For a certain project I am currently working on, I need to be able to find PA cuts in nonstandard models of PA, in desirable intervals. For example, I wonder if the following is true, where $\newcommand\PA{\text{PA}}\PA_k$ refers to the theory with only $\Sigma_k$ induction.

**Question.** If $M$ is a model of $\PA$ in which $\PA_{k-1}$ is consistent, but $\PA_k$ is not (so $k$ is nonstandard), then is there a $\PA$ cut in $M$ above $k$ in which $\PA_k$ is consistent?

That is, I want to cut $M$ below the first proof of a contradiction in $\PA_k$, but above $k$, and have $\PA+\text{Con}(\PA_k)$.

Alternatively, is there some other $\Sigma_1$ property of $k$, other than $\neg\text{Con}(\PA_k)$, such that I can always find a $\PA$ cut in $M$ between $k$ and the witness of that property? Kameryn Williams suggested that the Paris-Harrington result may provide this, since it is designed to ensure $\PA$ cuts below the corresponding PH-Ramsey number. But I would need, however, that one can always end-extend the model so as to make the $\Sigma_1$ property true. Does the PH construction have both these features?

With the consistency statements, for example, for any nonstandard $k$ in any model $M$ of $\PA$, there is always an end-extension of $M$ to a model of $\PA$ with $\neg\text{Con}(\PA_k)$.

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