[FOM] Cardinality in weaker set theories

[Andreas Blass]

Dear Colin,
	In your fom posting (copied below), items B and C are correct.  In A,  
you should require ordinals to be transitive sets; otherwise, any  
subset of an ordinal will be an ordinal.  Concerning  D, I believe  
you're right that there is no way to select one representative of each  
cardinality; nevertheless, Scott's trick provides a canonical  
definition of the alephs and indeed of cardinals in general: Define  
the cardinality of a set x to be the collection of those sets that are  
(1) equinumerous with x and (2) of least possible rank subject to (1).

Andreas Blass
>
> Message: 2
> Date: Mon, 17 Jan 2011 09:13:22 -0500
> From: Colin McLarty <colin.mclarty at case.edu>
> Subject: [FOM] Cardinality in weaker set theories
> To: Foundations of Mathematics <fom at cs.nyu.edu>
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> 	<AANLkTikYNuA26NXWdDEGffBT=x16Z03JNLbn6rDFy1_j at mail.gmail.com>
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>
> Hi,
>
> The foundations of cohomological number theory lead me to work on
> cardinals in Zermelo set theory and weaker theories and I want to
> check that I've got this right.
>
> Let ZC be the axioms of ZFC but with separation instead of
> replacement.  So this includes foundation.  Are the following all
> true?
>
> A) We can define von Neumann ordinals in ZC as sets linearly ordered
> by membership.
>
> B) ZC proves for every natural number n there is a von Neumann ordinal
> omega+n, and that is all it proves (if ZFC is consistent) since in ZFC
> the set of all sets of finite rank over the naturals is a model of ZC.
>
> C) So ZC does not prove every well ordered set is order-isomorphic to
> a von Neumann ordinal.
>
> D) ZC offers no canonical definition of the Alephs, that is no way of
> selecting one representative of each cardinality.
>
> Or have I missed something here?
>
> thanks, Colin