Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets.  is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

What L is
 can be thought of as being built in "stages" resembling the construction of von Neumann universe, . The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes  to be the set of all subsets of the previous stage, . By contrast, in Gödel's constructible universe , one uses only those subsets of the previous stage that are:

definable by a formula in the formal language of set theory,
with parameters from the previous stage and,
with the quantifiers interpreted to range over the previous stage.

By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.

Define the Def operator:

 is defined by transfinite recursion as follows:
 
  
 If  is a limit ordinal, then  Here  means  precedes .
  Here Ord denotes the class of all ordinals.

If  is an element of , 
then . So  is a subset of , which is a subset of the power set of . Consequently, this is a tower of nested transitive sets. But  itself is a proper class.

The elements of  are called "constructible" sets; and  itself is the "constructible universe". The "axiom of constructibility", aka "", says that every set (of ) is constructible, i.e. in .

Additional facts about the sets 
An equivalent definition for  is:

For any finite ordinal , the sets  and  are the same (whether  equals  or not), and thus  = : their elements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC in which  equals ,  is a proper subset of , and thereafter  is a proper subset of the power set of  for all . On the other hand,  does imply that  equals  if , for example if  is inaccessible. More generally,  implies  =  for all infinite cardinals .

If  is an infinite ordinal then there is a bijection between  and , and the bijection is constructible. So these sets are equinumerous in any model of set theory that includes them.

As defined above,  is the set of subsets of  defined by  formulas (with respect to the Levy hierarchy, i.e., formulas of set theory containing only bounded quantifiers) that use as parameters only  and its elements.

Another definition, due to Gödel, characterizes each  as the intersection of the power set of  with the closure of  under a collection of nine explicit functions, similar to Gödel operations. This definition makes no reference to definability.

All arithmetical subsets of  and relations on  belong to  (because the arithmetic definition gives one in ). Conversely, any subset of  belonging to  is arithmetical (because elements of  can be coded by natural numbers in such a way that  is definable, i.e., arithmetic). On the other hand,  already contains certain non-arithmetical subsets of , such as the set of (natural numbers coding) true arithmetical statements (this can be defined from  so it is in ).

All hyperarithmetical subsets of  and relations on  belong to  (where  stands for the Church–Kleene ordinal), and conversely any subset of  that belongs to  is hyperarithmetical.

L is a standard inner model of ZFC 
 is a standard model, i.e.  is a transitive class and the interpretation uses the real element relationship, so it is well-founded.  is an inner model, i.e. it contains all the ordinal numbers of  and it has no "extra" sets beyond those in . However  might be strictly a subclass of .  is a model of ZFC, which means that it satisfies the following axioms:
 Axiom of regularity: Every non-empty set  contains some element  such that  and  are disjoint sets.
  is a substructure of , which is well founded, so  is well founded. In particular, if , then by the transitivity of , . If we use this same  as in , then it is still disjoint from  because we are using the same element relation and no new sets were added.
 Axiom of extensionality: Two sets are the same if they have the same elements.
 If  and  are in  and they have the same elements in , then by 's transitivity, they have the same elements (in ). So they are equal (in  and thus in ).
 Axiom of empty set: {} is a set.
 , which is in . So . Since the element relation is the same and no new elements were added, this is the empty set of .
 Axiom of pairing: If ,  are sets, then  is a set.
 If  and , then there is some ordinal  such that  and . Then . Thus  and it has the same meaning for  as for .
 Axiom of union: For any set  there is a set  whose elements are precisely the elements of the elements of .
 If , then its elements are in  and their elements are also in . So  is a subset of . Then . Thus .
 Axiom of infinity: There exists a set  such that  is in  and whenever  is in , so is the union .
 Transfinite induction can be used to show each ordinal  is in . In particular,  and thus .
 Axiom of separation: Given any set  and any proposition ,  is a set.
 By induction on subformulas of , one can show that there is an  such that  contains  and  and ( is true in  if and only if  is true in ), the latter is called the "reflection principle"). So  = . Thus the subset is in .
 Axiom of replacement: Given any set  and any mapping (formally defined as a proposition  where  and  implies ),  is a set.
 Let  be the formula that relativizes  to , i.e. all quantifiers in  are restricted to .  is a much more complex formula than , but it is still a finite formula, and since  was a mapping over ,  must be a mapping over ; thus we can apply replacement in  to . So  =  is a set in  and a subclass of . Again using the axiom of replacement in , we can show that there must be an  such that this set is a subset of . Then one can use the axiom of separation in  to finish showing that it is an element of 
 Axiom of power set: For any set  there exists a set , such that the elements of  are precisely the subsets of .
 In general, some subsets of a set in  will not be in  So the whole power set of a set in  will usually not be in . What we need here is to show that the intersection of the power set with  is in . Use replacement in  to show that there is an α such that the intersection is a subset of . Then the intersection is . Thus the required set is in .
 Axiom of choice: Given a set  of mutually disjoint nonempty sets, there is a set  (a choice set for ) containing exactly one element from each member of .
 One can show that there is a definable well-ordering of , in particular based on ordering all sets in  by their definitions and by the rank they appear at. So one chooses the least element of each member of  to form  using the axioms of union and separation in 

Notice that the proof that  is a model of ZFC only requires that  be a model of ZF, i.e. we do not assume that the axiom of choice holds in .

L is absolute and minimal 
If  is any standard model of ZF sharing the same ordinals as , then the  defined in  is the same as the  defined in . In particular,  is the same in  and , for any ordinal . And the same formulas and parameters in  produce the same constructible sets in .

Furthermore, since  is a subclass of  and, similarly,  is a subclass of ,  is the smallest class containing all the ordinals that is a standard model of ZF. Indeed,  is the intersection of all such classes.

If there is a set  in  that is a standard model of ZF, and the ordinal  is the set of ordinals that occur in , then  is the  of . If there is a set that is a standard model of ZF, then the smallest such set is such a . This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.

Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.

Because both " constructed within " and " constructed within " result in the real , and both the  of  and the  of  are the real , we get that  is true in  and in any  that is a model of ZF. However,  does not hold in any other standard model of ZF.

L and large cardinals 
Since , properties of ordinals that depend on the absence of a function or other structure (i.e.  formulas) are preserved when going down from  to . Hence initial ordinals of cardinals remain initial in . Regular ordinals remain regular in . Weak limit cardinals become strong limit cardinals in  because the generalized continuum hypothesis holds in . Weakly inaccessible cardinals become strongly inaccessible. Weakly Mahlo cardinals become strongly Mahlo. And more generally, any large cardinal property weaker than 0 (see the list of large cardinal properties) will be retained in .

However,  is false in  even if true in . So all the large cardinals whose existence implies  cease to have those large cardinal properties, but retain the properties weaker than  which they also possess. For example, measurable cardinals cease to be measurable but remain Mahlo in .

If  holds in , then there is a closed unbounded class of ordinals that are indiscernible in . While some of these are not even initial ordinals in , they have all the large cardinal properties weaker than  in . Furthermore, any strictly increasing class function from the class of indiscernibles to itself can be extended in a unique way to an elementary embedding of  into . This gives  a nice structure of repeating segments.

L can be well-ordered 
There are various ways of well-ordering . Some of these involve the "fine structure" of , which was first described by Ronald Bjorn Jensen in his 1972 paper entitled "The fine structure of the constructible hierarchy". Instead of explaining the fine structure, we will give an outline of how  could be well-ordered using only the definition given above.

Suppose  and  are two different sets in  and we wish to determine whether  or . If  first appears in  and  first appears in  and  is different from , then let  if and only if . Henceforth, we suppose that .

The stage  uses formulas with parameters from  to define the sets  and . If one discounts (for the moment) the parameters, the formulas can be given a standard Gödel numbering by the natural numbers. If  is the formula with the smallest Gödel number that can be used to define , and  is the formula with the smallest Gödel number that can be used to define , and  is different from , then let  if and only if  in the Gödel numbering. Henceforth, we suppose that .

Suppose that  uses  parameters from . Suppose  is the sequence of parameters that can be used with  to define , and  does the same for . Then let  if and only if either  or ( and ) or ( and  and ), etc. This is called the reverse lexicographic ordering; if there are multiple sequences of parameters that define one of the sets, we choose the least one under this ordering. It being understood that each parameter's possible values are ordered according to the restriction of the ordering of  to , so this definition involves transfinite recursion on .

The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction. The values of -tuples of parameters are well-ordered by the product ordering. The formulas with parameters are well-ordered by the ordered sum (by Gödel numbers) of well-orderings. And  is well-ordered by the ordered sum (indexed by ) of the orderings on .

Notice that this well-ordering can be defined within  itself by a formula of set theory with no parameters, only the free-variables  and . And this formula gives the same truth value regardless of whether it is evaluated in , , or  (some other standard model of ZF with the same ordinals) and we will suppose that the formula is false if either  or  is not in .

It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-order the proper class  (as we have done here with ) is equivalent to the axiom of global choice, which is more powerful than the ordinary axiom of choice because it also covers proper classes of non-empty sets.

has a reflection principle
Proving that the axiom of separation, axiom of replacement, and axiom of choice hold in  requires (at least as shown above) the use of a reflection principle for . Here we describe such a principle.

By induction on , we can use ZF in  to prove that for any ordinal , there is an ordinal  such that for any sentence  with  in  and containing fewer than  symbols (counting a constant symbol for an element of  as one symbol) we get that  holds in  if and only if it holds in .

The generalized continuum hypothesis holds in L 
Let , and let  be any constructible subset of . Then there is some  with , so  for some formula  and some  drawn from . By the downward Löwenheim–Skolem theorem and Mostowski collapse, there must be some transitive set  containing  and some , and having the same first-order theory as  with the  substituted for the ; and this  will have the same cardinal as . Since  is true in , it is also true in , so  for some  having the same cardinal as . And  because  and  have the same theory. So  is in fact in .

So all the constructible subsets of an infinite set  have ranks with (at most) the same cardinal  as the rank of ; it follows that if  is the initial ordinal for , then  serves as the "power set" of  within  Thus this "power set" . And this in turn means that the "power set" of  has cardinal at most . Assuming  itself has cardinal , the "power set" must then have cardinal exactly . But this is precisely the generalized continuum hypothesis relativized to

Constructible sets are definable from the ordinals 
There is a formula of set theory that expresses the idea that . It has only free variables for  and . Using this we can expand the definition of each constructible set. If , then  for some formula  and some  in . This is equivalent to saying that: for all ,  if and only if [there exists  such that  and  and ] where  is the result of restricting each quantifier in  to . Notice that each  for some . Combine formulas for the 's with the formula for  and apply existential quantifiers over the 's outside and one gets a formula that defines the constructible set  using only the ordinals  that appear in expressions like  as parameters.

Example: The set  is constructible. It is the unique set  that satisfies the formula:

where  is short for:

Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would yield. But the point remains, there is a formula of set theory that is true only for the desired constructible set  and that contains parameters only for ordinals.

Relative constructibility
Sometimes it is desirable to find a model of set theory that is narrow like , but that includes or is influenced by a set that is not constructible. This gives rise to the concept of relative constructibility, of which there are two flavors, denoted by  and .

The class  for a non-constructible set  is the intersection of all classes that are standard models of set theory and contain  and all the ordinals.

 is defined by transfinite recursion as follows:
 = the smallest transitive set containing  as an element, i.e. the transitive closure of .
 = 
If  is a limit ordinal, then .
.

If  contains a well-ordering of the transitive closure of , then this can be extended to a well-ordering of . Otherwise, the axiom of choice will fail in .

A common example is , the smallest model that contains all the real numbers, which is used extensively in modern descriptive set theory.

The class  is the class of sets whose construction is influenced by , where  may be a (presumably non-constructible) set or a proper class. The definition of this class uses , which is the same as  except instead of evaluating the truth of formulas  in the model , one uses the model  where  is a unary predicate. The intended interpretation of  is . Then the definition of  is exactly that of  only with  replaced by .

 is always a model of the axiom of choice. Even if  is a set,  is not necessarily itself a member of , although it always is if  is a set of ordinals.

The sets in  or  are usually not actually constructible, and the properties of these models may be quite different from the properties of  itself.

See also 
 Axiom of constructibility
 Statements true in L
 Reflection principle
 Axiomatic set theory
 Transitive set
 L(R)
 Ordinal definable

Notes

References 
 
 

 
Works by Kurt Gödel