Re: Non-denumerable ordinals

On Tue, 7 Jul 2009 02:00:39 -0700 (PDT), Marc Alcobé García
<malcobe@xxxxxxxxx> wrote:

On 6 jul, 15:31, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:
Yes. The countable ordinals are closed under pretty much any operation
you can think of.

So the method never collapses.

Which are the weakest assumptions needed in order to prove the
existence of an uncountable ordinal?

It seems that some sort of choice and replacement (so that P(omega)
can be well-ordered and shown to be order-isomorphic to some ordinal).

You don't need choice _or_ replacement to construct an
uncountable well-ordered _set_. I think then you need replacement
to get a (von Neumann) ordinal.

Say C is the set of pairs (E, <=), where E is a subset of omega and
<= is a well-ordering of E. Let W = C/~, where ~ is the equivalence
defined by isomorphism. Then W is uncountable, and W is well-
ordered by the relation "is isomophic to an initial segment of".


David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: Non-denumerable ordinals
    ... existence of an uncountable ordinal? ... I think then you need replacement ... David C. Ullrich ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: Non-denumerable ordinals
    ... So the method never collapses. ... Which are the weakest assumptions needed in order to prove the ... existence of an uncountable ordinal? ... It seems that some sort of choice and replacement ...
    (sci.logic)
  • Re: Fractional Transforms
    ... X = Y. Of course this doesn't quite rule out the existence ... "Understanding Godel isn't about following his formal proof. ... Thanks for the response - from both you and Robert. ...
    (sci.math)
  • Re: Fractional Transforms
    ... X = Y. Of course this doesn't quite rule out the existence ... At the very least i thought i was polite enough to post and get responses. ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)
  • Re: "Proper functions", Replacement and Zorns Lemma.
    ... be a function except that it's a proper class of ... Hausdorff maximality and Zorn's lemma in ZC, i.e., without the axiom ... Otoh giving a precise statement of replacement is going to be a pain ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)