*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 $\mathbb{N}$.)