nLab
big class

Big classes

Idea

A class LL in material set theory is big if for any set XLX \in L there exists a set YLY \in L such that XYX \in Y.

A metalanguage formulation

Consider a class LL as a formula ϕ(z)\phi(z) with a free variable zz; intuitively LL is the collection of all sets such that ϕ(z)\phi(z) is true. Then, in the metalanguage, LL is big (i.e., the formula ϕ(x)\phi(x) exhibits a big class) if

ϕ(X)(Y)(ϕ(Y)(XY)) \phi(X) \implies (\exists Y)(\phi(Y) \wedge (X\in Y))

Examples and properties

Gödel’s constructible universe is a transitive big class.

Revised on January 8, 2011 05:44:15 by Toby Bartels (98.19.48.164)