Implementation of mathematics in set theory: Difference between revisions

Revision as of 17:55, 21 December 2005

Revision as of 17:58, 21 December 2005

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We can show that <math>|A| \leq |B|</math> is almost a linear order. Reflexivity is obvious

and transitivity is proven just as for equinumerousness. The [[Cantor-SchröderSchroder-Bernstein theorem]], provable in either [[ZFC]] or [[NFU]] in an entirely standard way, establishes that

*<math>|A| \leq |B| \wedge |B| \leq |A| \rightarrow |A| = |B|</math>
(this establishes that

we have antisymmetry on cardinals (not yet defined), but we are now considering a relation on sets),

and

and <math>|A| \leq |B| \vee |B| \leq |A|</math> follows in a standard way in either theory from

*<math>|A| \leq |B| \vee |B| \leq |A|</math>

and <math>|A| \leq |B| \vee |B| \leq |A|</math> follows in a standard way in either theory from

the Axiom of Choice.