[FOM] Status of AC
Andreas Blass
ablass at umich.edu
Wed Mar 1 11:16:18 EST 2006
Mikey Nguyen wrote:
> what is the current status of AC among mathematicians and
> philosophers of mathematics?
>
> To me, it seems quite unthinkable to drop AC, since without it a large
> part
> of mathematics would collapse (e.g. can anyone think of model theory
> without
> Compactness or topology without Tychonoff?). I feel it is as central to
> modern mathematics as infinity axiom, and much more central to
> Replacement
> or Axiom of Foundation. Nonetheless, I heard many people support
> axioms like
> AD, which contradicts AC. Also, in most set theory courses AC is
> emphatically separated from ZF, and students are told that it remains
> very
> controversial, while nothing is said about Axiom of Foundation!
>
> So what's the deal with it? How controversial is it to you? And can
> anybody
> argue against it?
I won't try to describe philosophers' opinions of the axiom of choice,
but it seems to me that mathematicians generally accept the axiom of
choice. Some of them pay attention to whether a proof uses it --- I
often get questions about this from "core" colleagues --- but many
don't care. And I don't know any case of a core mathematician
refraining from publishing a theorem just because the proof used the
axiom of choice. I also don't see AD being used by core
mathematicians. It's certainly used by set theorists, but for the
purpose of describing a universe of definable sets, for example the
universe of sets constructible from the reals. AD for such a
subuniverse is consistent with AC for the full universe.
I do not tell my set theory students that AC "remains very
controversial", though I do tell them that it was once (about a hundred
years ago) very controversial. When I present the axioms of ZFC, I
generally divide them into three groups: (1) the axioms of
extensionality and foundation, which just describe what we mean by
"set" (as opposed to "ordered list" or "multiset" or "isomorphism type
of extensional pointed digraph" or ...), (2) axioms saying that certain
specific (definable) sets exist (union, power set, infinity,
replacement), and (3) the axiom of choice, asserting the existence of
some sets that we can't define. It is this classification, not the
existence of any controversy, that (for me) separates the axiom of
choice from the others.
In this connection, it may be worthwhile for me to repeat here what I
wrote about this issue about 20 years ago, in the introduction to the
set theory volume of the Omega Bibliography of Mathematical Logic:
"A digression may be in order here, to comment on the idea that
mathematicians accept AC because it is needed for many results in
modern mathematics. This may be true of some mathematicians, but I
believe it is false of most set theorists. Utility is no reason to
accept false axioms (though it may be a reason to study them). We
accept the axiom of choice because, on the basis of what we mean by
"set", the axiom is true. We accept the axiom of regularity for the
same reason, although its utility is so limited that many
mathematicians have never heard of it. We do not accept the continuum
hypothesis (as an axiom), despite its great utility for proving
theorems, because we do not see that it is true; the same applies to
various hypotheses that contradict CH."
Let me add a minor point about the specific examples that Nguyen
mentioned as making it "unthinkable to drop AC". The compactness
theorem for first-order model theory and the Tychonoff theorem for
Hausdorff spaces (the only spaces most of my "core" colleagues think
about) are both equivalent to the Boolean prime ideal theorem, which is
strictly weaker than the axiom of choice. (Non-Hausdorff topologies do
occur in core math, for example the Zariski topologies on schemes, but
the Tychonoff theorem doesn't seem to accompany them, probably because
the topology of a product of schemes is almost never the product
topology.)
Andreas Blass
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