Editing Constructible universe (section)

==What ''L'' is==
<math>L</math> can be thought of as being built in "stages" resembling the construction of the [[von Neumann universe]], <math>V</math>. The stages are indexed by [[ordinal number|ordinals]]. In von Neumann's universe, at a [[successor ordinal|successor]] stage, one takes <math>V_{\alpha+1}</math> to be the set of ''all'' subsets of the previous stage, <math>V_\alpha</math>. By contrast, in Gödel's constructible universe <math>L</math>, one uses ''only'' those subsets of the previous stage that are:

*definable by a [[Formula (mathematical logic)|formula]] in the [[formal language]] of set theory,
*with [[Parameters#Logic|parameters]] from the previous stage and,
*with the [[Quantifier (logic)|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:<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974). Accessed 20 February 2023.</ref>

<math display="block">
\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\}.
</math>

<math>L</math> is defined by [[transfinite recursion]] as follows:
* <math display="inline"> L_0 := \varnothing. </math>
* <math display="inline"> L_{\alpha + 1} := \operatorname{Def}(L_\alpha). </math> 
* If <math display="inline"> \lambda </math> is a [[limit ordinal]], then <math display="inline"> L_{\lambda} := \bigcup_{\alpha < \lambda} L_{\alpha}. </math> Here <math display="inline">\alpha<\lambda</math> means <math display="inline">\alpha</math> [[ordinal number#Successor and limit ordinals|precede]]s <math display="inline">\lambda</math>.
* <math display="inline"> L := \bigcup_{\alpha \in \mathbf{Ord}} L_{\alpha}. </math> Here '''Ord''' denotes the [[class (set theory)|class]] of all ordinals.

If <math>z</math> is an element of <math>L_\alpha</math>, 
then <math>z=\{y\in L_\alpha\ \text{and}\ y\in z\}\in\textrm{Def}(L_\alpha)=L_{\alpha+1}</math>.<ref>K. J. Devlin, ''Constructibility'' (1984), ch. 2, "The Constructible Universe, p.58. Perspectives in Mathematical Logic, Springer-Verlag.</ref> So <math>L_\alpha</math> is a subset of <math>L_{\alpha+1}</math>, which is a subset of the [[power set]] of <math>L_\alpha</math>. Consequently, this is a tower of nested [[transitive set]]s. But <math>L</math> itself is a [[Class (set theory)|proper class]].

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