# 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“?

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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/… –  Michael Joyce Apr 5 '13 at 12:53
I would go with a neutral term. I suggest "claim". –  arsmath 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." –  Lee Mosher 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. –  François G. Dorais Apr 5 '13 at 14:52
how about "conjecture"? –  Toink Apr 6 '13 at 16:30
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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.

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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)

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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. –  Henry Cohn Apr 5 '13 at 17:05
Dear Henry, I did not mean this to be a serious suggestion; but I found this unbelievably funny. –  Angelo Apr 5 '13 at 17:19
Looks like it's theorem number $-4$. –  Noam D. Elkies Apr 5 '13 at 19:17
To Noam: ok, I replaced the $-$ with a long dash. They look identical to me. –  Angelo Apr 6 '13 at 0:59
Nice idea for a talk, but I doubt that it is a suitable style for a journal. –  Joachim Breitner Apr 8 '13 at 10:46