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I understand the need for a solid foundation of mathematics. But it seems to me that we have definitely settled for ZFC and I would like to know if there are strong reasons for this or not. I am not completely sure if this is a valid question for MSE or not, but in any case please read the clarification below before deciding that.

If I understand the stories that I read correctly, Zermelo proposed his axioms in 1908 with the hope of formalizing naive set theory. Some flaws were found years after and people like Fraenkel helped solve them and complete this axiomatic system into what we know today as ZFC. I may be wrong about what I will say next, because I am sure that Zermelo was not a man locked inside a room without any contact with the exterior world (at least he kept correspondence with Fraenkel). I am sure he based his axiomatic system in the many ideas that were already floating in mathematics at that time. So the next question is an exageration and should be taken as such:

Why is it reasonable to base the foundations of mathematics in a bunch of axioms that were laid down by a single human being?

I know that current set theorists and foundational mathematicians investigate other possible axiomatic systems, for example with additional axioms such as the axiom of constructibility or with a whole different system such as homotopy type theory. But it seems that the average mathematician does not bother himself or herself with these foundational issues. So in practice, it seems to me that most of mathematics is developed taking as granted this system. Is this not worrying? So:

What is so different about foundations with respect to any other branch of mathematics to justify the general lack of emphasis, research and interest in this field?

Clarification: I do not want a discussion or a debate, I want to know if there is an answer to the questions in yellow. Philosophical, historical and mathematical answers are all welcome, but I expect a combination of all of them and I consider that the mathematical part must be present in the answer (and this is the reason to post the question here and not in Philosophy Stack Exchange). I am aware that there are already some questions addressing closely related issues, but I didn't find an answer to whether it is reasonable or not to settle for ZFC.

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    $\begingroup$ @Alephnull What are you talking about? If we could just tell that statements were "true of 'sets'", we wouldn't need ZFC in the first place. $\endgroup$ – Derek Elkins Jan 10 '18 at 0:18
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    $\begingroup$ ZFC is named after Zermelo, but it is not like he went up to the mountain alone and came back with stone tablets that have been used unchanged ever since. He was one link in a multiple-person effort to construct a theory that (a) didn't allow the paradoxes to be derived, and (b) allowed all of ordinary mathematics to be formalized in a natural way. The system received some tinkering from other workers (and not only Fraenkel, even though Fraenkel is the only one who made it into the acronym) before it reached the particular shape that is taught as ZFC today. $\endgroup$ – Henning Makholm Jan 10 '18 at 0:23
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    $\begingroup$ For example, Zermelo's original publication stated explicitly as part of one of the axioms that singleton sets exists; it was only realized later that this is a consequence of unordered pairs, so in the modern retelling we do without a singleton axiom. $\endgroup$ – Henning Makholm Jan 10 '18 at 0:25
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    $\begingroup$ @Pedro: The entire premise of the first part of the system is that ZFC as we know it today is "laid down by a single human being". But that is simply not the case. Zermelo was the first to get (or at least seriously champion) the crucial idea of restricted comprehension as embodied the Axiom of Separation supplemented with power sets, but the reason it is accepted is not that the Prophet said so. It is that this idea actually turned out to work. Everything else in Z (in the original or the modern form alike) is just tinkering, an axiomatic scaffolding to make this central idea work. $\endgroup$ – Henning Makholm Jan 10 '18 at 0:35
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    $\begingroup$ And even that central idea was not what carried the day as a foundation -- it needed to be modified by Fraenkel before it was ultimately accepted. $\endgroup$ – Henning Makholm Jan 10 '18 at 0:35
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I'll take a swing at answering the question that I think you are trying to ask. I'll formulate it as follows:

If foundations are important as the name suggests and the so-called "foundational crisis" suggests, why do so few mathematicians concern themselves much with them nowadays. If foundations aren't important, then why was there a "foundational crisis" and a significant effort to create foundations?

tl;dr "Foundations" and ZFC were created to solve a fairly specific problem (founding real analysis), which they did. Now we don't worry about the problem, so many mathematicians don't have much reason to "faff about" with foundations.

The first thing to note is the obvious statement that mathematics has been done before, during, and after the establishment of ZFC as a foundational system. Just as clearly, very little mathematics prior to the establishment of ZFC has been deemed "incorrect" since its establishment. (Even the parts that arguably may have been have often been "revitalized" in modern treatments, sometimes utilizing other foundational approaches, e.g. "infinitesimals".)

So the first point is "doing math" doesn't require a foundational system as witnessed by the fact that math was being done for thousands of years before the advent of ZFC or anything like it. This is also witnessed by the fact that you can learn quite a bit of math today without concerning yourself much with the details of ZFC.

My understanding of the situation near the "foundational crisis", which may well be wrong - I'm no math historian - is there was a fairly specific group that wanted something like set theory: real analysts (as we'd call them nowadays). My reading of the situation is that it was the controversies and vagaries in real analysis that sparked mathematical (as opposed to philosophical) interest in foundations. Intuitions about "real numbers", "functions", "continuous functions" were not enough for the mathematicians of the day to converge on questions like what the Fourier transform of the constant function should be or whether it should even exist. This also raised the possibility that the notion of "real numbers" itself might be incoherent.

This led to the early work on defining the reals and defining a notion of function. (There were also philosophical motivations for this work that were likely partially independent, but I suspect that without the issues in real analysis mathematicians would have largely ignored such work.) Of course, from there Russell's paradox scuttled the still largely intuitive conception of naive set theory. This likely also scuttled the idea that we could rely on mathematical intuition alone and reinforced the possibility that, e.g., the real numbers really could be built on quicksand. They certainly were in naive set theories. Then we had 40 years of many proposed foundational systems, modifications to those systems, critiques of those systems, meta-logical analyses of those systems, and hands-on work using the systems. Presumably the majority of mathematicians of the time were at most spectators to this. They continued to plod on doing math the way they'd always done it likely without much concern if they were, say, a (non-analytic) number theorist.

I would say the main go/no-go issue for a set theory of that time would be whether it could found real analysis, i.e. construct the real numbers, construct a notion of continuous function, and prove widely accepted results like Heine-Borel. Jumping to the modern day, yes, it is the case that the mere existence of any acceptable foundation removes much of the urgency of the "foundational crisis". Most mathematicians during the "foundational crisis" didn't care about sets, they cared about real numbers and continuous functions. Given some other (non-set-theoretic) framework to assuage their concerns, they would have had little interest in set theory. Nowadays, students (perhaps unfortunately...) don't have these concerns in the first place, so they have little reason to devote much or any time to foundations. Set theory is usually taught in a naive way with some warnings. The language and tools from set theory are useful even without it being a foundational system, so it's not ignored entirely.

The variety of foundational systems, the fact most of them are also capable of serving as a foundation for real analysis and most other branches of math, and the fact that ZFC itself goes far beyond what is needed by most mathematicians means most of mathematics doesn't really depend on the specific details of the underlying foundational system. For example, while finding an inconsistency in ZFC would be big news, it's hard to imagine that it would also impact all other foundational systems that are capable of supporting e.g. real analysis. It is likely that most results would be unaffected and the ones that were affected would be relatively easily adapted and still "morally true". Maybe an extra assumption is added, say.

Another aspect of this is that there are many results in more solidly grounded fields like number theory that have proofs that use mathematical objects in less solidly grounded fields like real analysis. To the extent these results have "elementary" proofs within the more solidly grounded field, we have a web of justification for the validity of non-trivial aspects of the less solidly grounded field. This puts some limits on how "wrong" we could be in those fields before we'd have to be "wrong about everything".

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  • $\begingroup$ Why doesn't foundations come up again when we enter new frontiers of mathematics? It does, but you can see the same dynamic happen in these newer instances. Category theory requires extensions to ZFC to capture it as used. Mathematicians have spent time see what assumptions are necessary to do this. However, once some suitable foundation was found, it stopped being something that a new, young categorist would spend much time thinking about. (And set theory is a lot more relevant to category theory than it is to many other areas of math.) $\endgroup$ – Derek Elkins Jan 10 '18 at 2:25

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