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Provably total functions of an arithmetical theory is one of the tools used in proof theoretic analysis of theories.

I am looking for early history of its development. In particular,

Where was the first uses of this concept?
Who coined the term "provably total functions"?

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One of the references I would recommend is P. Odifreddi's Classical Recursion Theory Volume II. On Page 324-326, it contains many references about the development of provably total functions (if you are seeking for the development within the Peano Arithmetic).

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Thanks. I will have look but AFAIR it does talk about the development of this one. – Kaveh Jun 28 '12 at 17:00

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