Constructible Universe

What Is L?

L can be thought of as being built in "stages" resembling the 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

$\begin{align} \text{Def}(X) := & \Bigl\{ \{y \mid y\in X \text{ and } \Phi(y,z_1,\ldots,z_n) \text{ is true in }(X,\in)\} \mid \\ & \qquad \Phi \text{ is a first order formula and } z_1,\ldots,z_n\in X\Bigr\}. \end{align}$

L is defined by transfinite recursion as follows:

If is a limit ordinal, then .
.

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.

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