I am new to logic but I believe this is not a difficult problem, yet I am still 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 PAT proves T(Godel numeric number of A) iff 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.
Notation: I mean T(Godel numeric number of A) as T() where the thing inside the bracket is the usual top left and top right square corner of A, hope it is clear.
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!