Say that a **long model** is an $\mathfrak{A}\models\mathsf{I\Sigma\_1}$ such that $\mathfrak{A}$ has strictly greater cardinality than each of its proper initial segments (in the case $\vert\mathfrak{A}\vert=\aleph\_1$ this is "$\omega\_1$-like"ness). If $\mathfrak{A}$ is a long model and $\vert\mathfrak{A}\vert$ is regular, then in fact $\mathfrak{A}\models\mathsf{PA}$ since the bounding scheme is trivially satisfied in $\mathfrak{A}$ (see Lemma 2.15 of [Hajek/Pudlak](https://projecteuclid.org/eBooks/perspectives-in-logic/Metamathematics-of-First-Order-Arithmetic/toc/pl/1235421926)). Moreover, by the MacDowell/Specker theorem (see [Kossak, *63 years of the MacDowell-Specker theorem*](https://arxiv.org/pdf/2208.03156.pdf)) the converse holds: if $\kappa$ is an uncountable cardinal and $\varphi$ is true in every long model of $\mathsf{PA}$ of size $\kappa$, then $\mathsf{PA}\models\varphi$ already. (Given $\mathsf{PA}\not\models\varphi$, let $\mathfrak{B}\models\mathsf{PA}+\neg\varphi$ be countable and iterate MD/S $\kappa$-many times.)


Interestingly, the first observation seems to need regularity:



> 
> Is there a singular cardinal $\kappa$ such that the common theory of long models of size $\kappa$ is strictly weaker than $\mathsf{PA}$?
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Can singular long models require less than PA?

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> The answer to the question is strongly in the positive, since the following theorem of Kaye implies that for *every* singular cardinal $\kappa$, and for any fixed natural number $n$, the common theory of $\kappa$-like models of $\mathrm{I}\Sigma\_{n}$ is strictly weaker than $\mathsf{PA}$.
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**Theorem.** *Let $\kappa$ be a singular cardinal, $n \geq 1$, and let* $M\models \mathrm{B}\Sigma\_n + \mathsf{exp} + \lnot\mathrm{I}\Sigma\_n$. *Then there is a $\kappa$-like model $K$ that is elementarily equivalent to $M$.*


The above theorem appears as Theorem 2.4 of Kaye's paper *Constructing κ-like models of arithmetic*. **J. London Math. Soc.** (2) 55 (1997), no. 1, pp.1–10. According to Kaye, the paper arose out of a question of Slaman: Are there $\kappa$-like models of arithmetic for singular $\kappa$ that do not
satisfy $\mathsf{PA}$?


A key ingredient of Kaye's proof is the technology of "doubly indexed indiscernibles" invented by Keisler, who showed that for any consistent extension $T$ of Zermelo set theory, and any given singular cardinal $\kappa$, $T$ has a $\kappa$-like model. This result of Keisler appears in his paper *Models with orderings*, **Logic, methodology and philosophy of science III** (eds. B. van Rootselaar and J. F. Staal; North-Holland, Amsterdam, 1968) 35–62.


It is worth noting that Kaye has another paper related to this topic: *The theory of $\kappa$-like models of arithmetic*. (English summary)
**Special Issue: Models of arithmetic.
Notre Dame J. Formal Logic** 36 (1995), no. 4, pp.547–559.


