Constructible universe

The constructible yuniverse, commonly denoted by \(L\), is a proper class defined by the union of a hierarchy \((L_{\alpha})_{\alpha \in \textrm{On}}\) of sets called constructible hierarchy indexed by the proper class \(\textrm{On}\) of ordinals. Since the equality \(V = L\) is independent of ZFC set theory as long as it is consistent, we sometimes explicitly assume that every set is constructible in the sense that it is an element of \(L\).

Hierarchy
For a set \(X\), we denote by \(\textrm{Def}(X)\) the set of subsets of \(X\) definable by a first-order formula with parameters from \(X\). A first-order formula with parameters from \(X\) is a first order formula in set theory of the form \(\Phi(y,z_1,...,z_n)\) where y is a free variable (i.e. a variable which is not quantified within the formula) and \(z_1,...,z_n\in X\). A set is defined by a formula if it coincides with the set of sets \(y\) satisfying that formula.

Be careful that googologists often refer to the "definability" without understanding the precise definition, and hence the intuitive meaning is usually different from the original terminology in mathematics. The definability here is the original one in mathematics, which is precisely formalised in set theory. For example, when googologists say something like "the least natural number greater than any natural number definable by a first order theory using at most \(10^{100}\) symbols", then it is irrelevant to the original terminology of the definability, because it cannot be referred to in that unformalisable way.

For a class \(A\), we denine a hierarchy \((L_{\alpha}(A))_{\alpha \in \textrm{On}}\) in the following transfinite inductive way: Then the constructible universe \(L(A)\) relativised to \(A\) is defined as \(\bigcup_{\alpha \in \textrm{On}} L_{\alpha}(A)\).
 * If \(\alpha = 0\), then \(L_{\alpha}(A)\) is the smallest transitive class containing \(A\). (A class \(X\) is said to be transitive if \(x \subset X\) for any \(x \in X\).)
 * If \(\alpha \neq 0\), then \(L_{\alpha}(A) = \bigcup_{\beta < \alpha} \text{Def}(L_{\beta}(A)) = \bigcup \{\textrm{Def}(L_{\beta}(A)) \mid \beta \in \alpha\}\).

When \(A = \emptyset\), then we simply denote \(L_{\alpha}(A)\) by \(L_{\alpha}\). The constructible universe \(L\) is defined as \(L(\emptyset)\), and hence is the union of the hierarchy \((L_{\alpha})_{\alpha \in \textrm{On}}\).

By the definition, we have \(L_{\alpha} \subset V_{\alpha}\) for any ordinal \(\alpha\). Note that even if we assume \(V = L\), the equality \(L_{\alpha} = V_{\alpha}\) does not necessarily hold for a given ordinal \(\alpha\).

Application
The constructible hierarchy is frequently used to define models of set theories. For example, if \(\alpha\) is an inaccessible cardinal, then \(V_{\alpha}\) forms a model of \(\textrm{ZFC}\) set theory. By the soundness of first order logic, it implies that \(\textrm{ZFC}\) set theory augmented by the existence of an inaccessible cardinal proves the formalised consistency \(\textrm{Con}(\textrm{ZFC})\) of \(\textrm{ZFC}\) set theory itself. Therefore by Goedel's incompleteness theorem, the existence of an inaccessible cardinal is not provable under \(\textrm{ZFC}\) set theory as long as it is consistent.

Although it is not obvious from the definition, \textrm{KP} set theory is deeply related to the constructible hierarchy. The relation appears in the connection between computation theory and set theory, and is studied in mathematics. It is also used to define the notion of the admissibility of an ordinal.