# How to refer to a theorem that you have shown to be wrong

I am writing a paper about a flaw that I found in a published paper. There, the statement is called “Theorem 2”. In my paper, I am reproducing the other paper’s definitions, and steps leading towards that statement, and now I’d like to reproduce the statement, immediately followed by the counter example that I found.

I am tempted to reproduce the statement labelled and styled as a theorem in my paper as well, so that the reader can easily find and recognize it, and so that I can continue to refer to it as “Theorem 2”. But is that really valid, given that only correctly proven statements are, by definition of theorem, theorems? Or can there be such things as “false theorems“?

• The assertion formerly known as Theorem 2? With some LaTeX work, you could then use the Prince symbol to refer to it in the paper. en.wikipedia.org/wiki/… Apr 5 '13 at 12:53
• I would go with a neutral term. I suggest "claim". Apr 5 '13 at 12:58
• There is a practical matter, which is how to refer to the result in an unambiguous matter. The reproduction you have in mind solves that problem in one way. But another solution is: first state the problem as a question, then say "Paper X states in Theorem 2 that the answer is 'Yes'. We prove here that the answer is 'No' by giving an explicit counterexample." Apr 5 '13 at 14:24
• I don't know the exact context and Joachim is probably already doing this, but I want to mention that it is customary to indicate how to patch the claimed "theorem". After all, a purported "proof" that has been published is unlikely to be completely wrong, the authors probably missed a hypothesis or failed to verify one part of the conclusion or something like that. Indicating how to fix that and proposing a counterexample to indicate how the patch is necessary is the proper thing to do. Apr 5 '13 at 14:52
• how about "conjecture"? Apr 6 '13 at 16:30

You can have a look to the paper "A counterexample to a 1961 “theorem” in homological algebra" by Neeman and use his style. By the way, I think that the paper is very very good.

I once saw a mathematician giving a talk about a theorem that he thought he had proved, for which a counterexample had later been found. He stated the "result" as follows:

Theorem (1983–1987): Let $A$ be $\dots$

(I made up the dates of birth and death)

• This might be confusing to someone stumbling across it in a paper (it might be read as "this theorem was proved between 1983 and 1987 but published later"), but I love it. Apr 5 '13 at 17:05
• Dear Henry, I did not mean this to be a serious suggestion; but I found this unbelievably funny. Apr 5 '13 at 17:19
• Looks like it's theorem number $-4$. Apr 5 '13 at 19:17
• To Noam: ok, I replaced the $-$ with a long dash. They look identical to me. Apr 6 '13 at 0:59
• Nice idea for a talk, but I doubt that it is a suitable style for a journal. Apr 8 '13 at 10:46

In my opinion, it would be a bad idea to label statements known to be false as theorems. If you really want to do this, maybe you could put inverted commas around the word "theorem", to indicate you explicitly cast doubt on its following from axioms by applying rules of deduction. Or you could call it an Assertion, followed by a bold declaration that the assertion is false and the demonstration of such.

• I vote for inverted commas. We label things to make them easy for the reader to find. Inverted commas preserve the exact reference while also making it crystal clear that the result is not to be trusted. Apr 5 '13 at 14:27
• Ironically the original sense of "theorem" allows for a theorem to be false, since it just meant "assertion". So Euler could say that a theorem must be true even though he cannot prove it. But yes, modern usage doesn't allow for this. Apr 5 '13 at 19:21

Another example is the paper "Affine semigroups and Cohen-Macaulay rings" by Hoa and Trung (Trans. AMS 298, 1986) in which they give a counterexample to a result of Goto and Watanabe. Rumor has it that the counterexample was known before their paper, but in any case, they use "counterexample to the result".

It might be that the proof in the paper you refer to did not prove the whole claim, but proved something weaker but still nontrivial. (Imagine for example, a statement of the Weierstrass extreme value theorem without assuming that the domain is not empty. Probably the proof covers the non empty domain case.) In that case, I could try to state a theorem with the additional assumption and mention in a remark or in a footnote that the statement is weaker than the one in the source and that the assumption I have added was needed at some point in the proof I precise.

By the way, when I find a counterexample to a published result, I find it a good practice to identify where the published proof is failing. (Most of the time, proofs fail in the parts left to the reader, but you can still find a counterexample to an intermediate statement in a proof.)