Re: I don't like the Axiom of Choice

On 2007-05-06, in sci.math, Bill Taylor wrote:
Aatu Koskensilta wrote:

I had in mind the
possibility, in absence of choice, that there does not exist an ordinal with
cofinality greater than omega.

Is this possible within ZF? There is always a set, *the* set, of
countable ordinals. If this had to be of cofinality only omega,
then it would be a countable union of countable sets.

It is consistent with ZF that omega_1 has cofinality omega.

be interesting to know whether there's any clear conception of the world of
sets that would justify it or make it plausible -- indeed it would be
interesting to know if there's any clear conception of the world of sets
justifying the usual axioms of set theory and the failure of choice.

Well yes, for the latter, I think so. One can imagine a world of sets
that include only those that are "definable" in some sense. If any sense
can be made of this, (and I suspect it can), then all of ZF would apply to
it, but not AC.

Why would the axioms of ZF apply? In particular, why would the axiom of
powersets and the axiom of replacement apply? The most straightforward
interpretation of 'definable' would give us something like L_omega_1^CK in
which not all axioms of set theory hold. Also, while it is clear that the
conception of sets as definable -- in whatever sense -- collections does not
justify choice, it's not clear how it justifies the failure of choice.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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