[FOM] Cardinality in weaker set theories

[Thomas Forster]

On Mon, 17 Jan 2011, 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.

Yes

>
> 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.

Yes.  ZC does not prove the existence of the von Neumann ordinal 
$\omega.2$ since, as you say, it cannot prove the existence of any sets of 
rank $\omega.2$, for the reason you give.


>
> C) So ZC does not prove every well ordered set is order-isomorphic to
> a von Neumann ordinal.

Indeed it does not: how true.

>
> D) ZC offers no canonical definition of the Alephs, that is no way of
> selecting one representative of each cardinality.
>

You have to be careful.  You say you are assuming foundation, so you do 
have Scott's trick, so in a sense you do have a representative.  What is 
certainly the case is that the collection of representatives is not going 
to be a set. In the obvious model (V_{\omega +\omega}) you have a 
Scott's trick aleph_n for each finite n, but the graph of the 
function enumerating them is not a set, so you can't really do much with 
them.  I can't see how to get *representatives* - even using choice 
(thougfh of course global choice would do it for you). In fact I think
ZC proves that for every proper initial segment $I$ of the cardinals there 
is a function sending any set whose cardinal is in $I$ to a representative 
set of that size (and choice is [probably] essential here).......But it 
might be that you don't need to know that for your purposes.

Without foundation (i think i am right in saying that) you can't even do that.

    tf

URL:  www.dpmms.cam.ac.uk/~tf; DPMMS ph: +44-1223-337981;
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