I think this should have been discussed on MO, but I don't see such a thread, so excuse me (and delete this question) in this case.
This is inspired by this discussion. I see that the tradition of arguingdebates about the validity of the applicationnecessity of the axiom of choice has not died even now, and this surprises me greatly, because I never understood the meaning of these disputes.
I remember that when I was a student, once in front of my eyes, mathematicians were arguing about the "constructiveness" of some statements in Functional Analysis. The factthis or that the proofs use the axiom of choice was cosidered as the evidence of "non-constructiveness" of these resultsstatement are still ongoing. And very quickly itIn this regard, I became clear that the participantsinterested in the dispute simply do not understandwhether there are textbooks that the analysis known to them at the most elementary level essentially relies ondescribe what mathematical disciplines turn into when the axiom of choice (at least countable), because without it the simplest theorems, such as Bolzano-Weierstrass or the equivalence of the definitions of the limit by Cauchy and by Heine, etc. cannot be provedis consistently excluded from them.
From what I have read about thisIt seems to me that, I have the impressiontheoretically, nothing prevents a logician from writing a book that there are no meaningful statementsdescribes algebra or topology or analysis in Analysis at all that do not usesome axiomatic set theory without the axiom of choice, for example, in ZF.
ThatSo my question is why I want to ask people
are there texts describing what remains of analysis (or algebra, or topology) if it is built consistently without the axoim of choice, for example, in ZF (not in ZFC)?
And in general I think such a text would likebe very helpful, because in my observation, people who argue about the application of the axiom of choice tend to knowfocus on particulars without seeing the big picture, and moreover, having a rather vague idea of the subject.
what remains of mathematics as a whole if it is built in ZF?
For example, I guesshave met analysts who believe that in analysis the axiom of choice appears only in a few statements, such as the Hahn-Banach theorem, and if you do not use them, you can consider your conscience "unstained by its application".
People on this thread have already explained to me that some elementary factsthings in analysis can be preserved, in algebraparticular (forgive my ignorance), butit was a surprise to me that the theory of real numbers is preserved in some form (although, for examplethe classical theorems of mathematical analysis mostly disappear). And apparently, something can be preserved in algebra and topology,. I would be very surprised if something intelligible was preserved. Even in set theory, as far as I know, theterribly interested to look at this picture sharply turns into a dense forest.
Where is all this written? I remember Boris Kushner's book on "Constructive mathematical analysis", but it's about something else, about a variant of intuitionistic mathematics, where "constructiveness", as the adepts understand it, is woven into the system of axioms of logic, not set theory. Is there any overview or a book explaining what is preserved in mathematics when it is built in ZF?
Also I would be grateful if somebody could clarify to me what I asked on a comment to the discussion that I mentioned from the beginning:
When a person asks whether the statement X is true without the axiom of choice, does he mean to remove the axiom of choice from the whole theory in which he formulates this statement, or only from the proof of X?
As I told there, in my understanding, in both cases the question becomes senseless. If the idea is to remove axiom of choice from everywhere, then there must be books on Algebra, Topology, Analysis "without axiom of choice". Where are those books? And what is the reason to discuss the possibility to remove AC from X, if you don't mean to insert X into a "Big theory withut AC"? On the other hand, if the idea is to remove axiom of choice only from the proof of the statement X, then how can this be interesting, if any proof of X that you suggest in "non-constructive theory" inevitably uses supplementary results and constructions (theorems, lemmas, definitions) which are based on AC?