Implementation of mathematics in set theory: Difference between revisions

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== Size of sets ==

In both [[ZFC]] and [[New Foundations|NFU]], two sets ''A'' and ''B'' are the same size (or are '''[[equinumerous]]''') if and only if there is a [[Bijective function|bijection]] ''f'' from ''A'' to ''B''. This can be written as <math>|A|=|B|</math>, but note that (for the moment) this expresses a relation between ''A'' and ''B'' rather than a relation between yet-undefined objects <math>|A|</math> and <math>|B|</math>. Denote this relation by <math>A \sim B</math> in contexts such as the actual definition of the [[Cardinal number|cardinals]] where even the appearance of presupposing abstract cardinals should be avoided.

Similarly, define <math>|A| \leq |B|</math> as holding if and only if there is an [[Injective function|injection]] from ''A'' to ''B''.

It is straightforward to show that the relation of equinumerousness is an [[equivalence relation]]: equinumerousness of ''A'' with ''A'' is witnessed by <math>i_A</math>; if ''f'' witnesses <math>|A|=|B|</math>, then <math>f^{-1}</math> witnesses <math>|B|=|A|</math>; and if ''f'' witnesses <math>|A|=|B|</math> and ''g'' witnesses <math>|B|=|C|</math>, then <math>g\circ f</math> witnesses <math>|A|=|C|</math>.

It can be shown that <math>|A| \leq |B|</math> is a [[linear order]] on abstract cardinals, but not on sets. Reflexivity is obvious and transitivity is proven just as for equinumerousness. The [[Cantor–Bernstein–Schroeder theorem|Schröder–Bernstein theorem]], provable in [[ZFC]] and [[New Foundations|NFU]] in an entirely standard way, establishes that

*<math>|A| \leq |B| \wedge |B| \leq |A| \rightarrow |A| = |B|</math>

(this establishes antisymmetry on cardinals), and