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Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms (like ZFC) such that all that can be proved is proved ?

please do not include the issue of undecidable problems within a set of axioms (like continuum hypothesis and ZFC) in your answers, all I am asking is whether the number of theorem that can be proved (decidable) is finite or infinite within the realm of a certain set of axioms (for instance ZFC) ?

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    $\begingroup$ Hint: 1+1=2, 2+1=3, 3+1=4, 4+1=5, etc. are all theorems. $\endgroup$ Aug 6 '14 at 15:17
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If $\varphi$ is provable, then $\varphi\land\varphi$ is also provable. Therefore there are infinitely many theorems which are provable from any theory (consistent or inconsistent, any consistent theory proves at least valid statements, and inconsistent theories simply prove everything).

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