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There's nothing special about PA here, I'm just focusing on it since it's strong enough to ignore lots of minor technical issues around foundations. If switching to some other theory would yield a better response, feel free to do so.

Let $$PA_{bd}=\{\varphi: PA\vdash\forall^\infty n([n]\models\varphi)\},$$ where $[n]=\{0,1,...,n\}$ (with $+$ and $\times$ interpreted as $$a+^{[n]}b=\min\{a+b, n\},\quad a\times^{[n]}b=\min\{a\times b, n\}$$ so that this actually makes sense - we could also switch to the relational version of PA). Note that (for fixed $\varphi$) the statement "$\forall^\infty n([n]\models\varphi)$" can in fact be expressed in the language of arithmetic (Skolem functions over finite objects are themselves finite objects), so the definition of PA$_{bd}$ does actually make sense.

Clearly PA$_{bd}$ is recursively axiomatizable; however, this only gives an axiomatization which "goes through PA" in some sense. My question is:

Is there a reasonably-simple axiomatization of PA$_{bd}$ which doesn't reference PA itself?

One natural guess would be [some basic algebra stuff] together with the full induction scheme. But in fact this is too weak - already this theory is contained in the analogous theory (I$\Sigma_1)_{bd}$ (since saying that a finite structure satisfies a complicated formula is still a very simple formula!).

Further evidencing some structure here, it's easy to show (e.g. I$\Sigma_1$ can prove) that PA and PA$_{bd}$ are equiconsistent:

  • If PA is inconsistent then PA proves everything, and in particular PA proves $\forall^\infty n([n]\models\perp)$ which gives $\perp\in$ PA$_{bd}$.

  • If PA$_{bd}$ is inconsistent, that means that for some $\varphi$ we have PA$\vdash \forall^{\infty}n([n]\models\perp)$. But PA can prove enough about the semantics of finite structures to prove that no $[n]$ satisfies $\perp$, and so we get a contradiction in PA.

(As an aside, the usual proof of Tennenbaum's theorem also shows that PA$_{bd}$ has no computable models either. Which isn't surprising, since any of its models are clearly proper end extensions of $\mathbb{N}$.)

Noah Schweber
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    My motivation here is partly philosophical (I'm putting this in a comment rather than the main question to hopefully avoid sidetracking). Namely, ultrafinitism is of course opposed to the concreteness of a theory like PA (or rather, of its putative "models" or those models' elements), but need not be opposed to their formal consistency - witness e.g. Essenin-Volpin's claim of an ultrafinitistic proof of the consistency of ZF + infinitely many inaccessibles. (cont'd) – Noah Schweber Feb 08 '19 at 22:10
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    It would be interesting to me to whip up a philosophical argument in favor of the "concreteness" of a particular theory $T$ together with a proof that $T$ is in fact PA$_{bd}$ - at the least this would then translate to a philosophical argument in favor of the formal consistency of PA. At this point I should make it clear that (to put it mildly) I have no doubts about the consistency of PA or indeed much more but that doesn't mean that I can't find the general problem of searching for "natural" ideas nontrivially implying Con(PA) - like the well-foundedness of $\epsilon_0$ - to be interesting. – Noah Schweber Feb 08 '19 at 22:14
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    Just to be clear, $\forall^\infty$ means "for all but finitely many", right? (That is, $\lnot\exists^\infty\lnot$.) – Alex Kruckman Feb 08 '19 at 22:33
  • @AlexKruckman Yes, that's correct. – Noah Schweber Feb 08 '19 at 22:33
  • I'm not convinced the equiconsistency argument is strong evidence of structure. It might have more to do with the particular specification. I can make a c.e. theory which is empty if PA is consistent and is ${\bot}$ if PA is inconsistent and now this theory is equiconsistent with PA, but it clearly has very little structure in its own right. – James E Hanson Feb 25 '20 at 01:20
  • @JamesHanson Sure, I certainly wouldn't call it strong evidence; but I don't know of any natural examples of the phenomenon you're describing (where a theory has much higher consistency strength than its "structure" suggests), and I also don't see a simpler presentation of this theory. – Noah Schweber Feb 27 '20 at 21:18
  • @JamesHanson Coming back to this much later, maybe something like Kripke's fulfillability helps here? – Noah Schweber May 05 '24 at 03:07
  • @NoahSchweber, why not go through $L_{\omega_1, \omega}$, upgrade the theory of hereditarily bounded sets of Emil Jeřábek as to define ${\sf H}_k$ for each $k \in \omega$, then we can easily have a theory joining all upward absolute sentences, this can be phrased in that language directly! I think this is a possible detour? – Zuhair May 05 '24 at 08:21
  • @Zuhair Because I want a computably axiomatizable first-order theory. – Noah Schweber May 05 '24 at 15:40
  • @NoahSchweber, Ah I see! So by "reasonably simple" you meant that in particular. Ok, then. But, besides this demand, I think that passing through to $L_{\omega_1,\omega}$ is not that complex. You can simply axiomatize this theory there and it is done with effective machinery since the theory has countably many axioms, and so there is effective proof system for it. – Zuhair May 05 '24 at 16:03

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