nLab constructible universe

Definition

Gödel’s constructible universe $L$ is a subclass of the von Neumann universe $V$ of well-founded pure sets. It is 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:

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

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.

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).

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

AnonymousCoward: Does there exist a structural set theory definition of a constructible set and the constructible universe?

Properties

$L$ is a transitive big class containing all the ordinals.

The sets in this class can be effectively enumerated by von Neumann ordinals.

As inner model

In fact, $L$ is a model of the set theory (consider ZF), namely the smallest transitive class containing all the ordinals. Indeed, we can well-order all sets and the axiom of choice holds for $L$ as a model. And even the generalized continuum hypothesis holds for $L$ as a model. Note that properties of sets in the set theory might not hold as seen from within the inner model.

Relation to the von Neumann universe

By $L=V$, set theories refer to the collapsing of the von Neumann universe $V$ to the “thinner” $L$. The question of whether $L\neq V$ can not be decided in ZF. If $L\neq V$ then we still do not know how, without an axiom of choice, to produce specific sets that are not constructible.

Given the independence, one may add $L=V$ as an axiom. However, note that formally stating that axiom is more technical (involving ordinals, etc.) than the standard set theory axioms. Moreover, by its restrictiveness, that axiom also rules out many other popular studied axioms (e.g. it is inconsistent with the existence of “large” large cardinals such as measurable cardinals and above).

Links

• The wikipedia entry constructible universe is pretty elaborate.

• Richard Matthews, Michael Rathjen, Constructing the Constructible Universe Constructively, arxiv