Can someone please please please help me on this question, I totally can't do it
Let A be a sentence of FOPA (i.e., a formula with no free variables). Consider the following three assertions:
FOPA l- A (Means is A), FOPA ~I- A (Means is not A), A is true.
On the face of it, there are 8 possibilities for these assertions, namely
TTT, TTF, TFT, TFF, . . . , FFF, where, for example, TFF means it is true that FOPA I- A, it is false that FOPA ~A, and A is not a true statement in number theory.
For each of the 8 possibilities, is there such a sentence A?
If so, find one. If not, why not?
I know for the first possibility, TTT, it is not possible because if FOPA can prove A is true, Not A is true and the statement is true. It is not possible because FOPA cannot both prove something that is right and something as wrong too. So TTT is not possible but I am not sure if this is the correct reason for it. But I can't do the rest and I don't know the correct reasons to back them up. Can someone please please please help me ? I would really appreciate your help.