The Continuum Problem and Constructible Sets

Abstract

1.1. In this section we introduce the subclass L ⊂ V —“Gödel’s constructible universe”—and establish its fundamental properties. Perhaps the shortest description of L is that it is the smallest transitive model of the axioms of L1Set that contains all the ordinals. But the working definition of L, from which the name “constructible universe” is derived, is rather different. We consider the following operations F ₁ , . . . , F ₈ on sets:

$$\begin{array}{l} F_1 (X,{\rm }Y{\rm }){\rm } = {\rm }\{ X,{\rm }Y{\rm }\} , \\ F_2 (X,{\rm }Y{\rm }){\rm } = {\rm }X\backslash Y, \\ F_3 (X,{\rm }Y{\rm }){\rm } = {\rm }X{\rm } \times {\rm }Y, \\ {\rm }F_4 (X){\rm } = {\rm }\{ U|\exists W(U,W{\rm } \in {\rm }X)\} {\rm } = {\rm dom }X, \\ {\rm }F_5 (X){\rm } = {\rm }\{ U,W|U,W{\rm } \in {\rm }X;{\rm }U{\rm } \in {\rm }W\} , \\ {\rm }F_6 (X){\rm } = {\rm }\{ \left. {\left\langle {U1,{\rm }U2,{\rm }U3} \right.} \right\rangle |\left. {\left\langle {U2,{\rm }U3,{\rm }U1} \right.} \right\rangle {\rm } \in {\rm }X\} , \\ {\rm }F_7 (X){\rm } = {\rm }\{ \left. {\left\langle {U1,{\rm }U2,{\rm }U3} \right.} \right\rangle |\left. {\left\langle {U3,{\rm }U2,{\rm }U1} \right.} \right\rangle {\rm } \in {\rm }X\} , \\ {\rm }F_8 (X){\rm } = {\rm }\{ \left. {\left\langle {U1,{\rm }U2,{\rm }U3} \right.} \right\rangle |\left. {\left\langle {U1,{\rm }U3,{\rm }U2} \right.} \right\rangle {\rm } \in {\rm }X\} . \end{array}$$

We say that a set (or class) Y is closed with respect to an operation F of degree r if we have F(Z ₈ , . . ., Z ₈) ? Y for all Z ₈ , . . . , Z ₄ ∈ Y such that F(Z ₁ , . . ., Z _r) is defined. For every X ∈ V we let J (X) denote the smallest set Y ⊃ X that is closed with respect to the operations F ₁ , . . ., F ₈. It will later be shown (Section 1.4) that J (X) actually is a set. The following construction is analogous to the definition of V.