interpretation and the model in the effective topos. By Infinity and Exponentiation,  is an uncountable set, while the class  or even  is then provenly not a set, by Cantor's diagonal argument. So this theory then logically rejects Powerset and of course .
Subcountability is also in contradiction with various large set axioms. (On the other hand, also using , some such axioms imply the consistency of theories such as  and stronger.)

One may of course add a Church's thesis.

In 1989 Ingrid Lindström showed that non-well-founded sets can also be interpreted in Martin-Löf type theory, which are obtained by replacing Set Induction in  with Aczel's anti-foundation axiom. 
The resulting theory  may be studied by also adding back the -induction schema or relativized dependent choice, as well as the assertion that every set is member of a transitive set.

Intuitionistic Zermelo–Fraenkel
The theory  is  adopting both the standard Separation as well as Power set and, as in , one conventionally formulates the theory with Collection below. As such,  can be seen as the most straight forward variant of  without .
So as noted, in , in place of Replacement, one may use the

While the axiom of replacement requires the relation  to be functional over the set  (as in, for every  in  there is associated exactly one ), the Axiom of Collection does not.
It merely requires there be associated at least one , and it asserts the existence of a set which collects at least one such  for each such .
In classical , the Collection schema implies the Axiom schema of replacement. When making use of Powerset (and only then), they can be shown to be classically equivalent.

While  is based on intuitionistic rather than classical logic, it is considered impredicative.
It allows formation of sets using the Axiom of Separation with any proposition, including ones which contain quantifiers which are not bounded. Thus new sets can be formed in terms of the universe of all sets, distancing the theory from the bottom-up constructive perspective. With this general Separation, it is easy to define sets  with undecidable membership, namely by making use of undecidable predicates defined on a set.
Further, the power set axiom implies the existence of a set of truth values. In the presence of excluded middle, this set has two elements. In the absence of it, the set of truth values is also considered impredicative. The axioms of  are strong enough so that full  is already implied by  for bounded formulas. In fact, it is ready implied by the particular formula , the principle that knowledge of membership of  shall always be decidable, no matter the set. See also the previous discussion in the section on the Exponentiation axiom.

Metalogic
As implied above, the subcountability property cannot be adopted for all sets, given the theory proves  to be a set.
The theory has many of the nice numerical existence properties and is e.g. consistent with Church's thesis principle as well as with  being subcountable. It also has the disjunctive property.

 with Replacement instead of Collection has the general existence property, even when adopting relativized dependent choice on top of it all. But just  as formulated does not. The combination of schemas including full separation spoils it.

Even without , the proof theoretic strength of  equals that of . And  proves them equiconsistent and they prove the same -sentences.

Intuitionistic Z
Again on the weaker end, as with its historical counterpart Zermelo set theory, one may denote by  the intuitionistic theory set up like  but without Replacement, Collection or Induction.

Intuitionistic KP
Let us mention another very weak theory that has been investigated, namely Intuitionistic (or constructive) Kripke–Platek set theory .
It has not only Separation but also Collection restricted to -formulas, i.e. it is similar to  but with Induction instead of full Replacement. The theory does not fit into the hierarchy as presented above, simply because it has Axiom schema of Set Induction from the start. This enables theorems involving the class of ordinals. The theory has the disjunction property.
Of course, weaker versions of  are obtained by restricting the induction schema to narrower classes of formulas, say . The theory is especially weak when studied without Infinity.

Sorted theories

Constructive set theory
As he presented it, Myhill's system  is a theory using constructive first-order logic with identity and three sorts, namely
sets, natural numbers, functions. Its axioms are:
 The usual Axiom of Extensionality for sets, as well as one for functions, and the usual Axiom of union.
 The Axiom of restricted, or predicative, separation, which is a weakened form of the Separation axiom from classical set theory, requiring that any quantifications be bounded to another set, as discussed.
 A form of the Axiom of Infinity asserting that the collection of natural numbers (for which he introduces a constant ) is in fact a set.
 The axiom of Exponentiation, asserting that for any two sets, there is a third set which contains all (and only) the functions whose domain is the first set, and whose range is the second set. This is a greatly weakened form of the Axiom of power set in classical set theory, to which Myhill, among others, objected on the grounds of its impredicativity.

And furthermore:
 The usual Peano axioms for natural numbers.
 Axioms asserting that the domain and range of a function are both sets. Additionally, an Axiom of non-choice asserts the existence of a choice function in cases where the choice is already made. Together these act like the usual Replacement axiom in classical set theory.

One can roughly identify the strength of this theory with a constructive subtheories of  when comparing with the previous sections.

And finally the theory adopts
 An Axiom of dependent choice, which is much weaker than the usual Axiom of choice.

Bishop style set theory
Set theory in the flavor of Errett Bishop's constructivist school mirrors that of Myhill, but is set up in a way that sets come equipped with relations that govern their discreteness.
Commonly, Dependent Choice is adopted.

A lot of analysis and module theory has been developed in this context.

Category theories
Not all formal logic theories of sets need to axiomize the binary membership predicate "" directly. A theory like the Elementary Theory of the Categories Of Set (), e.g. capturing pairs of composable mappings between objects, can also be expressed with a constructive background logic. Category theory can be set up as a theory of arrows and objects, although first-order axiomatizations only in terms of arrows are possible.

Beyond that, topoi also have internal languages that can be intuitionistic themselves and capture a notion of sets.

Good models of constructive set theories in category theory are the pretoposes mentioned in the Exponentiation section. For some good set theory, this may require enough projectives, an axiom about surjective "presentations" of set, implying Countable Dependent Choice.

See also

 Axiom schema of predicative separation
 Constructive mathematics
 Constructive analysis
 Constructive Church's thesis rule and principle
 Computable set
 Existence Property
 Diaconescu's theorem
 Epsilon-induction
 Hereditarily finite set
 Heyting arithmetic
 Impredicativity
 Intuitionistic type theory
 Law of excluded middle
 Ordinal analysis
 Subcountability

References

Further reading 
 
 Aczel, P. and Rathjen, M. (2001). Notes on constructive set theory. Technical Report 40, 2000/2001. Mittag-Leffler Institute, Sweden.

External links 
 Laura Crosilla, Set Theory: Constructive and Intuitionistic ZF, Stanford Encyclopedia of Philosophy, Feb 20, 2009
 Benno van den Berg, Constructive set theory – an overview, slides from Heyting dag, Amsterdam, 7 September 2012

Constructivism (mathematics)
Intuitionism
Systems of set theory