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I'm interested in the generalization of natural numbers and especially ordinals. I have searched through some set theory textbooks but unable to find Von Neumann definition of ordinal numbers.

I have read somewhere that Von Neumann definition of ordinal numbers is elegent, so please suggest me some textbooks that mention Von Neumann definition of ordinal numbers.

Thank you so much!

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    $\begingroup$ I think it's worth commenting that almost any set theory text from the last century (I know of only two exceptions off hand) will use von Neumann ordinals as its definition of ordinal. They may not use that name, because at this point it's standard enough that one can almost always assume "ordinal" and "von Neumann ordinal" are synonymous. $\endgroup$ – Malice Vidrine Jul 5 '18 at 17:31
  • $\begingroup$ @Malice Vidrine: That's exactly what I was thinking, which has me wondering exactly what books the OP is looking at. I just checked my bookshelves out of curiosity, and I found several set theory books that do not deal with von Neumann ordinals, but all of them were first published before 1960. $\endgroup$ – Dave L. Renfro Jul 5 '18 at 18:35
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Kenneth Kunen's Set Theory: an Introduction to Independence Proofs develops the Von Neumann ordinals pretty much right out of the gate.

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Any book on set theory will probably mention ordinals at length. One that I know of is Moschovakis' Notes on Set Theory, which does use the von Neumann ordinals.

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I learned the definition of the von Neumann ordinals (and a lot more) from Paul Cohen's book "Set theory and the continuum hypothesis".

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The books mentioned so far, Kunen and Cohen, are excellent but perhaps advanced. Moschovakis, also excellent, does not get to ordinals till quite late.

A perhaps more accessible book would be Devlin's "Joy of Sets."

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