Material set theory

Complement Axiom for classes

The Complement Axiom is a possible axiom of Material set theory with classes¹: nbg

(∀ℭ𝔩𝔰⁡𝑋)(∃ℭ𝔩𝔰⁡𝑍)(∀𝔈⁡𝑢)[𝑢∈𝑍⟺𝑢∉𝑋]

which is to say, the complement of a class 𝑋 (within the universe of everything except proper classes) exists, which by extensionality is unique and we denote the 𝑋𝑐. It follows immediately that we have a universal class

𝑉=∅𝑐

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tidy | en | SemBr

Footnotes

1. 2015. Introduction to Mathematical Logic, §4.1, p. 236, B3 ↩