Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

Question

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti.

But these all seem to me like pathological, esoteric, ad-hoc examples, that really only matter in foundations, and most non-foundational and applied mathematics wouldn't go anywhere near touching them.

Am I wrong here? If we were to do non-foundational math over naive set theory and just ignore the paradoxes, what problems might we face? Yes, I know that we can technically prove $0=1$ because logic, but I'm looking for more interesting examples, particularly ones that could arise without having to specifically look for them.

Question: Notwithstanding technicalities like explosion, are there any "natural" examples of contradictions arising in non-foundational or applied math due to the paradoxes of naive set theory?

Has anyone ever arrived at a false statement in, say, algebra or number theory, using naive sets?

edit: I'd like to be clear that I'm playing devil's advocate. I'm of course aware that relying on an inconsistent theory is in general a bad idea, but of course not all flawed structures collapse immediately. How far could we go in practice before we ran into problems?

edit: By "non-foundational" I basically mean anything outside of set theory or mathematical logic. If the question of a theory's consistency comes up at all (this thought experiment notwithstanding), then it's probably "foundational". But it's of course fuzzy.

Answer 1

With a strict enough definition of "non-foundational mathematics" I think the answer is probably "no" (although I would be very interested in seeing potential examples.) However, this shouldn't make mathematicians working on such mathematics feel safe about using unrestricted comprehension. The reason is that it's not always clear a priori what mathematics will turn out to be "foundational".

Indeed, people may start working on some mathematics that seems non-foundational but then turns in a foundational direction. For example, Cantor's development of set theory was a natural consequence of his study of sets of uniqueness in harmonic analysis.

If someone working in a supposedly non-foundational branch of mathematics ended up with a contradiction by using unrestricted comprehension, then with the benefit of hindsight we could say that he or she must have been working in an area related to foundations after all.

It might seem like cheating to make such a declaration after the fact, but perhaps it is not: It seems likely that, from a novel use of unrestricted comprehension to obtain a contradiction, one could obtain a novel use of replacement to obtain a theorem that could not have been obtained without replacement (i.e. using only restricted comprehension). I say this because replacement is a natural intermediate step between restricted and unrestricted comprehension.

Mathematics that uses replacement in an essential way is often considered ipso facto to be foundational. So I think it is likely that mathematics that uses unrestricted comprehension an an essential way (to the extent that it can be salvaged) would be considered foundational as well.

(This answer doesn't address the question of how long, on average, it would take people using unrestricted comprehension in non-foundational-seeming areas of mathematics to run into problems. I think that question is very interesting but probably also very hard to answer.)

Answer 2

Well, I wouldn't trust a building whose structure was shown to be flawed. Note that it isn't a suspicion, it is a certainty.

But I know what you mean by your question... consider the following "proof" (taken from another question at MSE):

Theorem: Let $K$ be a field, then $K$ has an algebraic closure $\bar{K}$ (i.e an algebraic extension that is algebraically closed).

"Proof": Define $A=\{ F \supset K | F \text{ is an algebraic extension of } K\}$ and inherit this with the usual partial order of inclusion. One can check that Zorn's lemma applies (union of a nested chain of algebraic extensions is itself algebraic). Thus take $\overline{K}$ to be a maximal element. It must be algebraically closed for otherwise there is an irreducible polynomial with root in some strictly bigger field. $\blacksquare$

Do you trust this, as it is?

Answer 3

The whole idea of using set theory as a foundational theory is that you want a theory that if you believe is consistent, the rest of mathematics is consistent.

Naive set theory is inconsistent. So you can't really continue, you cannot trust it to give you the rest of mathematics. And it is not important that you "don't seem to appeal to the paradoxes".

Axiomatic set theories, like $\sf ZFC$, come to make an effort to at least fight the problems that come up with naive set theory.

Why should you really care about the axioms of $\sf ZFC$? You shouldn't. You should care about the fact that $\sf ZFC$ is sufficient to develop basic model theory, and create the rest of mathematics inside its universe. And this makes set theory like the back-end architecture of your CPU. Do you really care that your computer is using an Alpha back-end or a SPARC back-end? No. You care that it is able to run Chicken Invaders, or YouTube, or $\LaTeX$.

Of course, if you are interested in mathematics, then you should learn at least a little bit how this processor works. Much like learning programming involves understanding the operating system, or the hardware design. So you should care because you want to know how your mathematics is being modeled, but in general you should care because naive set theory is provably bad for you.

Answer 4

I think:

• This answer on MathOverflow is relevant.
• This paper is pointing out the counterexample.
• This paper is mending the original statement.

The theorem is in algebra, and is about whether the $\lim^1$ functor vanishes on Mittag-Leffler sequences in abelian categories satisfying certain axioms. In order to correct the mistake, the author had to add a smallness condition.

I also think that the non-existence of a free complete Boolean algebra on a countably infinite set of generators could be considered, though perhaps complete Boolean algebras are more explicitly "foundational".

Answer 5

There is a counterintuitive counterexample from the theory of inverse problems of a non-measurable conductivity on a disc that is not distinguishable from a homogeneous one w/boundary circle electrical measurements. It involves a function $f(r):[0,1]\rightarrow R^+$, such that $f(r^n)=f(r)$ for all natural numbers $n$. The construction of such a function involes a non-measurable set and the Axiom of choice. The inverse problems are relatively applied mathematics w/apllications e.g. to medical imaging.