Is the set of all theorems countable or uncountable?
Maybe its a stupid question. I just wanted to know. I am led to think that since, we use a finite set of symbols and English letters, the set of theorems is countable.
EDIT: The set of theorems is not just a set of linguistic combinations, but a set of mathematical truths.
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Do you think mathematic is finite? I think the number of mathematical theorem is infinite but countable. Did you mean that we count the number of letters in the theorems, we do not count the number of the theorems? For me, it does not make sense.
Ps. Somebody voted down for this answer, may be there is a misunderstanding here. I did not understand the question clearly, so I asked the author again for sure and I gave my thinking on it. Sorry for my poor English.
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If you have countable many theorems $T_1,T_2,T_3...$, you can construct an uncountable set of new theorems by stating that every subset of {$T_1,T_2,T_3...$} is a theorem.
Additionally there are true finite statements which in a given formal system is only provable by a countable infinite number of theorems. (eg. Goodstein's theorem in PA).
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If by "theorem" you just mean any mathematical truth, it doesn't seem too hard to figure out that the number of mathematical truths is uncountable. For the real numbers, in addition to c=c as Henning points out, you can consider (x+y)=z (or any other binary operation on the reals). There exist uncountably many triples (x, y, z) for which "(x+y)=z" is true, where x, y, and z all belong to the set of all real numbers. Perhaps more simply, consider x<4. There exist uncountably many real numbers we can put in for "x" which makes x<4 into a true statement, so there exist uncountably many mathematical truths of the form x<4. This all takes for granted that say 2.9<4 qualifies as a mathematical truth, which at the very least I would believe it hard for anyone to seriously deny as a mathematical truth, no matter what you mean by that term exactly.
A theorem is, by definition, a finite string of symbols that can be derived by some specified proof system. Because there are countably many finite strings of symbols, there are at most countably many theorems in any given theory.
Speaking of "a set of mathematical truths", where the elements of that set is supposed to be something different from symbolic representations, is not well defined. What kind of object is "a mathematical truth" to you such that you can put them into a set and count them? There is no definition of such a thing in general use, except for the formulas of symbolic logic.
In model theory one can speak of theories where the set of possible symbols are uncountable, such as a symbol for each real number. Such a theory, of course, has uncountably many theorems of the form $c=c$. However, these theories are generally considered artificial objects of study. Studying them can be useful as a stepping stone in proving things about ordinary countable theories, but their formulas are not considered to directly represent "mathematical truth".