[FOM] Cardinality in weaker set theories

[John Burgess]

Maybe you have to be a bit careful about the formulation of D.
One could choose omega to represent beth-zero, its powerset to  
represent beth-one, and so on,
and since you can't disprove GCH or prove the existence of beth-omega,
you can't prove this isn't assigning a unique representative to every  
cardinal,
though of course you can't prove it is.
Some more knowledgeable person can probably tell us whether there is a  
known theorem to the effect that
there is no definable assignment of representatives that can be proved  
to assign a unique representative to every cardinal.
Generally in ZC one would use Scott equivalence classes for cardinals;
but these of course aren't representatives (that is, this approach  
doesn't make the cardinal kappa to be some one distinguished set of  
that cardinality).

On Jan 17, 2011, at 9:13 AM, Colin McLarty wrote:

> 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
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