If AC skepticism were not low-status, you would expect to find papers and textbooks actively rejecting AC results, rather than merely mentioning in a remark or footnote that AC is involved. (Such footnotes are for use at dinner parties.)
Not entirely. If the only known proof for a result assumes choice, then a proof that doesn’t use choice will almost certainly be publishable.
And also, texts just as frequently do not bother to make apologies of the sort you allude to. A fairly random example I recently noticed was on p.98 of Algebraic Geometry by Hartshorne, where Zorn’s Lemma is used without any more apology than an exclamation point at the end of the (parenthetical) sentence.
Using an exclamation mark like that is a pretty rare thing to do. You wouldn’t for example see this if one used the axiom of replacement. The only other axiom that would be in a comparable position is foundation but foundation almost never comes up in conventional mathematics. Hartshorne is writing for a very advanced audience so I think putting an exclamation mark like that is sufficient to get the point across especially when one is using choice in the form of Zorn’s lemma.
is most correlated not with interest in logic and foundations, but with working in finitary, discrete, or algebraic areas of mathematics where AC isn’t much used.
This seems to fit my impression as well.
Incidentally, for what it is worth, your claim that rejection of AC is low status seems to be possibly justified. I know of two prominent mathematicians who explicitly reject AC in some form. One of them does so verbally but seems to be fine teaching theorems which use AC with minimal comment. The other keeps his rejection of AC essentially private.
Using an exclamation mark like that is a pretty rare thing to do. You wouldn’t for example see this if one used the axiom of replacement. The only other axiom that would be in a comparable position is foundation but foundation almost never comes up in conventional mathematics.
Of course it’s worth noting that axiom of replacement doesn’t come up much either, though obviously the case there isn’t quite as extreme as with foundation.
Not entirely. If the only known proof for a result assumes choice, then a proof that doesn’t use choice will almost certainly be publishable.
Using an exclamation mark like that is a pretty rare thing to do. You wouldn’t for example see this if one used the axiom of replacement. The only other axiom that would be in a comparable position is foundation but foundation almost never comes up in conventional mathematics. Hartshorne is writing for a very advanced audience so I think putting an exclamation mark like that is sufficient to get the point across especially when one is using choice in the form of Zorn’s lemma.
This seems to fit my impression as well.
Incidentally, for what it is worth, your claim that rejection of AC is low status seems to be possibly justified. I know of two prominent mathematicians who explicitly reject AC in some form. One of them does so verbally but seems to be fine teaching theorems which use AC with minimal comment. The other keeps his rejection of AC essentially private.
Of course it’s worth noting that axiom of replacement doesn’t come up much either, though obviously the case there isn’t quite as extreme as with foundation.