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I was just thinking, is it possible to prove that certain mathematical field (e.g. linear algebra) has a finite number of theorems? If yes, do all the areas of mathematics have finite number of theorems or are there some areas with infinite number of theorems?

Ana S. H.
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  • Are you referring to "potential theorems" (including those that could be proven, but haven't yet), or just "actual theorems" (ones that people have already proven)? –  Nov 02 '16 at 22:34
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    Very ill-posed question. Strictly speaking, there are infinite axioms (every natural number has a successor, that is a natural number), hence infinite theorems (if $10$ is a natural number, $12$ is a natural number too, since it is the successor of the successor of $10$). Additionally, how do you decide to which "branch" of Mathematics some theorem belongs to? Is the fundamental theorem of Algebra a topic in Complex Analysis or not? – Jack D'Aurizio Nov 02 '16 at 22:34
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    Since your questions isn't stated very precisely, I'd like to only give the following hint to what the answer you are looking for might be: Not only do all 'reasonable' mathematical fields have an infinite number of theorems, but - as a corollary of Goedel's theorems - they can always be 'refined into narrower fields' that can prove - using additional assumptions - theorems that couldn't be proved in the previous setting. – Stefan Mesken Nov 02 '16 at 22:37
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    @JackD'Aurizio I think, implicitly, part of the question is how one might make the question precise: if there is some meaningful way of formalizing the ideas of "branches of mathematics," "non-trivial results," and "meaningfully distinct results." I don't see a clear way to formulate it, but there's enough relevant stuff out there in mathematical logic that I can't rule it out, either. –  Nov 02 '16 at 22:39
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    It depends on your definition of "a theorem". If it is the logic theory definition, then no. But this definition isn't very interesting from the human perspective where "every integer is odd or even" and "every integer $> 2$ is odd or even" are actually the same theorem. For making the difference between those two statements, we need a mathematical definition of what is a "meaningful concept" (or, as MikeHaskel said, of what is "non-trivial"), and we don't have such (yet). – reuns Nov 02 '16 at 23:18
  • @MikeHaskel Many branches are easily formalized/codified. They have their own axioms which can then be proved of certain objects within ZFC. – Jacob Wakem Nov 02 '16 at 23:37

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Let $\phi_n$ be the statement $x=x \land \ldots \land x = x$ with $n$ occurrences of the equation $x = x $. Every $\phi_n$ is a theorem of first-order logic, so that gives you an infinite number of distinct theorems in any first-order logic.

To give a better answer to your question, we would need to have a good notion of when two theorems "mean the same thing". That is a difficult (and, in this context) probably unanswerable problem.

Rob Arthan
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    you have to explain why you gave this trivial answer, otherwise it is useless – reuns Nov 02 '16 at 23:22
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    @user1952009: I don't really understand your obnoxious objection, but I have elaborated my answer. If my answer is still "trivial" and "useless", please supply a better one (or explain why you are unable to do so). (Sorry about the offensive and imperative mode, but you started it!!!!) – Rob Arthan Nov 02 '16 at 23:36