Constructible Universe - Relative Constructibility

Relative Constructibility

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

The class L(A) for a non-constructible set A is the intersection of all classes which 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 .
• .

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 L(R), the smallest model which contains all the real numbers, which is used extensively in modern descriptive set theory.

The class L 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 is exactly that of L only with Def replaced by Def_A.

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

It is essential to remember that the sets in L(A) or L are usually not actually constructible and that the properties of these models may be quite different from the properties of L itself.