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There are infinitely many extension to Robinson's $Q$ arithmetic many of which are defined by adding an axiom schema of induction for particular set of formulas.

I am confused about the theory $\text{IOPEN}$, it is defined by adding an axiom schema of induction for open formulas. But since open formulas are treated as universaly closed in the proof system due to universal generalisation, is this theory the same theory as $\text{I}\Pi_1$? Is there something I'm overlooking? By this argument $\text{I}\Sigma_1$ would be the same theory as $\text{I}\Pi_2$.

(Originally posted to MSE, without answer)

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    $\begingroup$ Open formulas are not treated as universally closed (nor would it make any sense in this context). Universal generalization does not allow you to insert universal quantifiers deep inside the induction axiom. $\endgroup$ Commented Sep 5, 2019 at 7:49
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    $\begingroup$ Also, universal formulas are a much smaller class that $\Pi_1$. Nevertheless, for deep reasons (formalized version of the MRDP theorem), it turns out that $I\forall_1=I\Pi_1$ (which equals $I\Sigma_1$, for simple reasons). $\endgroup$ Commented Sep 5, 2019 at 7:59
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    $\begingroup$ @EmilJeřábek The MSE version of the question is math.stackexchange.com/questions/3321469 where I made a comment essentially equivalent to your first comment here. There was a bit of discussion in further comments there, which ended with my suggesting that we wait for an expert to come along. Since you've come along now, you might want to look there and add some more clarification either here or there. $\endgroup$ Commented Sep 5, 2019 at 12:08
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    $\begingroup$ So, for the record: that $I\forall_1=I\exists_1=I\Sigma_1=I\Pi_1$ follows from results of Richard Kaye, Diophantine induction, Ann. Pure Appl. Logic 46 (1990), 1–40. Thus $I\forall_1$, i.e., $I\Sigma_1$, is a very strong theory. It proves all kinds of things unprovable in IOpen, such as the totality of every primitive recursive function, and the consistency of $Q$, or even stronger theories like $I\Delta_0+\mathit{Exp}$ (in fact, it proves the consistency of each finite subtheory of PRA). However, already $IU_1=IE_1$ (induction for bounded universal or existential formulas) ... $\endgroup$ Commented Sep 5, 2019 at 13:54
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    $\begingroup$ ... which is a very weak theory, proves various things unprovable in IOpen. The simplest I can think of is the statement $\forall u\,\forall v\,(v^2=2u^2\to u=0)$ (i.e., “$\sqrt2$ is irrational”). This can be proved by induction on the $U_1$ formula $\phi(x)=\forall u,v\le x\,(v^2=2u^2\to u=0)$ by mimicking the usual infinite descent argument. (This needs a few basic facts like $2\mid xy\to 2\mid x\lor 2\mid y$ or $2x=2y\to x=y$, which are provable already in IOpen.) $\mathrm{IOpen}\nvdash\forall u\,\forall v\,(v^2=2u^2\to u=0)$ as it fails in Shepherdson’s model of IOpen, ... $\endgroup$ Commented Sep 5, 2019 at 14:01

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