[FOM] Another challenge to Pratt concrning AC
Arnon Avron
aa at tau.ac.il
Sun Aug 23 07:57:02 EDT 2009
Dear Vaughan,
> We should agree to agree. I don't see any of the differences between us
> that you're pointing to as significant.
I think that there is a difference between us (which is significant)
comcerning how the "mathematicians in the street) view things here.
But before continuing this debate(?) I'll be interested how you
would treat the following case.
In all basic books on Set theory there is the theorem
that every infinite set has a (proper) countable subset.
The proof in all of them directly use AC (or some weaker version
of it), even though usually this is not noted by the authors.
Now for any specific infinite set I have encountered (and any
such set which is of interest to our mathematicians in the street)
one can easily show the existence of such a
countable subset without using AC. Do you think that
noting this last fact would suffice for the mathematicians in the street,
and they would say here that foundationalists just make things
unnecesarily harder by insisting on formulating and proving
the general theorem (that requires AC for its proof)? And how
would you reformulate the theorem (without trivializing it)
so the use of AC will be avoided?
Just to clarify: I do not like AC, and I remember
well how suspicious I was of this particular
proof when I first learn it as a first year student
in our introductory course on set theory (only later
I came to know that what had bothered me was the use
of AC in that proof - the lecturer of course has never mentioned AC).
However, I do recognize that "mathematicians in the street"
frequently use AC - sometimes without being aware that they do.
Arnon
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