Cardinal: Difference between revisions
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There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> However, in the context of axiom choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\).
Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is [[Aleph 0|\(\aleph_0\)]] - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\).
If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.
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