primitive recursively axiomatized consistent extension of $PA$. Give sufficient conditions to make statement true

Question

Problem:

Let $T$ be a primitive recursively axiomatized consistent extension of $PA$. Under what conditions are each of the following statements true?

1. If $T\vdash\Phi$ then $T\vdash Prov_T([\Phi])$.

2. $T\vdash\Phi$ $\rightarrow $ $Prov_T([\Phi])$.

3. If $T\vdash Prov_T([\Phi])$ then $T\vdash\Phi$.

4. $T\vdash Prov_T([\Phi])\rightarrow \Phi$.

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Solution:

We start with the last one, since this is the one I find less confusing. I believe the right answer is this

(4): Assume $T\vdash\phi$ then $T\vdash \psi\rightarrow \phi$ for any formula $\psi$. Take $\psi=Prov_T([\phi])$. Then the result follows and, hence, we have to add the condition $T\vdash \phi$.

Answer: $T\vdash \phi$ is an assumption which makes the statement true.

$\textbf{(1)}$ For this one, I actually saw a similar result (I think) in the book by Peter Smith, "An introduction to Gödel Theorems":

For any sentence $\varphi$, $T\vdash\varphi$, then $T\vdash Prov_T([\varphi])$.

I hope I am allowed to write his proof here. The proof ge gives is

Suppose $T\vdash\Phi$. Then there is a $T$ proof of the wff with g.n $\phi$. Let this proof have the super g.n. m. Then, by definition, $prfsef(m,q)$. Hence, since prseg for $T$ is captured by Prf, it follows that $T\vdash Prf(\bar{m},\phi)$. So by existential quantifier introduction $T\vdash\exists v Prf(v,\phi)$, i.e $T\vdash Prov(\phi)$.

This seems very much like mine? So I don't have to add any conditions?

Answer: No addition conditions.

(2) Can I use the same argument as in (4)? Probably not.

(3) I think I read somewhere, something about $\omega$-incompleteness, but I am not sure.

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I have a hard time putting any sufficient conditions on any of the numbers (1)-(3). I think this is really hard.

How do you do when you're trying to put conditions on a statement in the field of logic? If you could help me with any of the conditions it would make me really happy. Hopefully it would give me a better understanding at least. :)

Best wishes

Answer 1

For convenience, I shall write $\def\box{\square}$"$\box_T P$" to mean "$\text{Prov}_T([P])$". $ \def\nn{\mathbb{N}} \def\eq{\leftrightarrow} $

(1)

As you stated, it is always true. Intuitively, think of it this way. PA can verify any finite number of steps of a computation. If (from the perspective of the meta-system) $T$ really proves something, it means that there is a finite string encoding that proof, and there is a finite computation that can verify the correctness of the proof. PA can verify that, so PA can also see that the proof exists.

(2)

I do not know a nice equivalent characterization, but if here is a sufficient condition that is in fact necessary if $T$ is $Σ_1$-sound (every $Σ_1$-sentence that $T$ proves is true in $\nn$).

If $P$ is decided by $T$, namely that $T \vdash P$ or $T \vdash \neg P$, then $T \vdash P \to \box_T P$, by (1) and basic logic.

If $T$ is $Σ_1$-sound and (2) is true, then LEM gives $T \vdash P \lor \neg P$ and so $T \vdash \box_T P \lor \box_T \neg P$, which is a $Σ_1$-sentence and hence is true in $\nn$, implying that $P$ is decided by $T$.

(3)

Again, I am not aware of a nice equivalent, but it is clearly always true if $T$ is $Σ_1$-sound.

(4)

You can check up Lob's theorem that essentially states that ( $T \vdash \box_T P \to P$ ) iff ( $T \vdash P$ ). To prove it, it is convenient to first establish the modal fixed-point theorem and the Hilbert-Bernays provability conditions. Then we can use the fixed-point theorem to obtain a sentence $Q$ such that $T \vdash Q \eq ( \box_T Q \to P )$. $Q$ is supposed to represent the sentence "If this sentence is trueprovable then $P$ is true.", imitating Curry's paradox. Aiming to prove $P$ within $T$, we can simply follow the reasoning of the paradox, using the provability conditions along the way, and at one point we need $\box_T P \to P$ to continue. This gives the external form of Lob's theorem. It turns out that the same reasoning can be followed to prove $\box_T ( \box_T P \to P ) \to \box_T P$ within $T$, which gives the internal form of Lob's theorem. In the linked post I show how these two immediately give the external and internal forms of Godel's second incompleteness theorems.