Meaning of Provable recursiveness

Is there any difference between provably total function and provable recursiveness of a function in first order PA ?

From provably total I mean that the totality of the function itself is provable in the theory... I know this is true of every total recursive function in PA , it can prove the totality of a given total recursive function..

But I get confuse with the term provable recursiveness when reading about it in Stanford Plato website...

There is some variance of usage, but usually "provably total" and "provably recursive" are indeed synonymous. That said, this old answer of mine for a case where they aren't! Always check the precise definition in your source. Basically, the key point is that we usually restrict attention to "low-complexity" defining formulas, which prevent a silly trick from making all total recursive functions provably total via "pathological" definitions.

That said, there's a fundamental error in your question:

"I know this is true of every total recursive function in PA"

This is false, at least when we restrict attention to low-complexity definitions (which is usually done per the above): any total recursive function $$f$$ is (strongly) representable in PA, but that does not mean that PA can prove that $$f$$ is total. All PA can do is prove, for each specific $$n$$, that $$f(n)$$ is defined; PA cannot in general prove "For all $$n$$, $$f(n)$$ is defined." (This is exactly analogous to how PA can prove, for each specific $$n$$, that there is no contradiction in PA involving at most $$n$$ symbols, but PA cannot prove "For all $$n$$ there is no contradiction in PA involving at most $$n$$ symbols.")

• Can you suggest any easy to follow proof/material stating there are total recursive functions not provably total in PA..., I didn't knew about it before... I need to check it ... I actually must check it...
– user1203803
Commented Jan 23 at 18:15
• Also I still can't understand how will you represent the predicate of contradiction proofs in PA, can you show me that predicate which is true if there isn't a contradiction proof less than x in PA ?
– user1203803
Commented Jan 23 at 18:18