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I've never taken a formal course of Set Theory, but I've been wondering about this for some time now.

Are non-standard axioms, like $\mathbb{V}=\mathbb{L}$ and axioms about large cardinals and any other you can think of (which are independent of $ZFC$) used outside pure Set Theory?

Are there some non-trivial problems in other branches of math (like algebra, topology, analysis etc.) that can only be solved if these axioms are used? Can they permit pathological behaviour in those fields?

If not, why don't we choose the axioms to make sets behave in the nicest way possible like $\mathbb{V}=\mathbb{L}$, and use them to simplify things in the Set Theory itself? Why is $ZFC$ so universally used?

Also, are there still some branches of math, that reject axiom of choice?

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Set theoretic axioms are used in a few other areas, which are perhaps closely related to set theory but not, in my opinion, "pure set theory".

One area is in "set theoretic topology", which is a field of point-set topology that considers properties that are more sensitive to set theoretic axioms. One example is Fleissner's 1974 paper "Normal Moore Spaces in the Constructible Universe" (Proc. AMS 46:2) which made progress toward the normal Moore space conjecture under the assumption $V=L$. Similarly, Martin's axiom has applications in general topology. For one recent example, see this abstract from the 2018 Joint Math Meetings. Overall, these uses of $V=L$ or Martin's axiom are vaguely analogous to the use of the unproved Riemann hypothesis to obtain results in number theory.

Another branch of math that sometimes uses axioms outside ZFC is set-theoretic combinatorics, although this is probably closer to "pure set theory".

But why is ZFC so heavily used? One reason is certainly sociological: ZFC is heavily used because it is what most mathematicians agree to. But there are two important caveats to that. The first is that most mathematics requires much less than ZFC, so saying "this is provable in ZFC" is usually a vast overestimate of what is needed. In this sense, ZFC is used because it is a safe estimate.

The second caveat is that most mathematicians don't worry about foundations too much, and probably can't name the axioms of ZFC, so for them the use of ZFC to formalize their work is entirely hypothetical. The same goes for any foundational system - similarly few mathematicians could give a reasonable set of definitions for any formal foundational system, be it ZFC or a topos-theoretical system.

As for the axiom of choice, essentially the only place it is not accepted is in certain (but not all) branches of constructive (intuitionistic, non-classical) mathematics. Essentially every contemporary intro textbook to algebra or analysis uses AC when needed to obtain the standard, classical results.

Finally, why don't we assume $V=L$ all the time, to simplify things? The best answer is that, in a sense, the axioms of ZFC are agreed to match the usual conception of "set" (that is, "pure well founded set"), while there is no generally accepted argument that $V=L$ is implicit in the notion of "set". It is always possible that additional axioms could be accepted in the future, if enough mathematicians believed they were natural. But somehow, in practice, the ZFC axioms seem to be the limit of what the term "set" means to the majority of mathematicians who have looked at these things. Much more was written about this in a series of articles "Does mathematics need new axioms" in the December 2000 Bulletin of Symbolic Logic. Separate articles were written by Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel giving different opinions about this exact issue.

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You should get a better answer from a set theorist.

The Continuum Hypothesis is sometimes used in analysis, in a funny way. It happens that we have a conjecture C, and CH allows us to construct a counterexample. Years ago I wondered why anyone would care, since this doesn't prove that C is false, since we don't know that CH holds.

But I was wrong; "we" "should" care about that counterexample using CH. Although it doesn't prove that C is false, it does prove that we can't prove C is true (in ZFC of course), since we know that ZFC does not prove that CH is false.

(There's a traditional example in measure theory that uses CH and shows, regardless of whether one believes in CH, that one of the hypotheses in Fubini's Theorem cannot be omitted...)

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  • $\begingroup$ Can you provide a reference to "conjecture $C$"? You've peaked my interest... $\endgroup$ – Stefan Mesken Jan 17 '18 at 20:49
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    $\begingroup$ Sorry. Wasn't referring to a specific conjecture, just "some" conjecture that I called C so I could refer to it later. $\endgroup$ – David C. Ullrich Jan 17 '18 at 21:46
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    $\begingroup$ @Stefan: Interests get piqued, unless you talk about taking a loan and the interest rate is at its peak... $\endgroup$ – Asaf Karagila Jan 18 '18 at 4:48
  • $\begingroup$ @Asaf Thank you -- finally someone told me. I must have used that incorrectly for many years now. $\endgroup$ – Stefan Mesken Jan 18 '18 at 8:36
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Allow me to add a marginalia to Carl Mummert's excellent answer.

  1. There are places where these kind of axioms are useful. Category theory makes uses of inaccessible cardinals; the question "is it consistent that without choice all sets of reals are Lebesgue measurable?" has an answer which depends on the choice of large cardinals; the question "is every projective set Lebesgue measurable?" also has an answer rooted in large cardinal assumptions.

    On the other hand, forcing gave us a lot of tools to construct a lot of weird and interesting counterexamples, in algebra and functional analysis and topology, to say the least. Assuming $V=L$ essentially says that this is not the case. But it does give rise to other strange counterexamples, as would many other "canonical" inner models of set theory (not just $L$ is at fault here).

  2. One of the reasons that $V=L$ or large cardinal axioms don't make it down to "everyday mathematics" very often is the same reason that Quine's New Foundation didn't succeed as a foundational theory. There's too much logic (and in set theory) involved. In order to understand $V=L$ as an axiom you need to understand basic model theory, you need to understand transfinite recursion, you need to have a bunch of set theory under your belt.

    If you don't understand $V=L$, what use is it for you? The consequences are often a combination of $\sf GCH$, and various combinatorial principles like $\lozenge$ or $\square$ principles. But those are consistent also with $V\neq L$. So assuming this axiom doesn't actually give you any benefit here.

    The same can be said on most "sufficiently large" large cardinal axioms. You need to understand elementary embeddings and ultrapowers if you want to understand measurable cardinals; you need a hell of a lot of set theory to understand very large axioms, e.g. $I0$; and you even need to have some set theory under your belt to fully understand what is a Mahlo cardinal.

    Since most people don't want to work with set theory directly, most people don't want to assume these axioms. And they won't.

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Grothendieck universes, developed by Grothendieck for use in algebraic geometry, and which have applications in category theory, are equivalent to strongly inaccessible cardinals.

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