Is the Goedel sentence of Peano arithmetic provable in true arithmetic?

Question

I'm struggling to answer the following past paper question, which does not have any solution.

It asks simply whether $TA \vdash \gamma_{PA}$, where $TA$ is true arithmetic and $\gamma_{PA}$ is the Goedel sentence of Peano arithmetic.

A previous question asked whether $PA \vdash \gamma_{PA}$ holds. It is clearly not because PA is consistent and axiomatizable, and no such theory proves its own Goedel sentence.

However, for this question, I'm unsure what tools we have available since all theorems are about a theory and their own Goedel sentence. There is some way of showing that if it were to hold, then PA would prove its Goedel sentence. I can see intuitively why that might be the case, but unsure how to show it is.

Any suggestions or pointers?

Answer 1

If $PA$ is consistent, then $\gamma_{PA}$ is true in standard model of natural numbers (as there is no number encoding proof of $PA$ inconsistency), so $TA \vdash \gamma_{PA}$ (as $\gamma_{PA}$ is even axiom of $TA$).

Answer 2

It‘s a mean examiners' trick [which I've used myself!] to set a bunch of related questions, slipping in one which is just there to check understanding -- i.e. a question where the examiner isn't looking for a fancy proof, and is inviting you to show that you realise just why a fancy proof isn't required!

This is a case in point. What is True Arithmetic? The theory which consists in all true sentences of the language of (standardly) first-order PA. So to say that $TA \vdash \gamma_{PA}$ is in effect just to say that $\gamma_{PA}$ is true. Which it is.