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Let $K$ be a kripke model and $k$ be one of its node, then $\mathcal{M}_k$ is classical structure of $k$.

What is the strongest theory of arithmetic like $T$ such that for every kripke model $K\Vdash HA$ and for every nodes of it like $k$, $\mathcal{M}_k\models T$?.

It is not hard to see that $T\vdash I\Delta_0$, because $HA$ proves least number principle for $\Delta_0$ formulas.

I found two paper that partially answered this question:

Finite Kripke models of HA are locally PA

Every Rooted Narrow Tree Kripke Model of HA is Locally PA

I want to know is there any other results on this question? Also can $T$ be a stronger theory than $I\Delta_0$ like $I\Sigma_1$?

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    $\begingroup$ The abstract of the first paper you mention (by van Dalen et al.) states that $T$ includes at least $I\Delta_1$; this seems to follow from the results in section 4 of the paper (by the way, $I\Delta_1$ is known to be equivalent to $B\Sigma_1$ in the presence $I\Delta_0$ + "exponentiation is a total function"). $\endgroup$
    – Ali Enayat
    Commented Dec 6, 2015 at 10:19
  • $\begingroup$ @AliEnayat: Thank you for your comment. By " $I\Delta_1$ is known to be equivalent to $B\Sigma_1$ in the presence $I\Delta_0+EXP$ " you mean $$I\Delta_0+EXP\vdash B\Sigma_1 \equiv I\Delta_1$$? $\endgroup$
    – Erfan Khaniki
    Commented Dec 6, 2015 at 10:40
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    $\begingroup$ Yes, for more detail see Slaman's 2004 paper available at: ams.org.ezproxy.ub.gu.se/journals/proc/2004-132-08/… $\endgroup$
    – Ali Enayat
    Commented Dec 6, 2015 at 10:48

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