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Talk:Cardinal number

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Revision as of 17:50, 10 January 2015 by Richard Pinch (talk | contribs) (Comparability of cardinals: Axiom of choice seems to have been assumed)
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Comparability of cardinals

The article deduces from the Schroder–Berstein theorem ($\mathfrak{a} \le \mathfrak{b}$ and $\mathfrak{b} \le \mathfrak{a}$ implies $\mathfrak{a} = \mathfrak{b}$) that cardinals are totally ordered. This seems wrong: all it proves is that $\le$ is indeed a partial order on cardinals. That any two cardinals are comparable is, I believe, a form of the axiom of choice. A similar assumption is made a little later when it is asserted that "Any cardinal number $\mathfrak{a}$ can be identified with the smallest ordinal number of cardinality $\mathfrak{a}$". Again this requires that any set can be well-ordered. Richard Pinch (talk) 18:50, 10 January 2015 (CET)

How to Cite This Entry:
Cardinal number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cardinal_number&oldid=36206