Re: Question regarding limit ordinals and transfinite cardinals.


"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote in message news:ae210af1-df10-4af7-b141-95733dbd1de2@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jan 24, 5:35 pm, "Peter Webb"
<webbfam...@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

The Axiom of Infinity directly proves the existence of a single limit
ordinal, w = {N}. It does not prove the existence of any other limit
ordinals. For this you need some additional machinery. You need to prove
that card (P(x)) > card (x) to prove that there are sets of larger
cardinality than N. You need to prove that ordinals are totally ordered.
Then you need the Axiom of Choice (in practice Zorn's Lemma) to prove that
there is a lowest member of the set of ordinals with a given cardinality,
which must hence be a limit ordinal. The existence of limit ordinals except
w is hardly a direct corollary of the axiom of infinity, and indeed without
AxC I suspect that you can't prove the existence of any limit ordinal except
w.

In ZF, even without the axiom of choice, we prove the existence of
limit ordinals other than w. For example, w+w. So, the addition of the
axiom of infinity to the other axioms of ZF does prove the existence
of limit ordinals other than w.

MoeBlee

**** True, of course, and my mistake - you don't need AxC to prove the existence of countable ordinals. But I still don't think the existence of w+w or any limit ordinal (other than w) is a "immediate corollary" of ZFC. Of course, your idea of "immediate" may be different to mine ...


.

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