Has it ever happened that a theorem of the form

If $P$, then $Q$

was proven and published, perhaps with great difficulty, only for someone to realize later that the condition $P$ of the theorem is never satisfied, or, worse, that the conclusion $Q$ of the implication is false?

For example, if the Generalized Riemann Hypothesis were disproved tomorrow, I would have a large supply of examples on my hands, as so many results are conditional on GRH. But surely, this must have happened before, in the long history of mathematics?

  • $\begingroup$ You make it sound as if the proof of "if P then Q" was wasted effort. But maybe the proof of the stronger result "if P then 0 = 1" is based on the proof of the partial result "if P then Q"? $\endgroup$ – bof Nov 18 '14 at 22:47
  • $\begingroup$ Yes, I agree with bof. If $Q$ is known to be false, then proving $P\rightarrow Q$ gives you some useful information that is exactly what you mention: $P$ never being satisfied. $\endgroup$ – Hayden Nov 18 '14 at 22:50
  • $\begingroup$ I agree with both of you that many interesting things could still arise out of such a situation, all of which could add to the charm of such an example. Presumaby, the effort would be completely wasted only if $Q$ were later proven to be false. $\endgroup$ – Bruno Joyal Nov 18 '14 at 22:54
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    $\begingroup$ Possible urban myth, S. T. Yau once said (he really said it, that's not the myth part, I was there) that there had been a Ph. D. dissertation on Holder continuous functions with exponent above $1.$ The trouble being that all such functions are constant. $\endgroup$ – Will Jagy Nov 18 '14 at 22:57
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    $\begingroup$ The second example in this answer seems to qualify. $\endgroup$ – Semiclassical Nov 19 '14 at 0:03

In the proof of FLT, was essential the fact that nontrivial FLT solution $\implies$ $\exists$ weird elliptical curve. Wiles proved that there isn't weird elliptical curve. See A question on FLT and Taniyama Shimura.

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    $\begingroup$ Dear Martín: This is not really what I am looking for. In this case, what would be the theorem which turned out to be vacuous? See Will's comment for an example of the type of situation that I am curious about. Best, $\endgroup$ – Bruno Joyal Nov 18 '14 at 23:48

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