# The constructible universe

## Definition

Gödel’s constructible universe is a subclass $L$ of the von Neumann universe $V$ of well-founded pure sets, defined by transfinite induction? as $L = \bigcup_\alpha L_\alpha$ where

$L_0 = \emptyset$
$L_{\alpha +1} = P(L_\alpha) \cap I(L_\alpha \cup \{L_\alpha\})$

and if $\alpha$ is a limit ordinal:

$L_\alpha = \bigcup_{\beta\lt \alpha} L_\beta$

Alternatively, we may say that

$L_\alpha = \bigcup_{\beta \lt \alpha} P(L_\beta) \cap I(L_\beta \cup \{L_\beta\})$

for any ordinal $\alpha$ ($0$, successor, or limit).

Here, for $X$ a pure set, by $I(X)$ we denote the smallest set containing $X$ and closed with respect to the operations of Cartesian product, set difference, unordered pair, ordered pair, taking the domain of a binary relation, and performing a permutation of an ordered triple.

The elements of the constructible universe are called constructible sets; the idea is similar to the constructible sets in topology and algebraic geometry.

## Properties

$L$ is a transitive big class containing all the ordinals. In fact, it is the smallest transitive model of the set theory containing all the ordinals. On the other hand, the sets in this class can be effectively enumerated by von Neumann ordinals. The question weather $L\neq V$ can not be decided in ZF. If $L\neq V$ then still we do not know how, without an axiom of choice, to produce specific sets which are not constructible.

The wikipedia entry constructible universe is pretty elaborate.

Revised on September 21, 2012 10:18:34 by Urs Schreiber (82.169.65.155)