Re: Uncountable sets in CZF?

From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 09/01/04 

Date: 1 Sep 2004 11:22:35 GMT

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

>David McAnally wrote:

>> Herman Jurjus <h.jurjus@hetnet.nl> writes:
>>
>>
>>>Herman Jurjus wrote:
>>
>>
>>>>David McAnally wrote:
>>>>
>>>>
>>>>>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.
>>>>
>>>>
>>>>Sorry to bother you once more, but... can you provide a short argument
>>>>for this, as well?
>>
>>
>>>Moreover, what's the status of "there exists a transfinite cardinal not
>>>cofinal with omega" ?
>>
>>
>> In ZFC, all successor cardinals are regular, so none of them are cofinal
>> with omega.

>Oh, ok. Apparantly my set theory has become a bit rusty.
>What's your definition of 'cofinal with omega'?

I may not have the terminology exactly right. By omega, I mean the
smallest limit ordinal, so that omega is the order type of N. When
I describe an ordinal alpha as being cofinal with omega, then I mean
that there is an ascending function f : omega -> alpha such that
sup_{n < omega} f(n) = alpha. In the case of a cardinal in ZFC, I
mean that the cardinal kappa is cofinal with omega iff it is cofinal
with omega as an ordinal, i.e. iff there is an ascending function f
mapping omega to ordinals of cardinality less than kappa, such that
sup_{n < omega} f(n) = kappa.

The cofinality of the limit ordinal alpha is the smallest ordinal
sigma such that there is an ascending function f : sigma -> alpha
such that sup_{tau < sigma} f(tau) = alpha. The cofinality is
necessarily a regular transfinite cardinal (i.e. the cofinality
of sigma is sigma itself). The cofinality of an aleph is the
cofinality of its initial ordinal.

In ZFC, if A has cardinality lambda, then the cofinality of the
power set of A necessarily exceeds lambda.

David

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