[FOM] RE: FOM New Axioms
Matt Insall
montez at fidnet.com
Fri Jun 27 11:45:09 EDT 2003
[Todd Eisworth]
->For a concrete question, what happened that makes the Axiom of Choice seem
->so much more reasonable to mathematicians now than 100 years ago?
[Bill Tait]
No, it *did* seem quite reasonable then,
[Matt Insall]
I agree with this. Historically, according to the reading I have done
on AC, it was used as if it were ``obvious'', at first. I am convinced
that this is due to the fact that it is obvious when applied only to
finite sets. (See my previous posting(s) on this issue.) The ability
to well-order any finite set was seen as an ability that was easily
extendable
to the case of infinite sets.
[Bill Tait]
just as it does now, to most mathies;
the only thing that's really changed since is that it is now in text-books
and undergraduate courses.
True, there *were* doubters back then, but there are also doubters now,
(including me).
[Matt Insall]
Among mathematicians, there are doubters of many things, but when pinned
down,
I seems to me that just about every mathematician would agree with the
axioms
of ZF. The fact that these axioms have found their way into many texts at
various
levels helps attest to this and increases all along the number of
mathematicians
who do believe the axioms of ZF. Many texts do still, however, attach some
special
significance to AC, and the way this is done attests to the fact that there
are
still some doubters of AC. But AC is so natural, it is common (see the
history
of AC), as you indicated, for doubters to use (some version of) it
unintentionally.
The fact that in many cases the actual usage can be seen as an application
of
a weak form of AC is interesting, but I think that the fact that a variety
of
forms of full AC are very natural (for example, ``every vector space has a
basis'')
is the reason that mainstream mathematics now adopts the full strength of
AC.
But the doubters get their word in, because many who make use of some form
of AC
include a comment about its use in their work. If AC were accepted as
completely
as say the axiom of infinity (proofs that use the axiom of infinity in
mathematics
typically are not labeled as such), there would be almost no proof labeled
that
it uses AC. We would just write our arguments and go on.
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