First Order Peano Arithmetic (FOPA) Question (1 Viewer)

[sarah666]

Hello,
Can someone please please please help me on this question, a friend of mind in uni totally can't do it and she has an exam this thursday and she needs to learn how to do it to help her prepare for the exam. Can someone do this question to help her ?

Let A be a sentence of First Order Peano Arithmetic (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?


She 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 is not sure if this is the correct reasoning. But she can't do the rest and don't know the correct reasons to back it up as well . Can someone please please please help ? I would really appreciate your help.

 

[Affinity]

Which university/course is this for? you should really look for more suitable forums than this one

I assume your "A is true" means that the sentence A is true in the standard model
that is
N |= A where N denotes the standard model

you can't have
PA |-- A
PA |-/- A

in fact precisely one of

PA |-- A
PA |-/- A

holds for every sentence A.

if
PA |-- A

we must also have
N |= A
by completeness of the deductive system at this point we need to assume that PA is consistent (which could be proven, but not within PA)

but you can also construct a sentence A such that
PA |-/- A
PA |-/- ~A
N |= A (or if you negate this A, N |=/= ~A)
by incompleteness of the axioms

so the following are possible

TFT
FTF
FTT