Re: Uncountable sets in CZF?

From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 08/31/04 

Date: 31 Aug 2004 01:10:56 GMT

Herman Jurjus <h.jurjus@hetnet.nl> writes:

>David McAnally wrote:
>> Herman Jurjus <h.jurjus@hetnet.nl> writes:
>[snip]

>>>Unless... we have just proved that there exist no models of ZFC in which
>>>N is the 'real' N and R is the 'real' R, a.l.f.o.
>>
>>
>> That might be the case.

>Sure.
>But: using the very same argument, it would not be possible for a model
>of ZFC to have an N that is countable-a.l.f.o, and an R-in-V that is
>uncountable-a.l.f.o. And i don't think that is true (you actually
>contradict it in the rest of your article, below).

>I must be missing something, but what?

I was giving it some thought, and I think that it may be that, while the
existence of a generic set or a generic ultrafilter is consistent relative
to ZFC, that does not mean that such a generic set or generic ultrafilter
must exist.

>>>That would be, well,
>>>perhaps not shocking, but quite disturbing.
>>
>>
>> A fact which could be considered disturbing is that for any transfinite
>> cardinal not cofinal with omega, there exists a model of ZFC in which the
>> cardinality of R is the given cardinal.

>Personally, i don't find it disturbing in itself that there are models
>of ZFC in which the 'counterparts of real sets' are not the same as the
>'real' Platonic ones. I got over that, long time ago. ;-)

On the other hand, the above suggests that the totality of the real
numbers exceeds in cardinality any set in the model under investigation.

>But so far i thought of forcing as something that only worked for
>countable models. You say your extension construction works in general,
>and doesn't add elements to sets.

I think that forcing (at least for a specific set of forcing conditions)
works, provided you can find a generic set or generic ultrafilter. I do
not believe that a model need be countable to allow that to happen.

I'll look at the rest later, when I have more time available.

David

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