[FOM] Status of AC

[Andreas Blass]

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