Implementation of mathematics in set theory: Difference between revisions

Revision as of 23:56, 21 December 2005

Revision as of 23:59, 21 December 2005

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external endomorphism of the ordinals.

A set A is said to be <strong>cantorian</strong> just in case <math>|A| = |P_1(A)| = T(|A|)</math>; the cardinal <strongmath>|A|</strongmath> is also said to be a cantorian cardinal.

A set A is said to be <strong>strongly cantorian</strong> (and its cardinal to be strongly cantorian as well) just in case the restriction of

the singleton map to A (<math>(x \mapsto \{x\})\lceil A</math>)is a set. Well-orderings

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since T fixes such cardinals and it is easy to show that a function space between cantorian

sets is cantorian (as are power sets, cartesian products, and other usual type constructors).

This offers further encouragement to the view that the "standard" cardinalities in [[NFU]] are the

cantorian (indeed, the strongly cantorian) cardinalities, just as the "standard" ordinals

seem to be the strongly cantorian ordinals.

Now the usual theorems of cardinal arithmetic with the axiom of choice can be proved, including