Can we add to PA a new predicate T such that for every sentence A of the old vocabulary the new theory proves $T(\ulcorner A\urcorner)\equiv A$

Question

I believe this is not a difficult problem, but I am soo confused, and the reason for that is because there are so many gaps in my knowledge or maybe I have overlooked so many "obvious" argument. I truly appreciate any explanations.

I am thinking of a problem whether we can add to Peano Arithmetic a new predicate T such that for every sentence A of the old vocabulary, the new theory $PA^T$ proves $T(\ulcorner A\urcorner)\equiv A$. In other words, can we consistently extend Peano Arithmetic with a truth predicate for sentences in the old vocabulary? I am trying to find any ways to show we can or show why is it impossible.

My reasoning might be too short or maybe even incorrect, but I will try my best:

We cannot consistently extend Peano Arithmetic with a truth predicate, since consistent deductively defined extensions of Peano Arithmetic are incomplete, so the predicate might be neither true nor false.

I am really doubtful about my approach, I will really appreciate any helps! Thanks!

Answer 1

You can add truth predicates over the old language, but you can't add a predicate such that it's a truth predicate for all $A$ in the current language.

Answer 2

There is a substantial literature on adding truth-predicates to arithmetics developing the fundamental facts that Alex Kocurek notes. A very useful, and freely accessible, entry-point is provided by Volker Halbach in his encyclopedia essay http://plato.stanford.edu/entries/truth-axiomatic/