# [FOM] Countable choice

Robert Black Mongre at gmx.de
Mon Jun 9 12:17:04 EDT 2008

I don't know any literature, but one thought might be that in order
to construct a countable choice set all we have to do is perform a
imagine because of the separability of R. The same sort of reasoning
might lead one to think that second-order logic over a countable
domain is determinate (i.e. that '*arbitrary* subset of N' is a
determinate and absolute notion) but that second-order logic over
uncountable domains might not be. I'm not myself terribly impressed
by this, because as a good platonist I don't think sets have anything
to do with possible constructions in time and even if they do it
doesn't seem necessary that time should have the structure of R. If
there's a better reason for thinking there's an important distinction
I'd be interested to hear it.

Robert

>I have a feeling that there is a literature on the idea that
>countable choice is not just a weak form of full choice, but
>is somehow a fundamentally different principle: there might
>be good reasons for adopting AC_\omega that aren't special
>cases of arguments for adopting AC.   Can anyone give me
>any pointers to it..?
>
>           tf