Re: Cardinality of Real Numbers

On 31 Aug 2005 07:17:42 -0700, "Ross A. Finlayson"
<raf@xxxxxxxxxxxxxxx> wrote:

>Virgil wrote:
>> In article <1125458081.555392.175280@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>> "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> wrote:
>>
>> > Jonathan Hoyle wrote:
>> > > >> Without the Axiom of Choice, not every set can be mapped
>> > > >> onto some cardinal.
>> > >
>> > > I misspoke here. I meant to say mapped to some aleph. Sorry for the
>> > > confusion.
>> >
>> > Is there not an ordinal for each of those cardinals?
>>
>> For all but finite cardinalities there are infinitely many ordinals
>> having each cardinality.
>
>Yes, that's what I thought. So, in ZF does every set have a cardinal?

Yes.

>Is each ordinal well-ordered?

Yes.

> Where that is so, Choice is a theorem of
>ZF. No?

No. Unless I'm mistaken, in ZF, we cannot prove that every cardinal
has any ordinals as representatives.

>Well-order the reals. Given the recent discussion of Cantor's first,
>are the reals a set?

Of course.

> If so, what does the existence of a well-ordering
>of them imply?

That they can be well-ordered.

Martin
.

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