Constructible universe

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 V_α+1 to be the set of all subsets of the previous stage, V_α. By contrast, in Gödel's constructible universe L, 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

${\displaystyle \operatorname {Def} (X):={\Bigl \{}\{y\mid y\in X{\text{ and }}(X,\in )\models \Phi (y,z_{1},\ldots ,z_{n})\}~{\Big |}~\Phi {\text{ is a first-order formula and }}z_{1},\ldots ,z_{n}\in X{\Bigr \}}.}$

L is defined by transfinite recursion as follows:

• ${\displaystyle L_{0}:=\varnothing .}$
• ${\displaystyle L_{\alpha +1}:=\operatorname {Def} (L_{\alpha }).}$
• If ${\displaystyle \lambda }$ is a limit ordinal, then ${\displaystyle L_{\lambda }:=\bigcup _{\alpha <\lambda }L_{\alpha }.}$ Here α<λ means α precedes λ.
• ${\displaystyle L:=\bigcup _{\alpha \in \mathbf {Ord} }L_{\alpha }.}$ Here Ord denotes the class of all ordinals.

If z is an element of L_α, then z = {y | y ∈ L_α and y ∈ z} ∈ Def (L_α) = L_α+1. So L_α is a subset of L_α+1, which is a subset of the power set of L_α. Consequently, this is a tower of nested transitive sets. But L itself is a proper class.

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.

An equivalent definition for L_α is:

For any ordinal α, ${\displaystyle L_{\alpha }=\bigcup _{\beta <\alpha }\operatorname {Def} (L_{\beta })\!}$.

For any finite ordinal n, the sets L_n and V_n are the same (whether V equals L or not), and thus L_ω = V_ω: 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, L_ω+1 is a proper subset of V_ω+1, and thereafter L_α+1 is a proper subset of the power set of L_α for all α > ω. On the other hand, V = L does imply that V_α equals L_α if α = ω_α, for example if α is inaccessible. More generally, V = L implies H_α = L_α for all infinite cardinals α.

If α is an infinite ordinal then there is a bijection between L_α 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]

Another definition, due to Gödel, characterizes each L_α+1 as the intersection of the power set of L_α with the closure of ${\displaystyle L_{\alpha }\cup \{L_{\alpha }\}}$ under a collection of nine explicit functions, similar to Gödel operations. This definition makes no reference to definability.

All arithmetical subsets of ω and relations on ω belong to L_ω+1 (because the arithmetic definition gives one in L_ω+1). Conversely, any subset of ω belonging to L_ω+1 is arithmetical (because elements of L_ω can be coded by natural numbers in such a way that ∈ is definable, i.e., arithmetic). On the other hand, L_ω+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 L_ω+1 so it is in L_ω+2).

All hyperarithmetical subsets of ω and relations on ω belong to ${\displaystyle L_{\omega _{1}^{\mathrm {CK} }}}$ (where ${\displaystyle \omega _{1}^{\mathrm {CK} }}$ stands for the Church–Kleene ordinal), and conversely any subset of ω that belongs to ${\displaystyle L_{\omega _{1}^{\mathrm {CK} }}}$ is hyperarithmetical.[3]

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:

• Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.

(L,∈) is a substructure of (V,∈), which is well founded, so L is well founded. In particular, if y ∈ x ∈ L, then by the transitivity of L, y ∈ L. If we use this same y as in V, then it is still disjoint from x 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 x and y are in L and they have the same elements in L, then by L's transitivity, they have the same elements (in V). So they are equal (in V and thus in L).

• Axiom of empty set: {} is a set.

{} = L₀ = {y | y∈L₀ and y=y} ∈ L₁. So {} ∈ L. Since the element relation is the same and no new elements were added, this is the empty set of L.

• Axiom of pairing: If x, y are sets, then {x,y} is a set.

If x ∈ L and y ∈ L, then there is some ordinal α such that x ∈ L_α and y∈L_α. Then {x,y} = {s | s ∈ L_α and (s = x or s = y)} ∈ L_α+1. Thus {x,y} ∈ L and it has the same meaning for L as for V.

• Axiom of union: For any set x there is a set y whose elements are precisely the elements of the elements of x.

If x ∈ L_α, then its elements are in L_α and their elements are also in L_α. So y is a subset of L_α. y = {s | s ∈ L_α and there exists z ∈ x such that s ∈ z} ∈ L_α+1. Thus y ∈ L.

• Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union ${\displaystyle y\cup \{y\}}$.

From transfinite induction, we get that each ordinal α ∈ L_α+1. In particular, ω ∈ L_ω+1 and thus ω ∈ L.

• Axiom of separation: Given any set S and any proposition P(x,z₁,...,z_n), {x | x ∈ S and P(x,z₁,...,z_n)} is a set.

By induction on subformulas of P, one can show that there is an α such that L_α contains S and z₁,...,z_n and (P is true in L_α if and only if P is true in L (this is called the "reflection principle")). So {x | x ∈ S and P(x,z₁,...,z_n) holds in L} = {x | x ∈ L_α and x ∈ S and P(x,z₁,...,z_n) holds in L_α} ∈ L_α+1. Thus the subset is in L.

• Axiom of replacement: Given any set S and any mapping (formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z), {y | there exists x ∈ S such that P(x,y)} is a set.

Let Q(x,y) be the formula that relativizes P to L, i.e. all quantifiers in P are restricted to L. Q is a much more complex formula than P, but it is still a finite formula, and since P was a mapping over L, Q must be a mapping over V; thus we can apply replacement in V to Q. So {y | y ∈ L and there exists x ∈ S such that P(x,y) holds in L} = {y | there exists x ∈ S such that Q(x,y)} is a set in V and a subclass of L. Again using the axiom of replacement in V, we can show that there must be an α such that this set is a subset of L_α ∈ L_α+1. Then one can use the axiom of separation in L to finish showing that it is an element of L.

• Axiom of power set: For any set x there exists a set y, such that the elements of y are precisely the subsets of x.

In general, some subsets of a set in L will not be in L. So the whole power set of a set in L will usually not be in L. What we need here is to show that the intersection of the power set with L is in L. Use replacement in V to show that there is an α such that the intersection is a subset of L_α. Then the intersection is {z | z ∈ L_α and z is a subset of x} ∈ L_α+1. Thus the required set is in L.

• Axiom of choice: Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.

One can show that there is a definable well-ordering of L which definition works the same way in L itself. So one chooses the least element of each member of x to form y using the axioms of union and separation in L.

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, L_α is the same in W and V, for any ordinal α. And the same formulas and parameters in Def (L_α) produce the same constructible sets in L_α+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 L_κ 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 L_κ. 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 L_κ and the V of L_κ are the real L_κ, we get that V = L is true in L and in any L_κ that is a model of ZF. However, V = L does not hold in any other standard model of ZF.

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 L_α+1 and y first appears in L_β+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 L_α+1 = Def (L_α) uses formulas with parameters from L_α 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.

Suppose that Φ uses n parameters from L_α. Suppose z₁,...,z_n is the sequence of parameters that can be used with Φ to define x, and w₁,...,w_n does the same for y. Then let x < y if and only if either z_n < w_n or (z_n = w_n and ${\displaystyle z_{n-1}<w_{n-1}}$) or (z_n = w_n and ${\displaystyle z_{n-1}=w_{n-1}}$ and ${\displaystyle z_{n-2}<w_{n-2}}$) 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 L to L_α, so this definition involves transfinite recursion on α.

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 L_α+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,...,z_k) with z₁,...,z_k in L_β and containing fewer than n symbols (counting a constant symbol for an element of L_β as one symbol) we get that P(z₁,...,z_k) holds in L_β if and only if it holds in L.

Let ${\displaystyle S\in L_{\alpha }}$, and let T be any constructible subset of S. Then there is some β with ${\displaystyle T\in L_{\beta +1}}$, so ${\displaystyle T=\{x\in L_{\beta }:x\in S\wedge \Phi (x,z_{i})\}=\{x\in S:\Phi (x,z_{i})\}}$, for some formula Φ and some ${\displaystyle z_{i}}$ drawn from ${\displaystyle L_{\beta }}$. By the downward Löwenheim–Skolem theorem and Mostowski collapse, there must be some transitive set K containing ${\displaystyle L_{\alpha }}$ and some ${\displaystyle w_{i}}$, and having the same first-order theory as ${\displaystyle L_{\beta }}$ with the ${\displaystyle w_{i}}$ substituted for the ${\displaystyle z_{i}}$; and this K will have the same cardinal as ${\displaystyle L_{\alpha }}$. Since ${\displaystyle V=L}$ is true in ${\displaystyle L_{\beta }}$, it is also true in K, so ${\displaystyle K=L_{\gamma }}$ for some γ having the same cardinal as α. And ${\displaystyle T=\{x\in L_{\beta }:x\in S\wedge \Phi (x,z_{i})\}=\{x\in L_{\gamma }:x\in S\wedge \Phi (x,w_{i})\}}$ because ${\displaystyle L_{\beta }}$ and ${\displaystyle L_{\gamma }}$ have the same theory. So T is in fact in ${\displaystyle L_{\gamma +1}}$.

So all the constructible subsets of an infinite set S have ranks with (at most) the same cardinal κ as the rank of S; it follows that if δ is the initial ordinal for κ⁺, then ${\displaystyle L\cap {\mathcal {P}}(S)\subseteq L_{\delta }}$ serves as the "power set" of S within L. Thus this "power set" ${\displaystyle L\cap {\mathcal {P}}(S)\in L_{\delta +1}}$. And this in turn means that the "power set" of S has cardinal at most ||δ||. Assuming S itself has cardinal κ, the "power set" must then have cardinal exactly κ⁺. But this is precisely the generalized continuum hypothesis relativized to L.

There is a formula of set theory that expresses the idea that X = L_α. It has only free variables for X and α. Using this we can expand the definition of each constructible set. If s ∈ L_α+1, then s = {y | y ∈ L_α and Φ(y,z₁,...,z_n) holds in (L_α,∈)} for some formula Φ and some z₁,...,z_n in L_α. This is equivalent to saying that: for all y, y ∈ s if and only if [there exists X such that X =L_α and y ∈ X and Ψ(X,y,z₁,...,z_n)] where Ψ(X,...) is the result of restricting each quantifier in Φ(...) to X. Notice that each z_k ∈ L_β+1 for some β < α. Combine formulas for the z's with the formula for s and apply existential quantifiers over the z's outside and one gets a formula that defines the constructible set s using only the ordinals α that appear in expressions like X = L_α as parameters.

Example: The set {5,ω} is constructible. It is the unique set s that satisfies the formula:

${\displaystyle \forall y(y\in s\iff (y\in L_{\omega +1}\land (\forall a(a\in y\iff a\in L_{5}\land Ord(a))\lor \forall b(b\in y\iff b\in L_{\omega }\land Ord(b)))))}$,

where ${\displaystyle Ord(a)}$ is short for:

${\displaystyle \forall c\in a(\forall d\in c(d\in a\land \forall e\in d(e\in c))).}$

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.

L(A) is defined by transfinite recursion as follows:

• L₀(A) = the smallest transitive set containing A as an element, i.e. the transitive closure of { A }.
• L_α+1(A) = Def (L_α(A))
• If λ is a limit ordinal, then ${\displaystyle L_{\lambda }(A)=\bigcup _{\alpha <\lambda }L_{\alpha }(A)\!}$.
• ${\displaystyle L(A)=\bigcup _{\alpha }L_{\alpha }(A)\!}$.

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).

A common example is ${\displaystyle L(\mathbb {R} )}$, the smallest model that contains all the real numbers, which is used extensively in modern descriptive set theory.

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 Def_A (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 y ∈ A. Then the definition of L[A] is exactly that of L only with Def replaced by Def_A.

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.

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

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