# What are some examples of theorem requiring highly subtle hypothesis?

I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almost absent from the proof but which are actually hidden behind some really abstract or technical argument. It would be even more interesting if this unnoticed hypothesis was not noted at first but later it had to be added in another paper or publication not because the proof of the theorem were wrong but because the author did not notice that this or that condition was actually playing a role behind the scene and needed to be added. And, finally, an extra point if this hidden hypothesis led to some important development or advance in the area around the theorem in the sense that it opened new questions or new paths of research. This question might be related with this other but notice that it is not the same as I am speaking about subtleties in proof that were not exactly incorrect but incomplete in the sense of not mentioning that some object or result had to be use maybe in a highly tangential way.

In order to put some order in the possible answers and make this post useful for other people I would like you to give references and to at least explain the subtleties that helps the hypothesis to hide at a first sight, expose how they relate with the actual proof or method of proving, and tell the main steps that were made by the community until this hidden condition was found, i.e., you can write in fact a short history about the evolution of our understanding of the subtleties and nuances surrounding the result you want to mention.

A very well known and classic example of this phenomenon is the full theory of classical greek geometry that although correctly developed in the famous work of Euclides was later found to be incompletely axiomatized as there were some axioms that Euclides uses but he did not mention as such mainly because these manipulations are so highly intuitive that were not easy to recognize that they were being used in an argument. Happily, a better understanding of these axioms and their internal respective logical relations through a period of long study and research lasting for millennia led to the realization that these axioms were not explicitly mention but necessary and to the development of new kinds of geometry and different geometrical worlds.

Maybe this one is (because of being the most classic and expanded through so many centuries and pages of research) the most known, important and famous example of the phenomena I am looking for. However, I am also interested in other small and more humble examples of this phenomena appearing and happening in some more recent papers, theorems, lemmas and results in general.

Note: I vote for doing this community wiki as it seems that this is the best way of dealing with this kind of questions.

• To make the question community wiki, you can flag your own post for moderator attention. ('flag' is one of the 4–6 small grey options at the bottom of the post text.) – LSpice Jun 1 '20 at 3:30
• I have the sense that the spectral theorem for unbounded operators is perhaps a good example, with the "subtle hypothesis" being self-adjointness (not merely symmetric). But I don't know the history well enough to write a proper answer. – Nate Eldredge Jun 1 '20 at 4:22
• Probably a lot of the basic results in analysis require somewhat "subtle" hypotheses, as evinced by the famous (mythical?- see mathoverflow.net/questions/176693/…) story of how Cauchy got tripped up by pointwise vs. uniform convergence – Sam Hopkins Jun 1 '20 at 4:35
• Just as an observation, this is essentially asking for a particular kind of badly written mathematics, which one would hope a referee picks up. I've seen examples of this; maybe the worst being a statement 'proved by induction' followed by a calculation. In this example the n=-1 and 0 cases of the induction were trivial, and the n\ge2 cases were proved by induction with the given calculation. And the calculation did not work in the n=1 case; here one had to notice that this was a hypothesis of the lemma. The final version of this paper made this point explicit. – user36212 Jun 1 '20 at 9:13
• The Axiom of Choice was (unconsciously) used in tons of proofs before it was realized to be an axiom. – BlueRaja Jun 1 '20 at 23:52

## 7 Answers

The convergence conditions for the Fourier series of a function $$f:S^1 \to \mathbb{R}$$ are a good example. The investigation of convergence conditions for Fourier series was a major motivation for Cantor's set theory and Lebesgue's measure theory. Depending on what kind of convergence you want, the conditions can be very subtle. For example, if you want the Fourier series of a continuous function to converge pointwise everywhere, then I don't think that there is any nice set of necessary and sufficient conditions known. Various sufficient conditions are known, e.g., the Dirichlet conditions, which are fairly subtle.

Nowadays, I think it is generally considered that asking for convergence everywhere is the "wrong question"; one should ask for convergence almost everywhere. Then the most famous theorem is Carleson's theorem that the Fourier series of a function in $$L^2$$ converges almost everywhere. The hypothesis here is easy to state, but the way that the hypothesis is used is subtle. There are various proofs known now but none of them is easy. Note for example that Kolmogorov's first paper gave an example of a function in $$L^1$$ whose Fourier series diverges almost everywhere.

• Do you have a reference for this investigation being a motivation for Cantor's set theory? – Wojowu Jun 1 '20 at 13:40
• @Wojowu : It's sort of clear if you just look at Cantor's papers, where he started off studying Fourier series, and naturally segued into set theory. But if you want a historical account that spells this out, see for example this and this. That's what I found after a quick search; there may be better references. – Timothy Chow Jun 1 '20 at 15:20
• Interesting, I had no idea this is what sparked Cantor's work! – Wojowu Jun 1 '20 at 15:44
• @Wojowu: See The trigonometric background to Georg Cantor's theory of sets by Joseph Warren Dauben. Among other things, Cantor proved the uniqueness theorem for trigonometric series, which says that if two trigonometric series agree for all $x \in [-\pi,\,\pi],$ then their corresponding coefficients are equal (i.e. the two trigonometric series are the same). (continued) – Dave L Renfro Jun 2 '20 at 20:12
• Cantor then managed to prove this in which a finite exceptional set is allowed (i.e. change "for all" to "for all but possibly finitely many"). After this, Cantor managed to allow certain types of infinite exceptional sets, first to sets each of whose set of limit points is finite (I'm not sure if this was separately published), and eventually to sets each of which after a finite application of the operation of taking limit points leaves a finite set (equivalently, leaves an empty set, by taking one more limit point operation). From this it's a small step to ordinal numbers (I'm out of space). – Dave L Renfro Jun 2 '20 at 20:17

There is Euler's formula $$V - E + F = 2.$$ Today, we might not think of the hypotheses as being especially tricky. But Lakatos's classic Proofs and Refutations makes an entertaining case for its subtlety.

If Lakatos does not convince you, consider Euler's Theorem for Tilings. Suppose we have a tiling of the plane; take a finite portion of it, apply the standard Euler formula, and divide by $$F$$. Intuitively, as we take larger and larger portions, $$V/F$$ and $$E/F$$ approach limiting values $$v$$ and $$e$$ respectively, and we obtain Euler's Theorem for Tilings: $$v - e + 1 = 0.$$ However, even if the limits $$v$$ and $$e$$ exist, they do not necessarily satisfy Euler's Theorem for Tilings unless the tiling satisfies certain subtle hypotheses. For example, in the heptagonal tiling below (taken from Grünbaum and Shephard's book Tilings and Patterns), the heptagons get skinnier and skinnier as one moves out from the center, creating a "singularity at infinity." It is not hard to see that $$v=7/3$$ and $$e=7/2$$, so $$v-e+1 = -1/6$$ and not zero. In the notes to Chapter 3, Grünbaum and Shephard write:

Euler's Theorem for Tilings and its various corollaries are often quoted and used—usually without any indication of restrictions that must be imposed on a tiling to give meaning and validity to this procedure. In contrast to many other cases—in which a cavalier attitude towards mathematical rigor is an aesthetic shortcoming that does not affect the outcome—here many authors have claimed to have proved statements that are actually false. As recent examples we may mention Walsh (Geometriae Dedicata 1 (1971), 117–124) and Loeb (Space Structures: Their Harmony and Counterpoint, especially Chapter 9).

This is one which I've seen trip up a number of students when first learning the material: the hypothesis of admissibility (or acceptability - I learned the latter, but the former seems more common) in the context of numberings of unary partial computable functions (or equivalent objects like c.e. sets).

Results like Rice's Theorem and the Recursion Theorem are generally presented for a specific numbering whose details are quickly forgotten; the motto "all reasonable numberings work the same" is introduced somewhere around this point, and is mostly true. However, the right notion of "reasonability" is not usually obvious, since presentations tend to focus on the following two features of the canonical numbering $$\Phi:=(\varphi_e)_{e\in\mathbb{N}}$$:

• The numbering construed as a partial binary function $$\langle e,x\rangle\mapsto\varphi_e(x)$$ should itself be computable.

• For every unary partial computable $$f$$ there should be some $$e$$ with $$f\simeq \varphi_e$$.

By themselves these properties are not enough to get the standard results to apply: the usual extreme counterexample is a Friedberg numbering, which is a numbering satisfying the two properties above such that every partial computable $$f$$ has exactly one index (so Rice's Theorem and the Recursion Theorem each fail basically trivially).

Instead, we need to strengthen the second bulletpoint above as follows:

• (Admissibility/acceptability): For every binary partial computable $$f$$ there is some total computable unary $$g$$ such that for each $$e$$ we have $$f(e,-)\simeq \varphi_{g(e)}.$$

This amounts to a kind of "universality" of the numbering in question; roughly speaking, every other numbering needs to be translatable into it. This turns out to be exactly what we need to deduce all the basic results about the usual numbering, and indeed so far as I'm aware there really are no essential differences between admissible numberings. Moreover, once this sort of universality occurs to us as something important we're led to consider general comparisons between numberings of various systems, and this leads to several interesting topics (see especially Rogers semilattices).

• I was unfamiliar with most of this — this is a wonderful example, thank you! – Steven Stadnicki Jun 2 '20 at 17:30
• In fact, I would expect most uses of the term "admissibility" in math describe conditions that are at least a little bit subtle. – Will Sawin Jun 2 '20 at 21:39
• This reminds me that in computational complexity theory, hierarchy theorems need a hypothesis that the functions in question be constructible or proper. Otherwise weird things can happen, e.g., the gap theorem. – Timothy Chow Jun 12 '20 at 13:48

This example has been mentioned elsewhere on MO but seems worth reproducing here. The abstract of Amnon Neeman's paper A counterexample to a 1961 “theorem” in homological algebra says:

In 1961, Jan-Erik Roos published a “theorem”, which says that in an [AB4∗] abelian category, lim1 vanishes on Mittag–Leffler sequences. … This is a “theorem” that many people since have known and used. In this article, we outline a counterexample. We construct some strange abelian categories, which are perhaps of some independent interest.

It turns out that the theorem can be repaired by adding some relatively weak hypotheses that are usually satisfied in practice. That the need for such hypotheses apparently went unnoticed for so long is perhaps evidence that they are "highly subtle."

• Interesting! Does someone know if the hypotheses needed for this result to be true were actually used unnoticeably in the 'proof' of Jan-Erik Roos? – Hvjurthuk Jun 2 '20 at 18:24
• I asked Prof. Neeman, and he responded, "Yes. As long as one can guarantee that the map from the inverse limit to the $n\,$th term in the sequence has the same image as the map from the $N\,$th term (for all $N\gg n$), then Roos' argument works. So the counterexample amounts to constructing an abelian category, and inside it an inverse sequence of epimorphisms whose inverse limit vanishes. It's counter-intuitive, but turns out to exist." – Timothy Chow Jun 2 '20 at 22:47

This is not a perfect example because the subtle hypotheses in question were not "unnoticed"; nevertheless I think it fulfills several of your other criteria. Let us define the "Strong Fubini theorem" to be the following statement:

If $$f:\mathbb{R}^2 \to \mathbb{R}$$ is nonnegative and the iterated integrals $$\iint f\,dx\,dy$$ and $$\iint f\,dy\,dx$$ exist, then they are equal.

The Strong Fubini theorem looks innocent enough, but without any measurability hypotheses, it is independent of ZFC. For example, Sierpinski showed that Strong Fubini is false if the continuum hypothesis holds.

In the other direction, a paper by Joe Shipman investigates a variety of interesting hypotheses that imply Strong Fubini, e.g., RVM ("the continuum is real-valued measurable"), which is equiconsistent with the existence of a measurable cardinal. Here's another one: Let $$\kappa$$ denote the minimum cardinality of a nonmeasurable set, and let $$\lambda$$ denote the cardinality of the smallest union of measure-zero sets which covers $$\mathbb{R}$$. Then the assertion that $$\kappa < \lambda$$ implies Strong Fubini.

Theorem. Assuming the axiom of choice, the countable union of countable sets is countable.

Proof. Let $$\{A_n\mid n\in\Bbb N\}$$ be a family of countable sets, and so we can write $$A_n$$ as $$\{a_{n,m}\mid m\in\Bbb N\}$$.

Let $$A$$ be the union, and define $$f(a) = 2^n3^m$$ such that $$n$$ is the least such that $$a\in A_n$$, and $$a=a_{n,m}$$. Easily, this is an injection so the union is countable.

The trained eye, of course will notice the use of the axiom of choice immediately. We choose an enumeration of each $$A_n$$. But this is very subtle and usually people will not notice that at first.

And of course, this use of choice is necessary. Indeed, it is consistent that the real numbers are a countable union of countable sets! (Still uncountable, though.)

• "Usually people will not notice that": As far as I know, "the countable union of countable sets is countable" was proved by Cantor, and his proof was accepted by the community, long before Zermelo pointed out the axiom of choice. – Andreas Blass Jun 2 '20 at 22:28
• Well, hardly gets any subtler than that. – Asaf Karagila Jun 2 '20 at 22:29
• A related example: The result that $\varepsilon$-$\delta$-continuity implies sequential continuity in metric spaces has some use of choice. – Qi Zhu Jun 9 '20 at 8:27
• @QiZhu: Indeed. Most indeedity. – Asaf Karagila Jun 9 '20 at 8:45

Some of Euclid's theorems rely on axioms of betweenness that he was not aware of.

Hilbert's axioms: https://www.math.ust.hk/~mabfchen/Math4221/Hilbert%20Axioms.pdf

• This is already mentioned in the question itself. – Emil Jeřábek Jun 2 '20 at 14:38