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The completion of the classification of finite simple proofs was first announced in 1983. However, as late as 2008 minor gaps were still found and closed.

How certain can we be that

  1. The proof of the classification theorem is correct
  2. Even though there may be mistakes in the proof, at least the result is correct, i. e. our list of finite simple groups is the right one?
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Personally, I feel assuming classification should still be a tentative business until a classification program more amenable to widespread peer review manifests in a later-generation proof. Not that I can stand anywhere near the experts in perspective on the group-theoretic details, I just believe in human fallibility's tendency to increase proportional to sheer extent of work and effort, and inversely to the level of open peer review feasible. –  anon Apr 1 '13 at 23:44
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Here are two relevant MathOverflow questions. –  Zev Chonoles Apr 1 '13 at 23:45
    
I wouldn't bet the farm on it. –  Baby Dragon Apr 1 '13 at 23:45
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It would not be too surprising to find a gap or a mistake in the thousands of pages that are involved in the classification. But I'm pretty sure most group theorists would agree that if someone discovered a new finite simple group now, it would be extremely shocking! –  spin Apr 2 '13 at 13:24
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We're pretty sure. This is a good read on the status by Aschbacher. At this point, we're trying to simplify the proof because it's pretty messy and not very elegant, not because we aren't sure that it's correct. –  Alexander Gruber Apr 2 '13 at 16:16

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