[FOM] Countable choice
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Sat Jun 14 04:32:19 EDT 2008
Robert,
Thank you for your thoughts on this. I'm not sure if what i reconstruct
from it below is what you meant, but it might be worth playing back to you
anyway.
What i found illuminating about your post can be summed up in the
one word `supertask'. You seemed to me to be saying that AC_\omega is
simply the allegation that a certain kind of supertask is in principle
performable. Nice! OK, so why not AC_\omega_1? you go on to ask (by
implication) - and your answer is that it's something to do with the
separability of the real line.
I take you to be saying this: an ordinary (finite non-super) task is
something we imagine ourselves to be performing *in time*. A supertask is
an omega-sequence of atomic tasks and we can imagine performing it *in
time* because we can embed omega in R and time is like R. Cooking up a
selection function for a countable family of sets (or an omega-sequnce as
in DC) looks like a supertask of the kind familiar from the philosophical
literature (Thompson's lamps etc). Now the tasks corresponding to
AC_\omega_1 are not supertasks like the Thompson's lamp because we cannot
embed the countable ordinals in R. (This is what i take your remark about
separability of R to mean).
I like the sound of this: it sounds plausible. Now note that one can
find this story plausible and interesting without thinking that there is
genuinely a good reason for believing AC_\omega that isn't also a reason
for believing AC. What you have given us is a story (a *narrative*?!)
that runs: ``if you are the kind of person who thinks a lot about
supertasks, thinks they are important and ought to be completeable etc
etc, then you will probably think of AC_\omega as a principle that tells
you what you want to hear, and will therefore be inclined to accept it.
And this line of thought won't tell you to accept AC''
Am i reading you correctly?
tf
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