axiom of choice

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### Context
#### Foundations
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[[!include foundations - contents]]
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# The axiom of choice
* table of contents
{: toc}

## Statement

The __axiom of choice__ is the following statement:

* _Every [[surjection]] in the category [[Set]] of [[sets]] [[split epimorphism|splits]]._

This means: for every [[surjection]] $f\colon A \to B$ of sets, there is a function $\sigma\colon B \to A$ (a [[section]]), such that
$$
  (B \stackrel{\sigma}{\to} A \stackrel{f}{\to} B)
  =
  (B \stackrel{Id_B}{\to} B)
  \,.
$$

Note that a surjection $A \to B$ of sets can be regarded as a $B$-indexed [[family]] of [[inhabited sets]], while the existence of a section is equivalent to a choice of one element in each set of this family.  This reproduces the more classical form of the axiom of choice.

When the full axiom of choice fails, it may still be valid for some restricted class of objects $A$ and/or $B$.  An object $B$ such that any [[epimorphism]] $A \to B$ [[split epimorphism|splits]] is called [[projective object|projective]]; this means that one can make choices 'indexed by' $B$.  Dually, an object $A$ such that one can make choices 'with values in' $A$ is called a [[choice object]] (this is not quite equivalent to every epimorphism $A \to B$ splitting).

## In other categories

More generally, we may consider analogous statements in [[categories]] $C$ other than $Set$.

### External form
 {#ExternalForm}

We say that $C$ satisfies the **external axiom of choice** if every [[epimorphism]] in $C$ [[split epimorphism|splits]].

In this form, the axiom of choice may look less mysterious than in its original formulation. For instance, it is clear that it fails in contexts such as $C =$ [[Top]] and $C = $[[Diff]], due to the existence of nontrivial topological and smooth [[fiber bundles]].

If $C$ is not [[balanced category|balanced]], such as a [[regular category|regular]] or [[coherent category]] which is not a [[pretopos]], it may be more appropriate to replace in this statement "epimorphism" by [[regular epimorphism]] (or [[extremal epimorphism]], [[effective epimorphism]], etc.)  In $Set$ (and in any [[topos]]), all of these notions of epimorphism are the same.

More generally still, if $C$ is a [[site]], then the axiom of choice for $C$ may be taken to say that any [[cover]] $U\to X$ admits a section.  Obviously this refers only to singleton covers, but if $C$ is [[superextensive site|superextensive]] then any covering family $(p_i\colon U_i \to X)_i$ can be replaced by a singleton cover $\coprod_u U_i \to X$.

### Internal form

However, when working in a category that has an [[internal logic]], we may want to "internalize" the axiom of choice by asserting, not that every epimorphism has a section, but that the statement "every epimorphism has a section" is true in the internal logic (or more precisely the [[stack semantics]]).  An equivalent statement is that every object is [[internally projective object|internally projective]].  We call this the **internal axiom of choice**.

This is generally a weaker statement: a [[topos]] satisfies the external AC if and only if it satisfies the internal AC and also (the external form of) [[supports split]].   Often, however, this is the more relevant notion to consider.

The following characterization can be found in Freyd-Scedrov ([1990](#FreydScedrov90), p.181)

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###### Theorem
A [[Grothendieck topos]] satisfies the internal axiom of choice iff it is a [[Boolean topos|Boolean]] [[étendue]].
=--

In particular, satisfaction of IAC entails Booleanness.

Classical examples of Boolean étendues are provided by toposes of group actions hence these all satisfy IAC but satisfy AC only in case the group is trivial (cf. Johnstone [1977](#JT77), p.144 or Freyd-Scedrov [1990](#FreydScedrov90), p.84, 179)).

If a topos $C$ satisfies IAC, then so do all of its [[slice categories]], although this may not be obvious.  See [this answer](http://mathoverflow.net/questions/139874/pullback-stability-of-internally-projective-objects/140262#140262).

### In toposes

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###### Theorem
A [[Grothendieck topos]] satisfies the external axiom of choice iff it is equivalent to the [[category of sheaves]] on a [[complete Boolean algebra]].
=--

## Equivalents

The following statements are all equivalent to the axiom of choice in $Set$ (although sometimes the proof in one direction requires [[excluded middle]]).  This is a *very* short list; much longer lists can be found elsewhere, such as at [Wikipedia](http://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents).  Some of the statements on this list, though, may be of interest to nLabbers but are not commonly mentioned as equivalents of choice.

* That any [[cartesian product]] of any [[family]] of [[inhabited sets]] is [[inhabited]].
* That given a [[surjection]] $f$ and an element $b$ in the [[codomain]] of $f$, the [[preimage]] $f^*(b)$ of $f$ at $b$ has a (local) [[choice operator]], a function $\epsilon:[f^*(b)] \to f^*(b)$ from the [[support of a set|support]] of $f^*(b)$ to $f^*(b)$ itself.
* The [[well-ordering theorem]] (that any set can be [[well-order|well-ordered]]),
* [[Zorn's lemma]],
* That ($L =$ [[monomorphism | monomorphisms]], $R =$ [[epimorphism | epimorphisms]]) is a [[weak factorization system on Set]].
* That [[Set]] is equivalent to its own [[free exact completion]].
* That there exists a [[group]] structure on every [[inhabited set]] (see [[Hartogs number]], or [this MO answer](http://mathoverflow.net/questions/12973/does-every-non-empty-set-admit-a-group-structure-in-zf/12988#12988)).

* {#EsoFunctorSplits} That every [[essentially surjective functor]] between [[small categories|small]] [[strict categories]] is [[split essentially surjective]].

* {#EsoFFFunctorEquivalence} That every [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]] functor between [[small categories|small]] [[strict categories]] is a strong [[equivalence of categories]].

* That the [[nonabelian cohomology]] $H^1(X;G)$ is trivial for every set $X$ and every [[group]] $G$ (see [this post](http://golem.ph.utexas.edu/category/2013/07/cohomology_detects_failures_of.html)).
* If $X$ and $Y$ are infinite sets and the two [[free groups]] $F(X)$ and $F(Y)$ have equal [[cardinality]], then $X$ and $Y$ also have equal cardinality (see [Kleppmann14](#Kleppmann14)).
* That every [[vector space]] is [[free object|free]].
* That every [[subset]] $A \subseteq B$ in a universe $\mathcal{U}$ comes with a choice of [[injection]] $i:A \hookrightarrow B$. Constructive mathematicians usually use subsets equipped with the structure of an injection, as those are usually more useful than general subsets with the mere property of being a subset.

## Variants
 {#Variants}

There are a number of weaker axioms which are implied by the full axiom of choice.  Some of these are valid or accepted more generally than the full AC, and/or suffice for some of the usual applications of choice.  In particular, the full axiom of choice is generally rejected in [[constructive mathematics]], whereas some of these weaker forms of choice may be accepted, such as (in order of increasing strength) [[countable choice]], [[dependent choice]], and [[COSHEP]].

* Many applications of choice in [[logic]], [[topology]], and [[algebra]] require only the [[ultrafilter principle]] (UF), or equivalently the *Boolean prime ideal theorem*.

* From the perspective of [[constructive mathematics]], the principle of [[excluded middle]] (EM) may be seen as a weak form of the axiom of choice; EM is equivalent to the statement that every [[Kuratowski-finite set]] is projective. See at _[[Diaconescu-Goodman-Myhill theorem]]_.

* A very weak form of choice (which follows from EM) is the statement that [[supports split]] in $Set$.

* The axioms of [[countable choice]] (CC) and [[dependent choice]] (DC) suffice for many of the usual applications of choice in the analysis of [[separable space | separable spaces]].  CC states that the set $\mathbb{N}$ of [[natural number | natural numbers]] is projective.  DC strenghtens CC by allowing the set of possible choices for $n+1$ to depend on the choice made for $n$.

* The axiom [[COSHEP]], also called the "presentation axiom," says that any set admits a surjection from a projective one (whereas full AC says that all sets are projective).  This implies CC and DC, and is moreover sufficient for the existence of [[projective resolution | projective resolutions]] and cofibrant replacements, as well as the usual theorems in algebra that (for example) [[Mod]] has enough projectives.  For example, see the [[canonical model structure on Cat]].

* The [[axiom of small violations of choice]] (SVC) asserts there is a set $S$ such that every set is a [[subquotient]] of $C\times S$ for some [[choice object|choice set]] $C$.  Intuitively, this says that the failure of AC is parametrized by a single set.  It can be regarded as a "dual" of COSHEP, since it deals with choice sets rather than projective ones, it implies the existence of (at least some) [[injective resolution | injective resolutions]], and together with COSHEP and EM it implies full AC.

* The [[axiom of multiple choice]] is a different way of saying that choice is violated in only a small way, which is more "local" than SVC.  It apparently follows from SVC, at least in [[ZF]].

* The [[small cardinality selection axiom]] is another similar axiom.  It asserts that there is a class function selecting for every set an isomorphic set (its "cardinality") such that among each isomorphism class of sets, the collection of all "cardinalities" forms only a set.

* A still weaker axiom along the lines of "AC fails in only a small way," which is implied by AMC, is [[WISC]], i.e. that for any set $X$, the full subcategory of $Set/X$ consisting of the surjections has a [[weakly initial set]] (under COSHEP it has a single weakly initial object, namely a projective cover of $X$).  Two similar assertions are that the [[free exact completion]] $Set_{ex/lex}$ of $Set$ is a [[topos]] (i.e. that $Set$ has a [[generic proof]]), and that $Set_{ex/lex}$ is [[well-powered category|well-powered]]; both of these imply WISC.

* In [[constructive mathematics]], there is also [[decidable choice]], where every [[set]] with [[decidable equality]] is projective.

The axiom of choice can also be strengthened in a few ways.

* While the ordinary axiom of choice says that any surjection of sets is split, the *axiom of global choice* says that this is also true for any surjection of [[proper classes]].  (Making this precise requires a bit of work.)  It is equivalent to the existence of a well-ordering of the class of all sets.

* One can also postulate a [[choice operator]], which gives a *specified* way to choose an element from any nonempty set.  This implies global choice, and conversely a choice operator can be defined from any well-ordering of the class of all sets.

Finally, one can instead adopt the *negation* of the axiom of choice, or a strengthened version of this negation:

* The assumption that every subset of the [[real line]] has the [[Baire property]] (BP) is consistent with DC but not AC; the same holds for the assumption that every subset of the real line is [[Lebesgue measure|measurable]] (LM) if at least one [[Grothendieck universe]] exists.  These assumptions leads to a very nice setting for analysis called [[dream mathematics]].

* The [[axiom of determinacy]] is a natural statement in [[game theory]] that is consistent with DC; in fact, it implies DC in certain models, such as in the constructible (in the sense of Goedel) closure of the set of reals. However, determinacy contradicts full AC (for example, it implies LM and BP, as in the previous entry).

* Any of the varieties of [[constructive mathematics]] that contradict excluded middle necessarily contradict choice, but they are usually (if not always) consistent with DC (and even COSHEP).

* The existence of a [[elementary embedding#Inconsistency|Reinhardt cardinal]] contradicts AC.

### In higher category theory

To formulate a version of the axiom of choice in a higher category, one has to make an appropriate choice of the meaning of "epimorphism".  In most cases, it is best to choose [[effective epimorphisms|effective epimorphism in an (infinity,1)-category]] or a related notion such as [[eso morphisms]].

Less obviously, we usually want to also impose [[truncated object|truncation]] requirements on at least some of the objects involved in the axiom of choice.  It seems usually necessary to require the codomain to be 0-truncated (axioms of choice without this requirement tend to be inconsistent); as for the domain we can choose to or not.

* An $(\infty,1)$-category satisfies the **axiom of $n$-choice**, or $AC_n$, if every $n$-[[truncated morphism]] with 0-truncated codomain has a section.  We write $AC_\infty$ for the **axiom of infinity-choice**: the statement that *every* morphism with 0-truncated codomain has a section.

These are stronger axioms as $n$ increases.  The "difference" between $AC_0$ and $AC_\infty$ is roughly the axiom that [[sets cover]].

For $(n,k)$-categories with $k\gt 1$ it is unclear whether it is sensible to allow the domain to be non-groupoidal.

* In [[2-category theory]], the _[[michaelshulman:axiom of 2-choice]]_ has been proposed to mean that every [[eso morphism]] with groupoidal domain and 0-truncated codomain has a section.

There are also "internal" versions of these axioms.

* In [[homotopy type theory]] (the internal logic of an $(\infty,1)$-topos), the internal version of $AC_n$ is "every surjection onto a set with $n$-type fibers has a section", or equivalently
  $$ \prod_{(X:Set)} \prod_{(Y:X\to n Type)} \Big( \prod_{(x:X)} \Vert Y(x)\Vert \to \Big\Vert \prod_{(x:X)} Y(x) \Big\Vert \Big) $$

* More generally, we can replace the $(-1)$-truncation by the $k$-truncation to obtain a family of axioms $AC_{k,n}$.

* We can also replace the $(-1)$-truncation by the assertion of $k$-connectedness, obtaining the **axiom of $k$-connected choice**.

### In type theory
 {#InTypeTheory}

In [[dependent type theory]], there are two interpretations of [[predicate logic]] in the type theory; [[propositions as some types]] and [[propositions as types]], resulting in two notions of the axiom of choice:

* [[propositional axiom of choice]]

* [[type theoretic axiom of choice]]

The former corresponds to the [[internal axiom of choice]] in [[categorical semantics]] and [[set theory]], while the latter is provable in any [[locally cartesian closed category]] or the [[set theory]] which interprets such categories.

### In spatial type theory

In [[spatial type theory]], which is [[dependent type theory]] with the [[sharp modality]] and the [[flat modality]], there exists a version of the axiom of choice called the **sharp axiom of choice**, which is given by the following [[inference rule]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ac}^{\sharp}_{A, B}:\left(\prod_{x:A} \sharp [B(x)]\right) \to \sharp\forall x:A.\sharp B(x)}$$

where $\sharp A$ is the sharp modality of type $A$, $[A]$ is the [[propositional truncation]] of $A$, and the [[universal quantifier]] $\forall x:A.B(x)$ is the propositional truncation of the [[dependent product type]] $\prod_{x:A} B(x)$.

## Consequences

* The axiom of choice implies the [[principle of excluded middle]] (due to Diaconesu 1975) see [McLarty, theorem 17.9](#McLarty), and see at _[excluded middle -- Relation to axiom of choice](excluded+middle#RelationToTheAxiomOfChoice)_

See also the [choice consequences data base](#ConsequencesDatabase)

## Related concepts

* [[propositional axiom of choice]]

* [[well-ordering]]

* [[anafunctor]]

* [[foundations]]

* [[Boolean topos]]

* [[finite choice]]

## References

### General

* [[HAF|Eric Schechter's analysis book]] surveys several variants of AC and its negation with a view to applications of [[analysis]], including this nice picture:

\begin{imagefromfile}
    "file_name": "ac-chart-7dd95d8c-7395-40f1-8c6d-9d3cbe59d957.gif"
\end{imagefromfile}

  (Here, UF, DC, CC, BP, and LM are as defined above.)

Discussion in [[toposes]] is in

* {#McLarty} [[Colin McLarty]], section 17.6 of _Elementary Categories, Elementary Toposes_

* {#ConsequencesDatabase} The [Consequences of the Axiom of Choice Project](http://consequences.emich.edu/conseq.htm) provides an interactive data base that can be used to search for implications between various (weakened) forms of the Axiom of Choice. [Choiceless grapher](http://cgraph.inters.co/) builds on this data and provides a graphical presentation.

* Gonçalo Gutierres da Conceição, _The Axiom of Countable Choice in Topology_, [pdf](http://www.mat.uc.pt/~ggutc/teses/teseingles.pdf)

  Despite the title, this covers more than countable choice, but the focus is on sequential aspects (metric spaces, first- and second-countable spaces, etc).

A classical reference for AC in toposes is section 5.2 (pp.140ff) in

* {#JT77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Reprinted by Dover Mineola 2014)

Relation to [[cohomology]] is discussed in

* [[Andreas Blass]], _Cohomology detects failures of the axiom of choice_, Trans. Amer. Math. Soc. 279 (1983), 257-269 ([web](http://www.ams.org/journals/tran/1983-279-01/S0002-9947-1983-0704615-7/))

* [[Mike Shulman]], _Cohomology_ on the [[homotopy type theory]] blog [here](http://homotopytypetheory.org/2013/07/24/cohomology/)

and to [[free groups]] in

* {#Kleppmann14} Philipp Kleppmann, *Generating sets of free groups and the axiom of choice*, Mathematical Logic Quarterly, 2014, [doi](https://doi.org/10.1002/malq.201300088)

An alternative form of AC in toposes using injectivity is explored in

* Toby Kenney, _Injective Power Objects and the Axiom of Choice_ , JPAA **215** (2011) pp.131–144.

The result on IAC in toposes stems from

* {#FreydScedrov90}[[Peter Freyd|P. J. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.

### In homotopy type theory
 {#ReferencesInHomotopy}

Discussion in [[homotopy type theory]]:

* {#HoTTBook} [[Univalent Foundations Project]], section 3.8 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

* [[Egbert Rijke]], section 17.4 of *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

### In spatial type theory

For the sharp axiom of choice in spatial type theory, see:

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

category: foundational axiom

[[!redirects axioms of choice]]
[[!redirects Choice]]
[[!redirects AC]]
[[!redirects full choice]]
[[!redirects full AC]]

[[!redirects axiom of global choice]]
[[!redirects global axiom of choice]]

[[!redirects axiom of infinity-choice]]
[[!redirects axiom of n-choice]]
[[!redirects axiom of k-connected choice]]

[[!redirects set-theoretic axiom of choice]]
[[!redirects set theoretic axiom of choice]]

[[!redirects external axiom of choice]]
[[!redirects internal axiom of choice]]

[[!redirects sharp axiom of choice]]


axiom of circle type localization

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#### Foundational axiom
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## Idea

In [[dependent type theory]], the [[0-truncation]] [[modality]] of a type $A$ can be defined by [[localization of a type]] $A$ at the [[circle type]] $S^1$. This means that a type $A$ is 0-truncated, i.e. a set, if it is $S^1$-local, which means that, in addition to [[axiom K]] and [[uniqueness of identity proofs]], there is another way to make the types of a universe into sets: by stipulating that every type is $S^1$-local, or that the canonical [[function]]
$$\mathrm{const}_{A, S^1} \equiv \lambda x:A.\lambda i:S^1.x:A \to (S^1 \to A)$$
which takes elements of $A$ to [[constant functions]] in $S^1 \to A$, is an [[equivalence of types]]. This is (tentatively) called the **axiom of circle type localization** or the axiom of $S^1$-localization.

Assuming that one has the function $\mathrm{const}_{A, S^1}:A \to (S^1 \to A)$ defined in the [[dependent type theory]], the syntactic rules for the axiom of $S^1$-localization in a universe $\mathcal{U}$ is given by:

$$\frac{\Gamma \vdash A:\mathcal{U} \quad \Gamma \vdash f:S^1 \to A}{\Gamma \vdash \mathrm{circlocal}_{A}:\mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$

There is also a definitional version of $S^1$-localization which says that $\mathrm{const}_{A, S^1}$ is a definitional [[isomorphism]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:S^1 \to A \vdash \mathrm{const}_{A, S^1}^{-1}(p):A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash \mathrm{const}_{A, S^1}^{-1}(\lambda i:S^1.x) \equiv x:A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:S^1 \to A \vdash \lambda i:S^1.\mathrm{const}_{A, S^1}^{-1}(p) \equiv p:S^1 \to A}$$

### Relation to axiom K

One can prove axiom K (positive copy induction rules) from the axiom of circle type localization (negative copy inference rules):

\begin{theorem}
Suppose that every type $A$ is definitionally $S^1$-local.

Then definitional [[axiom K (type theory)|axiom K]] holds: given any type $A$, and any type family $C(p)$ indexed by loops $p:S^1 \to A$ in $A$, and given any dependent function $t:\prod_{x:A} C(\lambda i:S^1.x)$ which says that for all elements $x:A$, there is an element of the type defined by substituting the constant loop of $x:A$ into $C$, $C(\lambda i:S^1.x)$, one can construct a dependent function $K_A(t):\prod_{z:S^1 \to A} C(z)$ such that for all $x:A$, $K_A(t, \lambda i:S^1.x) \equiv t(x):C(\lambda i:S^1.x)$.
\end{theorem}

\begin{proof}
$K_A(t)$ is defined to be

$$K_A(t) \equiv \lambda p:S^1 \to A.t(\mathrm{const}_{A, S^1}^{-1}(p))$$

and by the computation rules of loop types as negative copies, one has that for all $x:A$,

$$\mathrm{const}_{A, S^1}^{-1}(\lambda i:\mathbb{I}.x) \equiv x$$

and so by definition of $\mathrm{ind}_{S^1 \to A}(t)$ and the judgmental congruence rules for substitution, one has

$$K_A(t, \lambda i:S^1.x) \equiv t(\mathrm{const}_{A, S^1}^{-1}(\lambda i:S^1.x)) \equiv t(x)$$
\end{proof}

One has the following analogies between localization at a specific type and the type theoretic letter rule that it proves:

| localization rule | type theoretic letter rule |
|-------------------|----------------------------|
| [[I-localization|$\mathbb{I}$-localization]] | [[J-rule]] |
| [[S1-localization|$S^1$-localization]] | [[K-rule]] |

## Consequences

Since the [[boolean domain]] $\mathbb{2}$ is $S^1$-local, the axiom of $S^1$-localization implies that $S^1$ is [[compact connected]]:

\begin{theorem}
Assuming [[propositional truncation]], where $\Omega$ is the type of all $\mathcal{U}$-small propositions with type reflector $T$, and the axiom of $S^1$-localization, if the function $\mathrm{const}_{2, S^1}$ is an equivalence of types, then for all functions $P:S^1 \to \Omega$, if for all $x:S^1$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then either for all $x:S^1$, $T(P(x))$ is contractible, or for all $x:S^1$, $\neg T(P(x))$ is contractible.
\end{theorem}

\begin{proof}
If $P:S^1 \to \Omega$ is such that for all $x:S^1$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then there is a function $P':S^1 \to \mathbb{2}$ into the [[booleans type]] $\mathbb{2}$ with $\delta_{P'}^{1_2}(x):(P'(x) = 1_2)) \simeq T(P(x))$ and $\delta_{P'}^{0_2}(x):(P'(x) = 0_2)) \simeq \neg T(P(x))$. Since $\mathbb{2}$ is $S^1$-local, then, by the axiom of $S^1$-localization, $P'$ is constant, which implies that either for all $x:S^1$, $T(P(x))$ is contractible, or for all $x:S^1$, $\neg T(P(x))$ is contractible. Thus, $S^1$ is compact connected
\end{proof}

## In spatial type theory

In [[spatial type theory]], a crisp type $\Xi \vert () \vdash A$ is discrete if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]]. There is a variant of the above axiom called the **axiom of circle type cohesion** or **axiom $S^1 \flat$**, which states that given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $S^1$-local, or if $\mathrm{const}_{A, S^1}$ is an [[equivalence of types]].

$$\frac{\Xi \vert () \vdash A \; \mathrm{type}}{\Xi \vert () \vdash S^1 \flat\mathrm{ax}_A:\mathrm{isEquiv}(\lambda x:\flat A.x_\flat) \simeq \mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$

This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $S^1$-local, resulting in a flavor of [[cohesive homotopy type theory]] where the [[shape modality]] is the [[0-truncation]] [[modality]]. This rule is equivalent to the axiom of circle type localization if every type is discrete.

## See also

* [[set-level type theory]]

* [[axiom of set truncation]]

* [[axiom K]]

* [[uniqueness of identity proofs]]

* [[interval type localization]]

* [[axiom of cohesion]]

[[!redirects axiom of circle type localization]]
[[!redirects axiom of S1 localization]]
[[!redirects axiom of S1-localization]]

[[!redirects circle type localization]]
[[!redirects S1 localization]]
[[!redirects S1-localization]]

[[!redirects axiom of circle type cohesion]]
[[!redirects axiom of S1 cohesion]]
[[!redirects axiom of S1-cohesion]]

[[!redirects axiom circle type flat]]
[[!redirects axiom S1 flat]]
[[!redirects axiom S1-flat]]

[[!redirects circle type cohesion]]
[[!redirects S1 cohesion]]
[[!redirects S1-cohesion]]

[[!redirects circle type flat]]
[[!redirects S1-flat]]
[[!redirects S1 flat]]


axiom of cohesion

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## Idea
 {#Idea}

Axioms of [[cohesion]] are certain [[axioms]] added to any [[spatial type theory]] in order to define the [[shape modality]] for [[cohesive homotopy type theory]]. In particular, the *axiom of real cohesion* plays a role in defining [[real-cohesive homotopy type theory]] (the setting for [[classical homotopy theory]] and [[algebraic topology]]), and the *axiom of affine cohesion* plays a role in defining [[A1-cohesive homotopy type theory|$\mathbb{A}^1$-cohesive homotopy type theory]] (the setting for [[A1-homotopy theory|$\mathbb{A}^1$-homotopy theory]]), where the [[affine line]] $\mathbb{A}^1$ plays the role that $\mathbb{R}$ does in real-cohesive homotopy type theory. Affine cohesion could hypothetically also be used in defining a cohesive version of [[Mitchell Riley]]'s [[bunched logic|bunched]] [[linear homotopy type theory]] (the setting for [[stable homotopy theory]]) for the purposes of doing [[motivic homotopy theory]].

## Definition

We assume the presentation of [[spatial type theory]] using crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments. We also assume [[identity types]], the [[sharp modality]], and the [[flat modality]].

Given a type $A$, let us define $\mathrm{const}_{A, R}:A \to (R \to A)$ to be the type of all constant functions in $R$:
$$\delta_{\mathrm{const}_{A, R}}(a, r):\mathrm{const}_{A, R}(a)(r) =_A a$$
There is an equivalence $\mathrm{const}_{A, 1}:A \simeq (1 \to A)$ between the type $A$ and the type of functions from the [[unit type]] $1$ to $A$.
Given types $B$ and $C$ and a function $F:(B \to A) \to (C \to A)$, type $A$ is **$F$-[[localization of a type at a family of functions|local]]** if the function $F:(B \to A) \to (C \to A)$ is an [[equivalence of types]].

A crisp type $\Xi \vert () \vdash A$ is **discrete** if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]].

The **axiom of cohesion** for type $R$ states that there is a crisp type $\Xi \vert () \vdash R \; \mathrm{type}$ such that given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, R})$-local, or equivalently, if $\mathrm{const}_{A, R}$ is an [[equivalence of types]].

$$\frac{\Xi \vert () \vdash A \; \mathrm{type}}{\Xi \vert () \vdash R \flat\mathrm{ax}_A:\mathrm{isEquiv}(\mathrm{const}_{A, R}) \simeq \mathrm{isEquiv}(\lambda x:\flat A.x_\flat)}$$

This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, R})$-local, or equivalently, if $\mathrm{const}_{A, R}$ is an [[equivalence of types]].

Another consequence is that the [[shape]] of $R$ is [[contractible]].

\begin{theorem}
Assuming a type $R$ and the axiom of $R$-cohesion, the shape of $R$ is contractible.
\end{theorem}

\begin{proof}
The type $R$ is inhabited by $\kappa_R(\sigma_R)$, so it remains to show that for all $x:\esh R$, $x =_{\esh R} \kappa_R(\sigma_R)$. Since $\esh R$ is discrete, so is the identity type $x =_{\esh R} \kappa_R(\sigma_R)$, which means by $\esh$-induction, it suffices to prove $\sigma_R(x) =_{\esh R} \kappa_R(\sigma_R)$ for all $x:\esh R$. But this is true from the third introduction rule for $\esh R$.
\end{proof}

In addition, if the [[type of booleans]] $\mathbb{2}$ is discrete, then $R$ is [[compact connected]]:

\begin{theorem}
Assuming a type $R$, a [[type of all propositions]] $\Omega$ with type reflector $T$, and the axiom of $R$-cohesion, if the function $\mathrm{const}_{2, R}$ is an equivalence of types, then for all functions $P:R \to \Omega$, if for all $x:R$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then either for all $x:R$, $T(P(x))$ is contractible, or for all $x:R$, $\neg T(P(x))$ is contractible.
\end{theorem}

\begin{proof}
If $P:R \to \Omega$ is such that for all $x:R$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then there is a function $P':R \to \mathbb{2}$ into the [[booleans type]] $\mathbb{2}$ with $\delta_{P'}^{1_2}(x):(P'(x) = 1_2)) \simeq T(P(x))$ and $\delta_{P'}^{0_2}(x):(P'(x) = 0_2)) \simeq \neg T(P(x))$. But if $\mathbb{2}$ is discrete, then by $R$-cohesion $P'$ is constant, which implies that either for all $x:R$, $T(P(x))$ is contractible, or for all $x:R$, $\neg T(P(x))$ is contractible. Thus, $R$ is compact connected if $\mathbb{2}$ is discrete.
\end{proof}

## Examples

There are a number of axioms which in general could be called an axiom of cohesion for type $R$. The most general such axiom of cohesion is called **stable local connectedness** or **axiom C0**, which imposes no other restrictions on $R$. If we additionally assume that the type $R$ is pointed with point $0:R$, then the axiom becomes **punctual local connectedness** or **axiom C1**, and if we additionally assume that the type $R$ is a non-trivial bi-pointed set, with points $0:R$, $1:R$, and witnesses $\tau_0:\mathrm{isSet}(R)$ and $\mathrm{nontriv}:(0 =_{R} 1) \to \emptyset$, then the axiom becomes **contractible codiscreteness** or **axiom C2**. If we additionally assume that the type $R$ is a [[Dedekind complete]] [[Archimedean field|Archimedean]] [[ordered field|ordered]] [[lattice]] [[field]] (and usually written as $\mathbb{R}$), then the axiom becomes **real cohesion** or **axiom $\mathbb{R}$-flat**.

| number/symbol | name | associated [[shape modality]] | additional requirements |
|--------|------|------|-----------|
| $C_0$ | cohesion/stable [[locally ∞-connected (∞,1)-topos|local connectedness]] | [[localization of a type|localization]] at a [[type]] $R$ |  |
| $C_1$ | punctual cohesion/punctual [[locally ∞-connected (∞,1)-topos|local connectedness]] | [[localization of a type|localization]] at a [[pointed type]] $R$ |  |
| $C_2$ | contractible codiscreteness | [[localization of a type|localization]] at a non-trivial [[bi-pointed type|bi-pointed]] [[h-set]] $R$ |  |
|  | discrete cohesion | [[localization of a type|localization]] at a [[contractible type]] $R$ |  |
|  | affine cohesion/algebraic cohesion/$\mathbb{A}^1$-cohesion | [[localization of a type|localization]] at an [[affine line]] $\mathbb{A}^1$ |  |
|  | [[compact connectedness]] | [[localization of a type|localization]] at a [[type]] $R$ | the [[type of booleans]] $\mathbb{2}$ is discrete |
|  | [[continuum]] cohesion | [[localization of a type|localization]] at a [[Hausdorff space]] $R$ | $\mathbb{2}$ is discrete. |
|  | [[metric continuum]] cohesion | [[localization of a type|localization]] at a [[metric space]] $R$ | $\mathbb{2}$ is discrete. |
| $\mathbb{R} \flat$ | [[real number|real]] cohesion | [[localization of a type|localization]] at the [[real numbers]] $\mathbb{R}$ | A higher [[coinductive type]] representing the [[homotopy]] [[terminal]] [[Archimedean ordered field]] $\mathbb{R}$ |
| $\mathbb{R} \flat$ | [[Dedekind real number|real]] cohesion ([[impredicative mathematics|impredicative]]) | [[localization of a type|localization]] at the impredicative [[Dedekind real numbers]] or [[generalized Cauchy real numbers]] $\mathbb{R}$ | A [[type of all propositions]] $\Omega$ |
| $\mathbb{R}_{U} \flat$ | locally $U$-small real cohesion | [[localization of a type|localization]] at the $U$-[[Dedekind real numbers]] or $U$-[[generalized Cauchy real numbers]] $\mathbb{R}_U$ | A [[Tarski universe]] $(U, T)$ |
|  | [[unit interval]] cohesion | [[localization of a type|localization]] at the [[unit interval]] $[0, 1]$ | a higher [[coinductive type]] representing the [[homotopy]] [[terminal]] [[dyadic interval coalgebra]]. |
|  | $U$-small unit interval cohesion | [[localization of a type|localization]] at the $U$-small [[unit interval]] $[0, 1]_\mathbb{U}$ | A [[Tarski universe]] $(U, T)$. |
|  | smooth cohesion | [[localization of a type|localization]] at a [[local Artin algebra|local Artin $\mathbb{R}$-algebra]] | A [[local Artin algebra|local Artin $\mathbb{R}$-algebra]] $R$ |
| $S^1 \flat$ | [[circle type cohesion]] | [[localization of a type|localization]] at the circle type | The [[circle type]] $S^1$ |

## See also

* [[shape modality]]

* [[localization of a type at a family of functions]]

## References

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

* [[Mike Shulman]], *Homotopy type theory: the logic of space*, New Spaces in Mathematics: Formal and Conceptual Reflections, ed. Gabriel Catren and Mathieu Anel, Cambridge University Press, 2021 ([arXiv:1703.03007](https://arxiv.org/abs/1703.03007), [doi:10.1017/9781108854429](https://doi.org/10.1017/9781108854429))

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axiom of constructibility

An [[axiom]] in [[set theory]].

## References

* Wikipedia, _[Axiom of constructibility](http://en.wikipedia.org/wiki/Axiom_of_constructibility)_


axiom of determinacy

This [[axiom]] in [[set theory]] has important role implications in [[descriptive set theory]] (since regularity properties of certain classes of subsets of [[Polish space]]s are implied by corresponding statements of determinacy). There are well-established relationships between the variants of the axiom and reflection principles/[[large cardinal]] axioms.

## Definition

+-- {: .num_defn }
###### Definition

A [[game]] is called **determined** iff some player has a winning strategy. Typically in set theory Gale-Stewart games are considered, i. e. two players pick natural numbers (which might be required to be at most some $X$). The winning condition is then given as a subset $A$ of $X^\omega$ (for $X\le\omega$) and player 1 is considered to win iff the path chosen during the game is an element of $A$.

The [[set]] providing the winning condition is also called determined in the case that the game is determined.
=--

## Statement

The axiom of determinacy (AD) states that every Gale-Stewart game is determined. In ZF this axiom contradicts the axiom of choice.

## Restrictions and relations to reflection principles
Determinacy for [[Borel set]]s can be proven in ZFC, however it cannot be proven in Zermelo set theory (without the [[axiom of replacement]] which is equivalent to [[Levy-Montague reflection]]).

The existence of infinitely many [[Woodin cardinal]]s implies the axiom of projective determinacy (PD) stating that all [[projective set]]s are determined.

The existence of infinitely many Woodin cardinals is equiconsistent to AD.

## Reference

* Wikipedia, _[Axiom of determinacy](http://en.wikipedia.org/wiki/Axiom_of_determinacy)_

[[!redirects projective determinacy]]


axiom of extensionality

# The axiom of extensionality
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## Idea

In material [[set theory]], the _axiom of extensionality_ says that the global membership relation $\in$ is an [[extensional relation]] on the class of all [[pure sets]].

Since any relation becomes extensional on its [[extensional quotient]], one can interpret this axiom as a definition of [[equality]].  However, because the extensional quotient map need not reflect the relation, there is still content to the axiom: if two sets would be identified in the extensional quotient, then they must be members of the same sets and have the same sets as members.

If one models [[pure sets]] in structural [[set theory]], then this property may be made to hold by construction.

## Statements

### Weak extensionality

#### In unsorted set theories

In any [[unsorted set theory]], the **axiom of [[weak extensionality]]** states that given a set $A$ and a set $B$, $A = B$ if and only if for all $C$, $C \in A$ if and only if $C \in B$.

If the set theory does not have [[equality]] as a primitive, we could define equality as the predicate

$$A = B \coloneqq \forall C.(C \in A) \iff (C \in B)$$

The axiom of weak extensionality is a foundational axiom in most [[material set theories]], such as Zermelo set theory and Mac Lane set theory, both which do not have the [[axiom of foundation]], as well as [[ZFC]] which does have the [[axiom of foundation]].

In [[fully formal ETCS]], where the basic objects of the theory are functions, $0$ represents both the [[empty set]] and the [[identity function]] on the empty set, and $1$ represents the [[singleton]], the [[identity function]] of the singleton, and the sole [[element]] of the singleton all at the same time, there are three possible notions of sets:

* as [[identity functions]] $\mathrm{set}(a) \coloneqq \mathrm{dom}(a) = a$ or $\mathrm{set}(a) \coloneqq \mathrm{codom}(a) = a$

* as [[functions]] into the [[singleton]] $\mathrm{set}(a) \coloneqq (\mathrm{codom}(a) = 1)$

* as functions from the [[empty set]] $\mathrm{set}(a) \coloneqq (\mathrm{dom}(a) = 0)$

The elements are functions with domain $1$, $\mathrm{element}(a) \coloneqq (\mathrm{dom}(a) = 1)$

When sets are defined as functions into the singleton, the membership relation $a \in b$ is defined by the function $a$ being an element, the function $b$ being a set, and the codomain of $a$ being $b$
$$a \in b \coloneqq \mathrm{set}(a) \wedge \mathrm{element}(b) \wedge \mathrm{codom}(a) = b$$

Weak extensionality is a theorem in this case. By the universal property of the singleton, any two sets $A$ and $B$ with the same domain are equal to each other, which means that any proposition $P$ between $A$ and $B$ implies that $A = B$, and thus that $\forall C.(C \in A) \iff (C \in B)$ implies that $A = B$. The converse follows from [[indiscernibility of identicals]].

#### In two-sorted set theories

In any [[two-sorted set theory]], the **axiom of [[weak extensionality]]** states that given a set $A:Set$ and a set $B:Set$, $A =_{Set} B$ if and only if for all $a:Element$, $a \in A$ if and only if $a \in B$.

If the set theory does not have [[equality]] of sets as a primitive, we could define equality of sets as the predicate

$$A =_{Set} B \coloneqq \forall a:Element.(a \in A) \iff (a \in B)$$

### Material strong extensionality

Let $\sim$ be a [[bisimulation]], a binary relation such that for all sets $A$ and $B$ such that $A \sim B$, the following conditions hold:

* for all sets $C$ such that $C \in A$, there exists a set $D$ such that $D \in B$ and $C \sim D$
* for all sets $D$ such that $D \in B$, there exists a set $C$ such that $C \in A$ and $C \sim D$

The **axiom of strong extensionality** states that for every bisimulation $\sim$ and for every set $A$ and $B$, $A \sim B$ implies that $A = B$.

If the set theory does not have [[equality]] as a primitive, we could define equality as the [[terminal]] [[bisimulation]], as the bisimulation $=$ such that for every bisimulation $\sim$ and for every set $A$ and $B$, $A \sim B$ implies that $A = B$.

In any set theory with the [[axiom of foundation]], the axiom of weak extensionality implies the axiom of strong extensionality.

### Structural strong extensionality

In any [[structural set theory]], the axiom of strong extensionality states that for all sets $A$ and $B$ with an [[injection]] $i:A \hookrightarrow B$, the two definitions of $i$ being a [[bijection]] are logically equivalent to each other:

* there exists a function $i^{-1}:B \to A$ such that for all elements $a \in A$ and $b \in B$, $i^{-1}(i(a)) = a$ and $i(i^{-1}(b)) = b$
* for every element $x \in B$ there exists an element $y \in A$ such that $i(y) = x$.

Similar to how in material set theory one can use the axiom of extensionality to define [[equality]] of sets, in structural set theory one can use the axiom of strong extensionality to define [[bijection]] of sets.

## In type theory

In [[dependent type theory]], the *[[axiom of extensionality]]* is a property of [[power sets]], and states that given a [[type]] $A$ and [[subtypes]] $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, there is an [[equivalence of types]] between the [[identity type]] $B =_{\mathcal{P}(A)} C$ and the [[dependent function type]] $\prod_{x:A} (x \in_A B) \simeq (x \in_A C)$, where $(\Omega, \mathrm{El}_\Omega)$ is the [[type of all propositions]] and $x \in_A B \coloneqq \mathrm{El}_\Omega(B(x))$ is the local membership relation between elements $x:A$ and subtypes $B:\mathcal{P}(A)$. The axiom of extensionality holds in the [[dependent type theory]] if and only if [[function extensionality]] holds.

## See also

* [[extensionality]]
* [[identity of indiscernibles]]

## References

* [[Michael Shulman]], _Comparing material and structural set theories_ ([arXiv:1808.05204](https://arxiv.org/abs/1808.05204))

* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))

category: foundational axiom

[[!redirects Axiom of extensionality]]

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axiom of foundation

# The axiom of foundation
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In material [[set theory]], the __axiom of foundation__, also called the __axiom of regularity__, states that the membership relation $\in$ on the proper class of all [[pure set]]s is [[well-founded relation|well-founded]].  In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.

## Statement

Given a proper class $A$ of [[pure set]]s, suppose that $A$ has the property that, given any pure set $x$,
$$ \forall t,\; t \in x \Rightarrow t \in A ,$$
then $x \in A$.  Such an $A$ may be called a _membership-inductive_ class.  Then the __axiom of foundation__ states that the only membership-inductive class of pure sets is the class of all pure sets.  In this form, the axiom of foundation is also called *$\in$-induction*.

Although the statement here refers to proper classes, it can also be formulated as an axiom schema that makes no mention of classes.

## Alternative formulations

While the statement above follows how the axiom of foundation is generally *used* ---to prove properties of pure sets by [[transfinite induction]]---, it is complicated.  Two alternative formulations are given by the following lemmas:

+-- {: .num_lemma #InfiniteDescent}
###### Lemma
The axiom of foundation holds if and only if there exists no infinite descending [[sequence]] $\cdots \in x_2 \in x_1 \in x_0$.
=--
+-- {: .proof}
###### Proof
First, suppose the axiom of foundation holds, and suppose there were such a sequence.  Let $A$ be the class of all sets which are not equal to $x_i$ for any $i$.  Then for any $x$, if $x$ were equal to some $x_i$, then some $t\in x$ would also be equal to some $x_j$, namely $x_{i+1}$; hence if all $t\in x$ are in $A$, so is $x$.  Thus, by the axiom of foundation, all sets are in $A$, a contradiction.

Second, suppose there are no such sequences, and let $A$ be as in the statement of foundation.  Suppose that $A$ does not contain all sets.  Then there exists an $x_0\notin A$.  By hypothesis on $A$, there must exist an $x_1\in x_0$ such that $x_1 \notin A$.  And hence an $x_2\in x_1$ such that $x_2 \notin A$, and so on, producing an infinite descending $\in$-sequence in contradiction to our hypothesis; hence $A$ must contain all sets.
=--

This is essentially a version of Fermat\'s method of [[infinite descent]] modified to apply to [[transfinite induction]] instead of only to ordinary [[induction]].  It arguably provides the most direct intuitive picture of what the axiom means.  Note that as a special case, it implies that no set can contain itself, since then $\cdots \in x\in x\in x$ would be an infinite chain.

+-- {: .num_lemma #MembershipMinimal}
###### Lemma
The axiom of foundation holds if and only if every [[inhabited set|inhabited]] pure set $x$ has a member $y \in x$ such that no $t \in x$ satisfies $t \in y$.  (Such a $y$ is called a _membership-minimal_ element of $x$.)
=--
+-- {: .proof}
###### Proof
First, suppose the axiom of foundation holds, let $x$ be a pure set without a membership-minimal element, and let $A = \{ y | y\notin x \}$.  If $z$ is a set such that all $t\in z$ are in $A$, then we cannot have $z\in x$, since then we would have some $t\in z$ with $t\in x$ and hence $t\notin A$; hence $z\in A$.  So $A$ satisfies the assumptions of the foundation axiom, and hence $x$ is empty.

Second, suppose that every inhabited pure set has a membership-minimal element, and let $A$ be as in the statement of foundation.  Since every pure set is contained in a [[transitive set]], it suffices to show that $A \cap x = x$ for any transitive $x$.  Let $y = \{ z\in x \mid z \notin A \}$.  If $y$ is inhabited, then it contains a membership-minimal element $z$, i.e. we have $z\in x$, $z\notin A$, and for every $t\in z$ we have $t\notin z$---but $t\in x$ since $x$ is transitive, hence $t\in A$.  Thus this $z$ contradicts the assumption on $A$, so $y$ must be empty, as desired.
=--

This version is favoured by [[classical mathematics|classical]] set theorists as a statement of the axiom, since it uses neither higher-order reasoning (as our first definition does) or infinity (as infinite descent does).

However, neither of these is acceptable in [[constructive mathematics]], since both lemmas require at least the principle of [[excluded middle]] to prove at least one direction.    In particular, the nonexistence of infinite descending sequences is too weak to allow proofs by transfinite induction (except for special forms of $A$), while the requirement that every inhabited pure set have a minimal element is unnecessarily strong and itself implies excluded middle.

More precisely, the necessary set-theoretic axioms for the above proofs are the following.  (Is it known, in any case, that proofs using fewer axioms don't exist?)

* The "only if" direction of Lemma \ref{InfiniteDescent} requires only the axiom of infinity (for "infinite sequence" to make sense).

* The "if" direction of Lemma \ref{InfiniteDescent} evidently requires the principle of excluded middle, the axiom of infinity, and the axiom of [[dependent choice]].  It also appears to require the [[axiom of collection]] (since dependent choice, as usually stated, only chooses elements from a sequence of nonempty *sets*, rather than nonempty classes), and a principle of [[induction]] strong enough to recursively construct functions into a [[proper class]] (which is usually proven using the [[axiom of separation]]) in order to put together these nonempty sets into a sequence we can apply dependent choice to.

* The "only if" direction of Lemma \ref{MembershipMinimal} requires only excluded middle.

* The "if" direction of Lemma \ref{MembershipMinimal} requires excluded middle, also that every set is contained in a transitive one (the [[axiom of transitive closure]], which follows from [[axiom of replacement|replacement]]), as well as the [[axiom of separation]] in the form "the intersection of any class with a set is a set."

Another version of the axiom of foundation which *is* intuitionistically acceptable, but makes no reference to proper classes is:

+-- {: .num_lemma #WFTransitive}
###### Lemma
The axiom of foundation holds if and only if the relation $\in$ on any [[transitive set]] is a [[well-founded relation]].
=--
+-- {: .proof}
###### Proof
Suppose the axiom of foundation holds, let $x$ be a transitive set, and let $S\subseteq x$ be such that for any $y\in x$, if all $t\in y$ are in $S$, then $y\in S$.  Let $A$ be the class of all sets $y$ such that if $y\in x$, then $y\in S$; then $A$ satisfies the conditions in the axiom of foundation, so it contains all sets, and hence $S=x$.

Now suppose $\in$ is well-founded on any transitive set and let $A$ satisfy the conditions in foundation.  Since every set is contained in a transitive one, it suffices to show that $A\cap x = x$ for any transitive $x$, but this follows directly from the assumption.
=--

In this case:

* The "only if" direction requires no notable axioms at all, while

* The "if" direction requires [[transitive closure]], as in Lemma \ref{MembershipMinimal}, and also the axiom of separation.

Finally, another commonly cited version of foundation, equivalent to it at least over the other axioms of [[ZF]], is:

+-- {: .num_lemma #CumulativeHierarchy}
###### Lemma
The axiom of foundation holds if and only if every pure set is an element of $V_\alpha$ for some [[ordinal]] $\alpha$.
=--

Here the $V_\alpha$ are the [[cumulative hierarchy]] defined by [[transfinite recursion]] as $V_\alpha = P(\bigcup_{\beta\lt \alpha} V_\beta)$.

## Anti-foundation

Most of set theory works without the axiom of foundation, but not the deep study of well-founded pure sets.  However, one might want to do material set theory without assuming that all sets are well-founded, then one would not assume this axiom.

Alternatively, one can adopt the __axiom of anti-foundation__, which says:

* Given any [[extensional relation|extensional]] [[binary relation]] $\prec$ on any [[set]] $S$, there exists a unique [[transitive set]] $U$ such that $(U,\in)$ is [[isomorphism|isomorphic]] (necessarily uniquely) to $(S,\prec)$.

Since any relation has an [[extensional quotient]], we may also phrase the axiom thus:

* Given any binary relation $\prec$ on any [[set]] $S$, there exists a unique [[transitive set]] $U$ and [[surjection]] $f : S \to U$ such that $f(s_1) \in f(s_2)$ if and only if $s_1 \in s_2$, for $s_1, s_2$ in $S$.  (That is, $f$ is almost an isomorphism between $(S, \prec)$ and $(U, \in)$, but needn't be [[injection|injective]].)

Just as there are several versions of an [[extensional relation]], there are several versions of this axiom.  Note that the existence part of the statement is a set-formation axiom, while the uniqueness part is a strong version of the [[axiom of extensionality]] (which is equivalent to the usual one for well-founded sets).

If you include the hypothesis that $\prec$ be [[well-founded relation|well-founded]], then the statement is a theorem ([[Mostowski's collapsing lemma]]), while the converse is the axiom of foundation.

If you adopt the axiom of anti-foundation (with the strongest notion of extensional relation) instead of foundation, then the universe of [[pure sets]] becomes the [[corecursion|corecursively]] defined ill-founded sets instead of the [[recursion|recursively]] defined well-founded sets.

## Structural meaning

Since the axiom of foundation is about pure sets, there seems little point to it in a [[structural set theory]].  However, it does have a structural consequence: every set $S$ is the underlying set of elements of a well-founded model for a pure set (in any of the ways described at [[pure set]]).  If one assumes the [[axiom of choice]], however, then this statement follows from the [[well-ordering theorem]], since in that case $S$ is the underlying set of a model for a von Neumann [[ordinal number]].  But the axiom of foundation has no stronger structural consequence, since this statement already suffices to ensure that a model of structural set theory can be reconstructed from the material set theory consisting of its well-founded pure sets.

That this statement is the correct structural version of antifoundation may be justified by appeal to the [[material-structural adjunction]].

## See also

* [[Mostowski's principle]]

category: foundational axiom

[[!redirects axiom of foundation]]

[[!redirects axiom of anti-foundation]]
[[!redirects axiom of antifoundation]]
[[!redirects anti-foundation]]
[[!redirects antifoundation]]

[[!redirects axiom of regularity]]

[[!redirects membership induction]]
[[!redirects membership-induction]]


axiom of full comprehension

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## Definition

In an [[unsorted set theory|unsorted]] [[material set theory]], the [[axiom]] or rule of **full** or **unrestricted comprehension** says that for any property $P$, there exists an [[set]] $\{ x \mid P(x) \}$ of all objects satisfying $P$.

## Resolving the inconsistency

An [[unsorted set theory]] with the unrestricted [[comprehension]] rule is called [[naive set theory]], and is inconsistent due to [[Russell's paradox]] and [[Curry's paradox]]. Here we mention several approaches to this issue.

### Separating the sets and elements

One way to resolve the inconsistency is to require the set theory to have multiple [[sorts]], one for the [[elements]] and one for the [[sets]], and not allow any reflection rules which send sets to elements or elements to sets. This means that the membership relation is only between elements and sets, and there is no notion of $x \in x$ necessary to express [[Russell's paradox]] and [[Curry's paradox]]. This is the approach taken by [[structural set theories]].

### Restricted comprehension

Standard [[unsorted set theories]] such as [[ZFC]] avoid this [[paradox]] by replacing unrestricted comprehension with the [[axiom scheme of separation]] (or "restricted comprehension"), which restricts $x$ to lie in some previously specified set $X$.

### Stratified comprehension

Set theories such as [[New Foundations]] instead replace comprehension by a rule of "stratified comprehension".  This permits a "[[set of all sets]]" but still appears to avoid paradox.

### Substructural logics

It is also possible to retain full comprehension but avoid paradox by modifying the ambient logic.  Passing to [[constructive logic]] doesn't help, and indeed the root issue has nothing to do with [[negation]] as such, since [[Curry's paradox]] can be stated without any [[negation]].  One might think that [[paraconsistent logic]] would help, but many paraconsistent logics are still vulnerable to Curry's paradox.  Perhaps the most obvious culprit is the [[contraction rule]], and indeed [[linear logic]] (including some paraconsistent logics) can admit a full comprehension rule without explosion.

### Normal logics

Another possibility is to keep the contraction rule but restrict the use of the [[cut rule]].  It is not necessary to forbid all uses of cut, since many cuts can be normalized or eliminated.  Indeed, in ordinary consistent logic, *all* cuts can be eliminated; but in the presence of full comprehension they cannot all be.  Thus, another way to avoid paradox with full comprehension is to permit only proofs that can be [[normal form|normalized]].

Note that unlike a restriction on contraction, this is a "global" restriction: the proofs of two lemmas can independently be valid, but their combination may no longer be so.  Similar "global" restrictions on logic were investigated by [Fitch 1953](#Fitch52), [69](#Fitch69).

## References

### In linear logic

In [[linear logic]]:

* Grishin, V. N., "Predicate and set theoretic calculi based on logic without contraction rules" (Russian), _Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya_, 45(1): 47 – 68, 1981. English translation in Math. USSR Izv., 18(1): 41 – 59, 1982. ([math-net.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1547&option_lang=eng))

* [[Jean-Yves Girard]], Light Linear Logic, _Information and Computation_, 14(3):123-137, 2003. ([pdf.gz](http://iml.univ-mrs.fr/~girard/LLL.pdf.gz))

* [[Kazushige Terui]], Light Affine Set Theory: A Naive Set Theory of Polynomial Time, _Studia Logica: An International Journal for Symbolic Logic_, Vol. 77, No. 1 (Jun., 2004), pp. 9-40. ([jstor](http://www.jstor.org/stable/20016605)) ([pdf](http://www.kurims.kyoto-u.ac.jp/~terui/lastfin.pdf)).  See also Terui's slides, [Linear Logic and Naive Set Theory (Make our garden grow)](http://www.kurims.kyoto-u.ac.jp/~terui/summer3.pdf)

### Global restrictions

Several global restrictions were considered in

* {#Fitch52} [[Frederic Fitch]], *Symbolic Logic: An introduction*, Ronald Press, New York 1952

* {#Fitch69} [[Frederic Fitch]], *A method for avoiding the Curry paradox*, in *Essays in Honor of Carl G. Hempel*, Reidel, Dordrecht, Holland 1969, pp. 255--265 ([doi:10.1007/978-94-017-1466-2](https://link.springer.com/book/10.1007/978-94-017-1466-2))

The notation therein is somewhat difficult to follow for a modern reader, especially due to the somewhat confused treatment of what nowadays would be called free and bound variables.  A more modern explanation of Fitch's restrictions can be found in:

* Susan Rogerson, _Natural deduction and Curry's paradox_, Journal of Philosophical Logic (2007) 36: 155--179. [pdf](https://link.springer.com/content/pdf/10.1007/s10992-006-9032-0.pdf)

The normalizability restriction is also discussed philosophically in

* Tennant, N.: *Proof and paradox*, Dialectica 36 (1982), 265-296

and other references (someone add!).

category: foundational axiom

[[!redirects full comprehension]]
[[!redirects full comprehension rule]]
[[!redirects axiom of full comprehension]]
[[!redirects axioms of full comprehension]]
[[!redirects axiom scheme of full comprehension]]
[[!redirects axiom schemes of full comprehension]]
[[!redirects axiom schema of full comprehension]]
[[!redirects axiom schemas of full comprehension]]
[[!redirects axiom schemata of full comprehension]]

[[!redirects unrestricted comprehension]]
[[!redirects unrestricted comprehension rule]]
[[!redirects axiom of unrestricted comprehension]]
[[!redirects axioms of unrestricted comprehension]]
[[!redirects axiom scheme of unrestricted comprehension]]
[[!redirects axiom schemes of unrestricted comprehension]]
[[!redirects axiom schema of unrestricted comprehension]]
[[!redirects axiom schemas of unrestricted comprehension]]
[[!redirects axiom schemata of unrestricted comprehension]]


axiom of infinity

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## Idea

In the [[foundations]] of [[mathematics]], an [[axiom of infinity]] is any axiom that asserts that [[infinite set]]s exist. In [[set theory]] and [[set-level type theory]], infinite sets cannot be constructed from finite sets, so their existence must be posited as an extra axiom. Further axioms, in this vein which assert the existence of even larger sets that cannot be constructed from smaller ones are called [[large cardinal]] axioms.

## Statement

### Natural numbers

One common form of the axiom of infinity says that the particular set $N$ of [[natural number]]s exists.  In material [[set theory]] this often takes the form of asserting that the von Neumann [[ordinal number]] $\omega$ exists, where $\omega$ is characterized as the smallest set such that $\emptyset\in\omega$ and whenever $a\in \omega$ then $a\cup \{a\}\in \omega$.  In structural set theory the usual form of the axiom of infinity is the existence of a [[natural numbers object]].

In [[dependent type theory]], the axiom of infinity for a [[Tarski universe]] is given by the element
$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:T(\mathbb{N}) \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:T(\mathbb{N})} T(C(x)) \to T(C(s(x)))} \sum_{c:\prod_{x:T(\mathbb{N})} T(C(x))} (c(0) =_{T(C(0))} c_0) \times \prod_{x:T(\mathbb{N})} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$
or
$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:U} \prod_{c_0:T(C)} \prod_{c_s:T(C) \to T(C)} \exists!c:T(\mathbb{N}) \to T(C).(f(0) =_{T(C)} c_0) \times \prod_{n:T(\mathbb{N})} c(s(n)) =_{T(C)} c_s(c(n))$$
which states that there is a [[natural numbers type]] in the universe.

There is an alternate way to express the axiom of infinity in a Tarski universe, as the axiom of resizing the [[set truncation]] of the [[type of finite types]] in $U$, since $\mathrm{isFinite}$ and set truncations are definable from the [[type of propositions]] in $U$, $\sum_{A:U} \mathrm{isProp}(A)$, but they are all usually large, and so have to be resized to be small:

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} T(\mathbb{N}) \simeq \left[\sum_{A:U} \mathrm{isFinite}(T(A))\right]_0$$

### Integers

#### Inductive definition

Instead of a defining the natural numbers via its induction principle, one can instead define the [[integers]] via its induction principle, and then use the fact that disjoint unions are disjoint and $\mathbb{Z} \cong \mathbb{Z} \uplus \mathbb{Z}$ to construct the natural numbers.

#### Second-order definition

Alternatively, one can assume a (trichotomous) [[ordered integral domain]] $\mathbb{Z}$, such that every (trichotomous) ordered integral subdomain of $\mathbb{Z}$ is equivalent to the [[improper subset]] of $\mathbb{Z}$. This defines the [[integers]], since the integers are the [[initial object|initial]] (trichotomous) ordered integral domain and are [[strict initial object|strictly initial]]. Since the integers as defined automatically comes with a [[total order]] $\leq$ and a [[pseudo-order]] $\lt$, one can define the natural numbers as the set of non-negative integers.

The benefit of such a definition of infinity is that it allows for a [[strongly predicative mathematics|strongly predicative]] definition of the [[natural numbers]], the [[rational numbers]], and the [[real numbers]] in [[dependent type theory]], since one doesn't need arbitrary [[function sets]] or [[dependent function types]] to define the natural numbers, which one needs to characterize the natural numbers by its usual recursion or induction principle. Instead, one only needs dependent function types of a family of propositions, weak function extensionality, and the power set of the ordered integral domain $\mathbb{Z}$. Unlike the case for arbitrary ordered integral domains, being an ordered integral subdomain is only a property of an element of a power set, and then one can construct the type of ordered integral subdomains of $\mathbb{Z}$, from which one can characterize $\mathbb{Z}$ as having a contractible type of ordered integral subdomains of $\mathbb{Z}$.

### Rational numbers

If one has [[power sets]], one can assume an ([[trichotomous]]) [[ordered field]] $\mathbb{Q}$, such that every (trichotomous) ordered [[subfield]] of $\mathbb{Q}$ is equal to the [[improper subset]] of $\mathbb{Q}$. This defines the [[rational numbers]], since the rational numbers are the initial (trichotomous) [[ordered field]] and are [[strict initial object|strictly initial]]. The rational numbers are automatically infinite, and one can construct the [[integers]] $\mathbb{Z}$ as the [[intersection]] of all [[ordered integral domain|ordered integral subdomains]] of $\mathbb{Q}$, and since the integers as defined automatically comes with a [[total order]] $\leq$ and a [[pseudo-order]] $\lt$, one can define the natural numbers as the set of non-negative integers.

## Generalizations

In the form of an NNO, the axiom of infinity generalises to the existence of [[inductive type]]s or [[W-type]]s.  These can be constructed from a NNO if [[power set]]s exist, but in [[predicative mathematics|predicative]] theories they can be added as additional axioms.

One could also posit the existence of the set of [[extended natural numbers]] instead of the set of natural numbers, as the set of extended natural numbers have [[countable|countably infinite]] cardinality and is the categorical [[duality|dual]] of the natural numbers in Set, a [[terminal coalgebra]] for the endofunctor $F(X) = 1 + X$ in Set. This generalises to the existence of [[coinductive types]] or [[M-types]], which can be added as additional axioms.

One could also posit the existence of [[FinSet]], the collection of [[finite sets]]. In dependent type theory this is a [[type of finite types]], a [[universe]] $\mathcal{U}$ that satisfies the axiom of finiteness (see below).

## Alternatives

Broadly speaking, [[finite mathematics]] is mathematics that does not use or need the axiom of infinity; a finitist is an extreme breed of [[constructive mathematics|constructivist]] that believes that mathematics is better without the axiom of infinity, or even that this axiom is false.

A more extreme case is to *deny* the axiom of infinity with an __axiom of finiteness__: every set is [[finite set|finite]].  There is one of these for every definition of 'finite' given on that page; here is the strongest stated directly in terms of [[set theory]] as an axiom of [[induction]]:

* Any property of sets that is invariant under [[isomorphism]] and holds for the [[empty set]] must hold for all sets if, whenever it holds for a set $X$, it holds for the [[disjoint union]] $X \uplus \{*\}$.

In [[material set theory]], this is equivalent given the [[axiom of foundation]] (which guarantees that $X$ and $\{X\}$ are [[disjoint sets|disjoint]]):

* Any property of sets that holds for the empty set must hold for all sets if, whenever it holds for a set $X$, it holds for the [[union]] $X \cup \{X\}$.

In higher categorical terms, the above axiom of finiteness could be stated as follows: [[Set]] is an initial algebra of the 2-endofunctor $F(X) \cong X \coprod 1$ in the [[(2,1)-category]] [[Grpd]].

In [[dependent type theory]], given a [[Tarski universe]] $(U, T)$ that is closed under the [[empty type]], the [[unit type]], and [[sum types]], the axiom of finiteness for the universe states that

* For all type families $A:U \vdash C(A)$ such that $T(A) \simeq T(B)$ implies that $C(A) \simeq C(B)$, elements $c_0:C(\mathbb{0})$ and dependent functions $c_s:\prod_{A:U} C(A) \to C(A + \mathbb{1})$, there exists a unique dependent function $c:\prod_{A:U} C(A)$ such that $c(\mathbb{0}) =_{C(\mathbb{0})} c_0$ and for all $A:U$, $c(A + 1) =_{C(A + 1)} c_s(c(A))$.

In [[dependent type theory]] with [[dependent product types]], [[dependent sum types]], [[identity types]], [[function extensionality]], and a [[type of all propositions]], the *axiom of finiteness* for the entire type theory is an [[axiom schema]] which states that given a type $A$, one could derive a witness that the type is a [[finite type]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{finWitn}_A:\mathrm{isFinite}(A)}$$

where

$$
\mathrm{isFinite}(A) \equiv
\begin{array}{c}
\prod_{S:(A \to \mathrm{Prop}) \to \mathrm{Prop}} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}} \prod_{Q:A \to \mathrm{Prop}} (P \in S) \\
\times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S)
\end{array}
$$

The membership relation and the subtype operations used above are defined in the nLab article on [[subtypes]].

In particular, the axiom of finiteness for the entire type theory implies the [[principle of excluded middle]] for the [[type of all propositions]], since the only finite propositions are the [[decidable propositions]]. Furthermore, the axiom of finiteness implies that the type theory is a [[set-level type theory]] because every finite type is an [[h-set]].

## Related concepts

* [[finite set]], [[finite object]]

* [[Set]], [[FinSet]]

## References

In relation to [[classifying toposes]]:

* [[Andreas Blass]], *Classifying topoi and the axiom of infinity*, Algebra Universalis **26** (1989) 341-345 [[doi:10.1007/BF01211840](https://doi.org/10.1007/BF01211840)]

For constructing the [[natural numbers]] from the [[integers]]:

* {#Sattler23} [[Christian Sattler]], *Natural numbers from integers* ([pdf](https://www.cse.chalmers.se/~sattler/docs/naturals.pdf))

category: foundational axiom

[[!redirects axiom of infinity]]
[[!redirects axioms of infinity]]

[[!redirects axiom of finiteness]]
[[!redirects axioms of finiteness]]
[[!redirects axiom of finity]]
[[!redirects axioms of finity]]


axiom of materialization

# Axioms of materialization

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## Idea

An **axiom of materialization** states that every [[set]] is [[isomorphic]] to a [[pure set]] of a certain form.  They can be formulated either in [[structural set theory]] or [[material set theory]], and in reasonably strong set theories they are often provable.

## Versions

* The **axiom of well-founded materialization** says that every set is isomorphic to a well-founded pure set, or equivalently (at least in a strong enough set theory) that every set can be embedded into a well-founded extensional graph.  In material set theory this is also known as **Coret's Axiom B** ([Coret 64](#Coret64)).

* The **axiom of ill-founded materialization** says that every set can be embedded in an [[extensional relation|strongly extensional]] graph.

Since well-founded extensional graphs are strongly extensional, the well-founded axiom implies the ill-founded one.

## Provability

* The axiom of well-founded materialization follows from the [[axiom of choice]]: every [[well-ordered set]] is a well-founded extensional graph, so if every set is well-orderable then the axiom follows.

* In material set theory, the axiom of well-founded materialization follows from the  [[axiom of foundation]], since then every set *is already* a well-founded pure set.  Similarly, the axiom of ill-founded materialization follows from the [[axiom of anti-foundation]].

* In the presence of [[excluded middle]] and the [[axiom of replacement]], every strongly extensional graph can be embedded into a well-founded extensional graph.  Specifically, a graph is a coalgebra for the powerset functor, hence has a cone over the [[transfinite construction of free algebras|terminal coalgebra sequence]] for that endofunctor (which exists using the axiom of replacement, even though it does not converge to a terminal coalgebra).  The kernels of the maps in this cone are the approximations to the bisimilarity of the graph, and hence converge to it at some ordinal $\alpha$ (namely the [[Hartogs number]] of the lattice of binary relations; this uses excluded middle).  By strong extensionality, the bisimilarity is the identity, so the $\alpha^{\mathrm{th}}$ map in the cone is an injection into a well-founded set.

  Thus, given excluded middle and replacement, the axiom of ill-founded materialization implies the well-founded one, hence the two are equivalent.  In particular, if the axiom of foundation in ZF is replaced by the axiom of anti-foundation, then the axiom of *well-founded* materialization is also provable.

  It is unclear whether the two axioms of materialization are distinct in weaker theories lacking excluded middle or replacement, such as [[IZF]] or [[Zermelo set theory]] with the axiom of foundation removed.

## Consequences

Either axiom of materialization implies that the [[category of sets]] is [[equivalence of categories|equivalent]] to its subcategory of the relevant kind of pure sets.  Thus, for instance, [[Scott's trick]] can be used to prove isomorphism-invariant properties (see for instance [this MO question](https://math.stackexchange.com/questions/305859/scotts-trick-without-the-axiom-of-regularity)).

## See also

* [[Mostowski's principle]]
* [[Mostowski set theory]]

## References

* {#Coret64} J. Coret, *Formules stratifides et axiome de fondation*, Comptes Rendus hebdomadaires des seances de l'Academie des Sciences de Paris serie A, vol. 264 (1964)

* {#Shulman19} [[Mike Shulman]], *Comparing material and structural set theories*. Annals of Pure and Applied Logic 170(4), 2019, p465–504, [arxiv](https://arxiv.org/abs/1808.05204v2)

category: foundational axiom

[[!redirects axiom of well-founded materialization]]
[[!redirects axiom of ill-founded materialization]]
[[!redirects axioms of materialization]]
[[!redirects Coret's axiom B]]
[[!redirects Coret's Axiom B]]


axiom of multiple choice

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# The axiom of multiple choice
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This article is about an axiom of constructive mathematics.  Some set theory literature instead uses this name for an unrelated weakening of [[axiom of choice|AC]].  For that notion, see [[(classical) axiom of multiple choice]].

## Idea

The **axiom of multiple choice** (AMC) is a weaker version of the [[axiom of choice]], which can hold in [[constructive mathematics]].

## Statement

A set-indexed family $\{D_c\}_{c\in C}$ of sets is said to be a *collection family* if for any $c\in C$ and any [[surjection]] $E\twoheadrightarrow D_c$, there exists a $c'\in C$ and a surjection $D_{c'}\twoheadrightarrow D_c$ which factors through $E$.

Depending on the author, the *axiom of multiple choice* is one of the following statements:

1. for every set $X$, there exists a collection family $\{D_c\}_{c\in C}$ such that $X\cong D_c$ for some $c$ ([[Michael Rathjen]]'s formulation, attributed to [[Peter Aczel]] and [[Alex Simpson]]), or

1. for every set $X$, there exists a collection family $\{D_c\}_{c\in C}$, with $C$ inhabited, and a family of surjections $\{D_c \to X\}_{c\in C}$ (the formulation originally given by [[Ieke Moerdijk]] and [[Erik Palmgren]]), or

1. for every set $X$, the full subcategory $(Set/X)_{surj}$ of the slice category $Set/X$ consisting of the surjections has a weakly initial set (in [[Benno van den Berg]]'s formulation; this is also called [[WISC]]).

The nLab uses the initialization AMC to cover either the first two formulations.

The third is a weaker condition, and while some may refer to as a "weak axiom of multiple choice", van den Berg obviously does not; he calls his the AMC and the Moerdijk-Palmgren formulation rather the "strong axiom of multiple choice".

## Relationships to other axioms

* Note that $P$ is a [[projective set]] if and only if the singleton family $\{P\}$ is a collection family.  Therefore, since AC is equivalent to "all sets are projective," it implies AMC.

* An extension of this argument shows that [[COSHEP]] is sufficient to imply AMC.

* The [[Reflection Principle]] (RP) is equivalent to AMC (the one called strong AMC by van den Berg). RP is motivated by the [[regular extension axiom]] (REA) from constructive set theory. RP states that every map belongs to a representable class of small maps.

* However, AMC does not imply [[countable choice]] or any of the other usual consequences of AC.

* Rathjen proves that [[SVC]] also implies AMC.  It follows that AMC holds in "most" models of set theory.

* AMC implies [[WISC]], and therefore also implies that the category of [[anafunctors]] between two [[small categories]] is [[essentially small category|essentially small]]. Thus WISC may be termed "weak axiom of multiple choice".

* A [[ΠW-pretopos]] satisfying the (weak) axiom of multiple choice is a _[[predicative topos]]_, or removing the word "weak", we may speak of a strong predicative topos.

## References

* [[Ieke Moerdijk]], [[Erik Palmgren]], _Type theories, toposes and constructive set theory: predicative aspects of AST_ (2000) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.8934))

* Rathjen, "Choice principles in constructive and classical set theories"

In

* [[Benno van den Berg]], _Predicative toposes_ ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959))
 {#vdBerg}

[[WISC]] is called the "axiom of multiple choice".

* Jech, _The Axiom of Choice_ (1973), ISBN : 0444104844 (New York)

category: foundational axiom

[[!redirects axiom of multiple choice]]
[[!redirects multiple choice]]
[[!redirects AMC]]


axiom of pairing

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# The axiom of pairing
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## Idea

In [[material set theory]] as a [[foundation of mathematics]], the axiom of pairing is an important axiom needed to get the foundations off the ground (to mix metaphors).  It states that [[pair sets]] exist.

## Statement

### Pairing

The __axiom of pairing__ (or __axiom of pairs__) states the following:

**Axiom of pairing**: _If $x$ and $y$ are (material) sets, then there exists a set $P$ such that $x \in P$ and $y \in P$._

Using the axiom of separation ([[bounded separation]] is enough), we can prove the existence of a particular set $P$ such that $x$ and $y$ are the *only* members of $P$.  Using the [[axiom of extensionality]], we can then prove that this set $P$ is unique; it is usually denoted $\{x,y\}$ and called the __[[pair set]]__ of $x$ and $y$.  Note that $\{x,x\}$ may also be denoted simply $\{x\}$.

One could also assume that the [[material set theory]] has a primitive binary operation $P$ which takes of a material set $x$ and $y$ and returns a material set $P(x, y)$. Then the axiom of pairing becomes

**Axiom of pairing**: _If $x$ and $y$ are (material) sets, then $x \in P(x, y)$ and $y \in P(x, y)$._

### Unordered pairing

The __axiom of unordered pairing__ (or __axiom of unordered pairs__) states the following:

**Axiom of unordered pairing**: If $x$ and $y$ are (material) sets, then there exists a set $P$ such that $x \in P$ and $y \in P$ and for all sets $z$, $z \in P$ implies that $z = x$ or $z = y$.

Using the [[axiom of extensionality]], we can then prove that this set $P$ is unique; it is usually denoted $\{x,y\}$ and called the __[[pair set]]__ of $x$ and $y$.  Note that $\{x,x\}$ may also be denoted simply $\{x\}$.

One could also assume that the [[material set theory]] has a primitive binary operation $\{-,-\}$ which takes of a material set $x$ and $y$ and returns a material set $\{x, y\}$. Then the axiom of pairing becomes

**Axiom of unordered pairing**: If $x$ and $y$ are (material) sets, then $x \in \{x, y\}$, $y \in \{x, y\}$, and for all sets $z$, $z \in \{x, y\}$ implies that $z = x$ or $z = y$.

### Ordered pairing

Let us assume that the [[material set theory]] has a primitive binary operation $(-,-)$ which takes of a material set $x$ and $y$ and returns a material set $(x, y)$.

The __axiom of ordered pairing__ (or __axiom of ordered pairs__) states the following:

**Axiom of ordered pairing**: _If $x$ and $y$ are (material) sets, then $x \in (x, y)$, $y \in (x, y)$, and for all sets $a$ and $b$, $(a, b) = (x, y)$ if and only if $a = x$ and $b = y$._

$$\forall a.\forall b.\{a, b\} = \{x, y\} \iff (a = x \wedge b = y)$$

### With sets and elements different

In set theories where sets and elements are not the same thing, pairing becomes an operation on both the sets and the elements. One has to add a primitive ternary relation $p(X, Y, P)$ which says that $P$ is the [[Cartesian product]] of $X$ and $Y$, as well as primitive quaternary relations $\pi_1(X, P, c, a)$ and $\pi_2(Y, P, c, b)$ which says that element $a \in X$ is the left element of the pair $c \in P$ and element $b \in Y$ is the right element of the pair $c \in P$, and the following axiom:

**Axiom of ordered pairing**: _If $X$ and $Y$ are sets, then there exists a set $P$ such that $p(X, Y, P)$ and for every object $a$ and $b$, $a \in X$ and $b \in Y$ implies that there exists an object $c$ such that $c \in P$, $\pi_1(X, P, c, a)$, and $\pi_2(Y, P, c, b)$_

$P$ is usually denoted $X \times Y$ and called the __[[Cartesian product]]__ of $X$ and $Y$, while $c$ is usually denoted $(a, b)$ and called the __[[ordered pair]] of $a$ and $b$.

## Generalisation

The axiom of pairing is the binary part of a [[binary/nullary pair]] whose nullary part is the axiom stating the existence of the [[empty set]].  We can use these axioms and the [[axiom of union]] to prove every instance of the following __axiom__ (or rather theorem) __schema of finite sets__:

\begin{theorem}
If $x_1, \ldots, x_n$ are sets, then there exists a set $P$ such that $x_1, \ldots, x_n \in P$.
\end{theorem}

Again, we can prove the existence of specific $P$ such that $x_1, \ldots, x_n$ are the *only* members of $P$ and prove that this $P$ is unique; it is denoted $\{x_1, \ldots, x_n\}$ and is called the __[[finite set]]__ consisting of $x_1, \ldots, x_n$.

Note that this is a _schema_, with one instance for every (metalogical) [[natural number]].  Within axiomatic set theory, this is very different from the single statement that begins with a [[universal quantification]] over the (internal) set of natural numbers.  In particular, each instance of this schema can be stated and proved without the [[axiom of infinity]]. Of course, there is one proof for each natural number.

*  For $n = 0$, this is simply the axiom of the empty set.
*  For $n = 1$, we use the axiom of pairing with $x \coloneqq x_1$ and $y \coloneqq x_1$ to construct $\{x_1\}$.
*  For $n = 2$, we use the axiom of pairing with $x \coloneqq x_1$ and $y \coloneqq x_2$ to construct $\{x_1, x_2\}$.
*  For $n = 3$, we first use the axiom of pairing twice to construct $\{x_1, x_2\}$ and $\{x_3\}$, then use pairing again to construct $\big\{\{x_1, x_2\}, \{x_3\}\big\}$, then use the axiom of union to construct $\{x_1, x_2, x_3\}$.
*  In general, once we have $\{x_1, \ldots, x_{n-1}\}$, we use pairing to construct $\{x_n\}$, use pairing again to construct $\big\{\{x_1, \ldots, x_{n-1}\}, \{x_n\}\big\}$, then use the axiom of union to construct $\{x_1, \ldots, x_n\}$.  (A direct proof of a single statement for $n \gt 3$ can actually go faster than this; the length of the shortest proof is [[logarithmic]] in $n$ rather than linear in $n$.)

Note that these 'finite sets' are precisely the [[Kuratowski-finite sets]] in a [[constructive mathematics|constructive]] treatment.

## Related notions

* [[pairing structure]]

In the $n$Lab, the term '[[pairing]]' usually refers to *[[ordered pair|ordered]]* pairs.

## References

For the axiom of ordered pairing see:

* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))

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category: foundational axiom