I want to know what does it mean when we say for example $$f(x)=2^x$$ is provably total in Peano arithmetic? Also what's the diffrence between provably total and provably recursive?
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You can take a look at the answer to this question: http://math.stackexchange.com/questions/177535/learning-how-to-prove-that-a-function-cant-proved-total – Mohsen Shahriari May 30 '15 at 10:00