TITLE: How special is first-order $\mathsf{PA}$?
QUESTION [14 upvotes]: This is a modified version of a question which was asked and bountied at MSE without success.

Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic.
There are various "schematic" theories out there, like $\mathsf{PA}$ and $\mathsf{ZFC}$, which basically consist of three components: a "base" set of axioms which are more-or-less taken for granted (e.g. the  discrete nonnegative ordered semiring axioms for $\mathsf{PA}$, and Pairing-Union-Extensionality-Foundation-Powerset-Infinity-Choice for $\mathsf{ZFC}$), an informal idea(s) for a further set of rules indexed by formulas (e.g. induction for $\mathsf{PA}$, and separation/replacement for $\mathsf{ZFC}$), and a choice of logic for implementing the latter (first-order logic for $\mathsf{PA}$ and $\mathsf{ZFC}$, and indeed in general). For any such theory we can hope to produce interesting variants by fixing the first two parts and varying the third - see e.g. here for a question about the case of $\mathsf{ZFC}$.
However, $\mathsf{PA}$ seems remarkably stubborn here: every natural logic (= regular logic possibly without negation) I can think of lands in one of two extremes. To be precise, given a logic $\mathcal{L}$ let $\mathfrak{PA}(\mathcal{L})$ be the class of models of $\mathsf{PA}$ with no $\mathcal{L}$-definable nontrivial proper cuts, and say that $\mathcal{L}$ is:

strong for induction if $\mathfrak{PA}(\mathcal{L})$ consists of just $\mathbb{N}$ up to isomorphism;

weak for induction if every complete first-order extension of $\mathsf{PA}$ is satisfied by some element of $\mathfrak{PA}(\mathcal{L})$; and

intermediate for induction if neither of the above cases holds.


My question is:

Is there any natural logic which is intermediate for induction? (Here, by "natural" I mean "has appeared in at least two different papers whose respective authorsets are $\subseteq$-incomparable.")

In the MSE version I asked for an even stronger property, namely not pinning down $\mathbb{N}$ even up to elementary equivalence, but that seems overly optimistic in retrospect.

Here are some quick negative observations:

Uniformly across models of $\mathsf{PA}$, standardness is definable by a $\Pi^1_1$ formula or by a very simple infinitary formula. So neither $\Pi^1_1$ nor any reasonable infinitary logic will be intermediate for induction; in fact, they'll pin down $\mathbb{N}$ up to isomorphism, not just up to first-order-elementary-equivalence.

$\Sigma^1_1$ also pins down $\mathbb{N}$ up to isomorphism, although parameters now appear necessary; see here. For the same reason, first-order logic + the equicardinality quantifier pins down $\mathbb{N}$ up to isomorphism.

On the opposite end of things, $\Delta^1_1$ is weak for induction. (OK, $\Delta^1_1$ may not look like a logic since the $\Delta^1_1$-ness of a $\Sigma^1_1$ formula is structure-dependent, but we can handwave past this: define an abstract logic with a formula $\hat{\varphi}$ for each $\Sigma^1_1$ formula $\varphi$, where $\hat{\varphi}$ is interpreted in $\mathcal{M}$ as $\varphi$ if $\neg\varphi^\mathcal{M}$ is $\Sigma^1_1$ over $\mathcal{M}$ and as $\top$ otherwise.)

Similarly, $\mathsf{FOL}$ + the quantifier "At least as many $x$ satisfy $\varphi$ as satisfy $\neg\varphi$" is weak for induction, although its model class contains no countable nonstandard models; to see weakness, consider the $\omega_1$-like models of $\mathsf{PA}$.


The most promising approach to me at the moment is to look for fragments of second-order logic between $\Delta^1_1$ and $\Sigma^1_1$, since we see a genuinely interesting and nontrivial change in behavior there. However, I can't at the moment think of a good candidate fragment.
(I've added the set theory tag since, while set theory isn't built into the question a priori, it seems relevant to all the ideas I've had so far.)

REPLY [13 votes]: There are several notable papers, starting with a key paper of Angus Macintyre (Ramsey quantifiers in arithmetic, Model theory of algebra and arithmetic, Lecture Notes in Math., 834, Springer, 1980), which explore natural extensions of Peano Arithmetic formulated in "well-behaved" logics extending first order logic. The most well-known of these extensions are obtained by adding the so-called Ramsey quantifiers $Q^{n}$ (where $n\in \omega,~n>0$) to first order logic; these quantifiers are also known as Magidor-Malitz quantifiers.
The extension of PA formulated using Ramsey quantifiers is usually referred to as $PA(Q^{<\omega})$. Given a model $\mathcal{M}$ of PA, the semantics of $Q^{n}$ is guided by:
$\mathcal{M}\models Q^{n}x_1,...,x_n~ \varphi(x_1,...,x_n)$ iff there is an unbounded subset $H$ of $\mathcal{M}$ such that $\mathcal{M}\models \varphi(h_1,...,h_n)$ for every increasing sequence $h_1,...,h_n$ from $H$.
Macintyre proved various interesting results in his paper, including a completeness theorem (with the help of the combinatorial principle $\diamond_{\omega_1}$). This allowed him to show that $PA(Q^{<\omega})$ has an $\omega_1$-like model (an uncountable model every proper initial segment of which is countable). As also noted by Macintyre, the Paris-Harrington extension of the finite form of Ramsey's theorem is provable in $PA(Q^{<\omega})$, thus the 1-consistency of PA is provable in $PA(Q^{<\omega})$ [indeed proving the Paris-Harrington principle in a natural extension of PA is what led Macintyre to introduce $PA(Q^{<\omega})$].
Schmerl and Simpson refined Macintyre's work by eliminating the need for $\diamond_{\omega_1}$ in the results of Macintyre in the aforementioned paper (On the role of Ramsey quantifiers in first order arithmetic, Journal of Symbolic Logic, Vol.47, 1982). Their refinement allowed them to calibrate the arithmetical strength of $PA(Q^{<\omega})$ by proving the following theorem:
Theorem (Schmerl and Simpson). The following statements are equivalent for an arithmetical statement $\varphi$:
(a) $PA(Q^{<\omega}) \vdash \varphi$.
(b) $PA(Q^{2}) \vdash \varphi$.
(c) $\Pi^1_1-\mathrm{CA_0} \vdash \varphi$.
In a later paper by Schmerl, entitled PA(aa), in Notre Dame Journal of Formal Logic, Vol. 36, 1982, Schmerl studied the extension of PA with stationary quantifiers (originally introduced and studied in a landmark 1978 paper of Barwise, Kaufmann, and Mekler) and established that the purely arithmetical consequences of PA(aa) + Det [where Det is the so-called scheme of finite determinateness of stationary logic] coincides with the purely arithmetical consequences of second order arithmetic $Z_2$ (the first order theory whose subsystems are studied in reverse mathematics, as opposed to PA formulated in full second logic, which is a much stronger theory which only has one model up to isomorphism).
Another important paper of Schmerl (Peano arithmetic and hyper-Ramsey logic, Trans. Amer. Math. Soc., 1986) connects PA formulated in so-called hyper-Ramsey logic with subsystems of second order arithmetic.