Arriving at the same result with the opposite hypotheses

Question

I am pretty distant from anything analytic, including analytic number theory but I decided to read the Wikipedia page on the Riemann hypothesis (current revision) and there is some pretty interesting stuff there:

Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, (Ireland & Rosen 1990, p. 359) say

The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!

What is surprising is that both a statement and its negation are useful for proving the same theorem.

Do similar situations arise with other major, notorious conjectures in mathematics? I only care about algebraic geometry and algebraic number theory for the most part but I guess it will make little sense to have such questions devoted to each area of mathematics so post whatever you've got.

To give an initial direction: are there any interesting statements one can prove assuming both some hard conjecture about motives (e.g. motivic $t$-structure, the standard conjectures, Hodge/Tate conjectures) and its negation?

Answer 1

I think this is much less surprising today. It's not uncommon to argue that a statement about some structures is true because one can decompose structures into random-like and structured parts ('structure and randomness'), and in either case get to the desired conclusion. Usually one is trying to prove that a certain statement is true for a whole class of structures by this method; there is no one special structure for which we really want to prove it. Take Roth's theorem as an example: we want to prove any positive-density set of integers in $N$ (for some large $N$) contains a three-term AP. We can do this if the set looks random (small Fourier coefficients) easily enough, but if the set does not look random, then there is a large Fourier coefficient, so the set looks structured, namely it has a noticeably larger density on some long AP than on $[N]$. But then we can pass to the AP and iterate this argument.

It just happens that when we try to prove things about the primes, we really only want to know about the one structure, contained in a class of structures with prime-like properties (which might not have more than one member, depending on the properties used in the proof). But the proof method is still to show that the whole class of structures satisfies the desired conclusion.

In this case (G)RH is a statement that the primes behave in some random-like way, in a fairly strong quantitative sense. And so its negation is the assertion of some structure beyond the obvious - which of course can be useful.

Of course, if you want to find other examples of theorems which you can prove by appealing to some major open problem being either true or false, you should probably have some reason why either outcome helps. Quite a few major open problems (or major theorems) can be read as some kind of quasirandomness statement, and for that there is a reason why the conjecture failing might be useful, so in principle there should be a decent list of possible candidates.

Answer 2

Theorem. The Stone-Čech compactification of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two parts:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c\leq\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The assumptions $\neg$CH and CH are critical to the constructions. Also the types of continua constructed are very different in Case 1 vs Case 2. In fact, all continua of the type constructed for Case 1 are homeomorphic under CH.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting is that both cases are necessary to this proof because CH and $\neg$CH are each consistent with ZFC. This may be different from the use of RH and $\neg$RH, or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then you could throw away the other half of your proof.

Answer 3

This paper proves the existence of an essentially-deterministic algorithm for the following problem:

Input: an integer $n$

Output: An $n$-bit prime.

Which runs in time sub-exponential in $n$ for infinitely many inputs.

On a high level the argument works as follows:

1. Suppose a certain kind of pseudo-random generator exists. Then we can derive an algorithm to compute the above.
2. Suppose that kind of pseudo-random generator does not exist. Then we have a collapse of complexity classes $\mathsf{PSPACE} = \mathsf{ZPP}$ which then implies some circuit lower bounds, which in turn yields a pseudo-random generator with the desired properties.

Answer 4

Here is a baby answer.

Theorem: There exist two irrational numbers $p,q$ such that $p^q$ is rational.

Proof: In case $\sqrt 2^{\sqrt 2}$ is rational we can take $p=q=\sqrt 2$.

Otherwise take $p= \sqrt 2^{\sqrt 2}$ and $q=\sqrt 2$. We have $p^q = \big(\sqrt 2^{\sqrt 2}\big)^{\sqrt 2} = \sqrt 2^{\sqrt 2 \cdot \sqrt 2} = \sqrt 2^2=2$ is rational.

Answer 5

Every proof by contradiction can be seen as following the template identified in the theorem.

Namely, when we've proved a statement $S$ by contradiction, then $S$ follows from $S$ and also from $\neg S$.