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Proving a hypothesis often requires the development of new and powerful techniques and maybe even new branches of mathematics.

However disproving such a hypothesis could result from a single counterexample.

Are there any existing non-trivial mathematical hypotheses that cannot, even in principle, be disproved by counterexample?

Note

By existing, I mean that have been published or discussed in reputable mathematical journals.

By non-trivial, I mean hypotheses that are not explicitly designed for the purpose of not being susceptible to counterexample.

chasly - supports Monica
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    "For all sufficiently large $n$, $2n$ is the sum of two primes". – lulu Feb 26 '23 at 16:13
  • @lulu - surely you could could find a counterexample by exhaustive search on a particular 2n – chasly - supports Monica Feb 26 '23 at 16:15
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    But it wouldn't matter, because that would just prove that your example wasn't "sufficiently large". – lulu Feb 26 '23 at 16:17
  • Maybe it would be clearly to you if it was phrased, equivalently, as "there are only finitely many natural numbers $n$ such that $2n$ is not the sum of two primes." – lulu Feb 26 '23 at 16:18
  • What is your definition of sufficiently large? – chasly - supports Monica Feb 26 '23 at 16:18
  • "all but finitely many $n$". – lulu Feb 26 '23 at 16:19
  • @lulu - Hmm, I see. Can you suggest a modification to my question that excludes deliberately formulated stumbling blocks? P.S. I am not a mathematician. – chasly - supports Monica Feb 26 '23 at 16:20
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    There's nothing odd or forced about my example. Lots of interesting results in number theory have a small (finite) list of exceptions. Note, of course, you could disprove my hypothesis by giving an infinite list of examples (if, say, you were able to prove that $2\times 10^k$ was never expressible as the sum of two primes). But you will need an infinite list. – lulu Feb 26 '23 at 16:28
  • Maybe https://math.stackexchange.com/questions/710950/prove-that-a-counterexample-exists-without-knowing-one is what you actually are looking for. – Moishe Kohan Feb 26 '23 at 16:31
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    Existence-Theorems can usually not be disproven by counterexample. Say e.g. you have a function $f:\mathbb{R}\longrightarrow \mathbb{R}$ then the statement "There is an $x\in \mathbb{R}$ s.t. $f(x) = 0$" can not be disproven by counterexample. Actually only theorems which make statements about every object of some type can be disproven by counterexample. – StiftungWarentest Feb 26 '23 at 16:33

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Working in ZFC,

The Continuum Hypothesis is a famous example.

That there is no set with Cardinality greater than the naturals, and less than the Reals.

Michael Carey
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