Not true.
From PA we can construct a function that can search all possible proofs that can be constructed in PA. In fact I outlined one way to do this at the end of my answer.
With this function, we can construct a function will-return that analyzes whether a given function, with a given input, will return. This is kind of like an attempted solution to the Halting Problem. We know that it doesn't always work. But we also know that it works a lot of the time.
From will-return we can create a function opposite-return that tries to return if a given function with a given input would not, and doesn't return if that function would. This construction is identical to the one in the standard proof of the Halting Problem.
Now we consider (opposite-return opposite-return opposite-return). (Actually you need a step to expand the argument into this recursive form. I've left that out, but that is identical to the one in the standard proof of the Halting problem.)
PA can prove the following:
- PA proves that if PA can prove that opposite-return returns, then it doesn't. - PA proves that if PA can prove that opposite-return doesn't return, then it does. - PA proves that if it can prove everything that it proves that it can prove, then PA must have a proof of one of the two previous statements. - Therefore PA proves that if it can prove everything that it proves that it can prove, then PA is inconsistent.
This is a form of Gödel's second incompleteness theorem.
And, therefore, there must be a distinction to be made between "PA proves" and "PA proves that it proves".