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Do mathmatician ever prove that a theorem could not generalize into a much general theorem? Is there a historic mile-stone example refer to the above question?

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It's my feeling that any theorem can be generalized. There is also the old joke about a theorem so general it has no particular application. Examples are welcomed. – marty cohen Apr 7 '12 at 4:46
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This theorem is a nice example. – Quimey Apr 7 '12 at 4:49
I think this depends on what you put into the word generalize. There are often different ways to look upon things, one may for example try to generalize a statement within a universe, lift a statement into a larger universe or restrict a statement into a smaller universe. – AD. Apr 7 '12 at 5:59
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Why the two downvotes? Admittedly it's a rather open-ended question, but I do think there could be meaningful answers. – joriki Apr 7 '12 at 8:49
Suppose you have to theorems, $T_1$ and $T_2$. Although they seem to have nothing common, one can ask whether it is true if both $T_1$ and $T_2$ holds, and that is more general than $T_1$ or $T_2$. If you continue this and take all known theorems and ask whether all of them holds, I get an application of Russell's paradox. – Jaakko Seppälä Oct 23 '12 at 16:35

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