Definition talk:Cardinality

Cardinality of Infinite Sets
How will this article address Cardinality of Infinite Sets?
 * It already does;

I think that Cardinality should be defined in terms of equinumerosity and dominance relations, which themselves should be defined in terms of mappings (that is, for example, two sets are equinumerous if there exists a function that maps one set 1-1 onto the other set. -Andrew Salmon 14:17, 11 September 2011 (CDT)
 * It is, just it's all on other pages than this. Follow the links, it should go down that route.
 * This bit is not all rigorous yet, I never finished it. Feel free.
 * But before going ahead and altering the metastructure of ProofWiki too much, read around it and see what there already is. --prime mover 14:39, 11 September 2011 (CDT)

How Exactly are we Definining it?
Here, it seems that we are not so much defining $|X|$ as we are defining $|X| = |Y|$.

Takeuti/Zaring develop surprisingly much about cardinality without using the axiom of choice by defining cardinality as:


 * $\displaystyle |X| = \bigcap \{ x \in \operatorname{On} : x \sim X \}$

That way, for sets that fail to have a bijection with an ordinal (that is, if the axiom of choice fails), their cardinality is $\varnothing$. The case with natural numbers then becomes a special case of this definition. This definition becomes useful when proving that AH implies the Axiom of Choice (the aleph hypothesis says that the cardinality of the powerset of $\aleph_x$ is $\aleph_{x+1}$. It is equivalent to the Generalized Continuum Hypothesis in ZF.) --Andrew Salmon 00:17, 28 July 2012 (UTC)