Implementation of mathematics in set theory: Difference between revisions

Revision as of 00:41, 22 December 2005

Revision as of 00:45, 22 December 2005

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assertion).

Strongly cantorian sets have important properties which we reviewintroduce here. The type of any variable

restricted to a strongly cantorian set A can be raised or lowered as desired by replacing

references to <math>a \in A</math> with references to <math>\bigcup f(a)</math> (type of a raised)

or <math>f^{-1}(\{a\}</math> (type of a lowered) where <math>f(a) = \{a\}</math> for all <math>a \in A</math>, so it is not necessary to assign types to such variables for purposes of stratification.

Any subset of a strongly cantorian set is strongly cantorian. The power set of a strongly

cantorian set is strongly cantorian. The cartesian product of two strongly cantorian sets is strongly cantorian.

Introducing the Axiom of Counting means that we do not need to assign types to variables

restricted to N or to P(N), R (the set of reals) or indeed any set ever considered in classical

mathematics outside of set theory.