I want to prove the following.



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> For every $\Pi^0\_1$ statement $\forall x\phi(x)$, where $\phi(x)$ is a $\Delta^0\_1$ formula, there is $e\in\mathbb{N}$ such that $\forall x\phi(x)$ implies $W\_e=PA$\* and $PA+\text{$W\_e$ is consistent}\vdash\forall x\phi(x)$.
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The argument is inspire by an argument of Turing in his [PhD dissertation](https://www.princeton.edu/turing//alan/dissertation/) (lately rephrased in [Feferman 1962](https://www.jstor.org/stable/2964649)) to show that the transfinite progressions of adding consistency statements is sensitive on which branch we choose at the limit stage of Kleene's $\mathcal{O}$, and it is as follow.



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> Fix a computable function $\sigma$ such that the range of $\sigma$ is $PA$. By recursion theorem, we can construct a partial computable function $\varphi\_e$ such that
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> $\varphi\_e(n)=\begin{cases}\sigma(n),&\text{if $\forall x<n\phi(n)$},\\
> \sigma(n)\wedge\text{$W\_e$ is consistent\*},&\text{o.w.}\end{cases}$
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> Since $\forall x\phi(x)$ holds, $W\_e=PA$. So the statement "$W\_e$ is consistent" is really "$PA$ is consistent". In the next paragraph, we prove in $PA+$"$W\_e$ is consistent".
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> Assume $\forall x\phi(x)$ fails, than $W\_e=PA+$"$W\_e$ is consistent". By Goedel's second incompleteness theroem, $W\_e$ is not consistent. A contradiction.
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> This finishes the argument.
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My question is (1) is this argument valid, or if something is missed or misunderstood? (2) is there other reference on this and related issues?


\*$W\_e$ is the range of $\Phi\_e$ by fixing an enumeration of Turing machines $\{\Phi\_e\}\_{e\in\mathbb{N}}$. $PA$ stands for Peano Arithmetic. Please forgive my abusing of notation.



Is every true $\Pi^0_1$ statement entailed from a consistency statement of $PA$?
The statement you want to prove is true, and your argument works - but there's a simpler one: drop all reference to the recursion theorem and Godelian incompleteness, and just change the second clause in your definition of $\varphi\_e(n)$ to "$\sigma(n)\wedge\exists x(x\not=x)$." That is, introduce an inconsistency *directly* rather than playing with deeper facts about the theory and computability. 


As you say, the only way $W\_e$ can be consistent is if that second case doesn't occur, and if $\forall x\phi(x)$ is indeed true then that second case never occurs and $W\_e$ "is" PA.


Given the directness of this construction, I suspect that there is no explicit reference.


