Material set theory

Complement Axiom for classes

The Complement Axiom is a possible axiom of Material set theory with classes1: #m/def/set/nbg

(โˆ€โ„ญ๐”ฉ๐”ฐโก๐‘‹)(โˆƒโ„ญ๐”ฉ๐”ฐโก๐‘)(โˆ€๐”ˆโก๐‘ข)[๐‘ขโˆˆ๐‘โŸบ๐‘ขโˆ‰๐‘‹]

which is to say, the complement of a class ๐‘‹ (within the universe of everything except proper classes) exists, which by extensionality is unique and we denote the ๐‘‹๐‘. It follows immediately that we have a universal class

๐‘‰=โˆ…๐‘


#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2015. Introduction to Mathematical Logic, ยง4.1, p. 236, B3 โ†ฉ