Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular class of sets that can be described entirely in terms of simpler sets. L is the union of the constructible hierarchy Lα. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".[1] In this, 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.

L can be thought of as being built in "stages" resembling the construction of von Neumann universe, V. The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes Vα+1 to be the set of all subsets of the previous stage, Vα. By contrast, in Gödel's constructible universe L, one uses only those subsets of the previous stage that are:

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.

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

For any finite ordinal n, the sets Ln and Vn are the same (whether V equals L or not), and thus Lω = Vω: their elements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC in which V equals L, Lω+1 is a proper subset of Vω+1, and thereafter Lα+1 is a proper subset of the power set of Lα for all α > ω. On the other hand, V = L does imply that Vα equals Lα if α = ωα, for example if α is inaccessible. More generally, V = L implies Hα = Lα for all infinite cardinals α.

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

As defined above, Def(X) is the set of subsets of X defined by Δ0 formulas (with respect to the Levy hierarchy, i.e., formulas of set theory containing only bounded quantifiers) that use as parameters only X and its elements.[2]

All arithmetical subsets of ω and relations on ω belong to Lω+1 (because the arithmetic definition gives one in Lω+1). Conversely, any subset of ω belonging to Lω+1 is arithmetical (because elements of Lω can be coded by natural numbers in such a way that ∈ is definable, i.e., arithmetic). On the other hand, Lω+2 already contains certain non-arithmetical subsets of ω, such as the set of (natural numbers coding) true arithmetical statements (this can be defined from Lω+1 so it is in Lω+2).

L is a standard model, i.e. it is a transitive class and it uses the real element relationship, so it is well-founded. L is an inner model, i.e. it contains all the ordinal numbers of V and it has no "extra" sets beyond those in V, but it might be a proper subclass of V. L is a model of ZFC, which means that it satisfies the following axioms:

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

If W is any standard model of ZF sharing the same ordinals as V, then the L defined in W is the same as the L defined in V. In particular, Lα is the same in W and V, for any ordinal α. And the same formulas and parameters in Def (Lα) produce the same constructible sets in Lα+1.

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

If there is a set W in V that is a standard model of ZF, and the ordinal κ is the set of ordinals that occur in W, then Lκ is the L of W. If there is a set that is a standard model of ZF, then the smallest such set is such a Lκ. 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 the L of L and the V of L are the real L and both the L of Lκ and the V of Lκ are the real Lκ, we get that V = L is true in L and in any Lκ that is a model of ZF. However, V = L does not hold in any other standard model of ZF.

Since Ord ⊂ LV, properties of ordinals that depend on the absence of a function or other structure (i.e. Π1ZF formulas) are preserved when going down from V to L. Hence initial ordinals of cardinals remain initial in L. Regular ordinals remain regular in L. Weak limit cardinals become strong limit cardinals in L because the generalized continuum hypothesis holds in L. 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 L.

However, 0# is false in L even if true in V. So all the large cardinals whose existence implies 0# cease to have those large cardinal properties, but retain the properties weaker than 0# which they also possess. For example, measurable cardinals cease to be measurable but remain Mahlo in L.

If 0# holds in V, then there is a closed unbounded class of ordinals that are indiscernible in L. While some of these are not even initial ordinals in V, they have all the large cardinal properties weaker than 0# in L. 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 L into L. This gives L a nice structure of repeating segments.

There are various ways of well-ordering L. Some of these involve the "fine structure" of L, 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 L could be well-ordered using only the definition given above.

Suppose x and y are two different sets in L and we wish to determine whether x < y or x > y. If x first appears in Lα+1 and y first appears in Lβ+1 and β is different from α, then let x < y if and only if α < β. Henceforth, we suppose that β = α.

The stage Lα+1 = Def (Lα) uses formulas with parameters from Lα to define the sets x and y. If one discounts (for the moment) the parameters, the formulas can be given a standard Gödel numbering by the natural numbers. If Φ is the formula with the smallest Gödel number that can be used to define x, and Ψ is the formula with the smallest Gödel number that can be used to define y, and Ψ is different from Φ, then let x < y if and only if Φ < Ψ in the Gödel numbering. Henceforth, we suppose that Ψ = Φ.

The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction. The values of n-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 L is well-ordered by the ordered sum (indexed by α) of the orderings on Lα+1.

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

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 V (as we have done here with L) 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.

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

By induction on n < ω, we can use ZF in V to prove that for any ordinal α, there is an ordinal β > α such that for any sentence P(z1,...,zk) with z1,...,zk in Lβ and containing fewer than n symbols (counting a constant symbol for an element of Lβ as one symbol) we get that P(z1,...,zk) holds in Lβ if and only if it holds in L.

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 s and that contains parameters only for ordinals.

Sometimes it is desirable to find a model of set theory that is narrow like L, 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 L(A) and L[A].

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

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

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

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

The sets in L(A) or L[A] are usually not actually constructible, and the properties of these models may be quite different from the properties of L itself.