*Below, I've focused on PA when lots of other theories would do. If replacing PA with a different theory leads to a more answerable question, feel free to do so.*




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The *standard system* of a nonstandard model $M$ of PA is the set of sets of natural numbers coded by elements of $M$: $$SS(M)=\{X\subseteq\omega: \exists a\in M\forall x\in\mathbb{N}(x\in X\iff p\_x\vert a)\}.$$ *(Here "$p\_i$" denotes the $i$th prime.)* An easy overspill argument shows that $SS(M)$ is always a Scott set,$^1$ and [Scott proved](https://mathscinet.ams.org/mathscinet-getitem?mr=0141595) that every countable Scott set is the standard system of some nonstandard model.$^2$


We can define an analogous notion$^3$ of standard system with $\mathbb{N}$ replaced with more general initial segments: if $M\subsetneq\_{end} N$ are models of PA, then we let $SS\_M(N)$ be the set of elements of $M$ coded by elements of $N$: $$SS\_M(N)=\{X\subseteq M: \exists a\in N\forall x\in M(x\in X\iff p\_x\vert a)\}.$$ *(Here "$p\_i$" denotes the $i$th prime in the sense of $N$, or equivalently in this case of $M$.)* The same overspill argument shows that the second-order structure $(M, SS\_M(N))$ is a model of WKL.


My question is whether the analogue of Scott's theorem holds, at least for countable $M$:



> 
> **Question.** Suppose $M$ is a countable model of PA and $\mathcal{X}$ is a countable family of subsets of $M$ such that $(M, \mathcal{X})\models$ WKL. Is there an $N\supsetneq\_{end} M$ such that $SS\_M(N)=\mathcal{X}$?
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The problem here is that in the usual case, we don't need to worry about $\subseteq$ versus $\subseteq\_{end}$, whereas that poses a real problem here.




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$^1$A set of sets of natural numbers closed under join and Turing reducibility such that for every infinite binary tree in the set, an infinite path through that tree is also in the set. Equivalently, the second-order part of an $\omega$-model of WKL.


$^2$The generalization of Scott's theorem to *uncountable* Scott sets is wildly open. [Knight and Nadel](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/models-of-arithmetic-and-closed-ideals/02FDD187324ED5298CFBBA6F7DA002EC) generalized Scott's theorem to Scott systems of cardinality $\aleph\_1$, thus solving the problem under the assumption of CH, but their argument breaks down immediately for models of cardinality $\aleph\_2$ or greater. Meanwhile, [Gitman has recently shown](https://arxiv.org/abs/0801.4364) that the question has an affirmative answer for a class of Scott sets characterized in terms of forcing, assuming the set-theoretic hypothesis PFA (which contradicts CH); however, my understanding is that her arguments really only apply to that particular class of Scott sets and that the hypothesis of PFA is currently necessary.


$^3$I've had trouble finding much literature on this, especially compared to the usual notion of a standard system, but [Kossak and Schmerl's book](https://rads.stackoverflow.com/amzn/click/0198568274) does contain some information about them (especially in Chapter $7$).



Are all generalized Scott sets realized as generalized standard systems?

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> The question has a positive answer, not only when $M$ is a model of $PA$, but even when $M$ is a model of the fragment $I\Sigma\_1$ of $PA$. 
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The positive answer alluded to above follows from [Tanaka's self-embedding theorem](https://www.sciencedirect.com/science/article/pii/S0168007295000585), which states that if $(M,\cal{X})$ is a countable nonstandard model of $WKL\_0$, then $(M,\cal{X})$ is isomorphic to a proper initial segment of itself, i.e. to a model of the form $(I, \cal{X}\_{I})$, where $I$ is a proper initial segment of $M$, and $\cal{X}\_I$ is the restriction of $\cal{X}$ to $I$, consisting of sets of the form $X\cap I$, as $X$ ranges over $\cal{X}$; note that $\cal{X}\_I$ coincides with $SSy\_I(M)$.


So in this case, the $N$ you are asking for can be arranged to be isomorphic to $M$. For further work and references and more info, see [this paper](https://gup-server.ub.gu.se/v1/asset_data/167548) of mine (It appeared in New Studies in Weak Arithmetics, ed. by Patrick Cégielski, Charalampos Cornaros, and Costas Dimitracopoulos, CSLI Lectures Notes, No. 211, 2013). 



> 
> If the end extension $N$ is only required to satisfy $I\Delta\_0$, then result even holds if $M$ is a model of $I\Delta\_0$ + $Exp$ + $B\Sigma\_1$.
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The above result appears as Theorem 4.6 in [this recent paper](https://www.sciencedirect.com/science/article/pii/S016800721630183X?via%3Dihub) of Tin Lok Wong and myself, where we prove that if $(M,\cal{X})$ is a countable model of $WKL^\*\_0$, then there is an end extension $N$ of $M$ such that $SSy\_M(N)=\cal{X}$, and $N$ is a model of $I\Delta\_0$. Here $WKL^\*\_0$ is the well-known subsystem of $WKL\_0$ that is conservative over $I\Delta\_0$ + $Exp$ + $B\Sigma\_1$. 



> 
> In case $(M,\cal{X})$ is a countable model of $ACA\_0$, we can arrange the desired $N$ to be an *elementary* end extension of $M$. 
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The above follows from the following two facts:


**(1)** if $(M,\cal{X})$ is a model of $ACA\_0$, then the expansion of $M$ with predicates for each $X\in\cal{X}$, i.e., the model $(M,X)\_{X\in\cal{X}}$, satisfies the theory often referred to as $PA^\*$, i.e., the natural extension of $PA$ in which the predicates $X\in\cal{X}$ can appear. 


**(2)** The extension of the McDowell-Specker Theorem due independently to Phillips and Gaifman, that states that *every countable model $M$ of $PA^\*$ has a conservative elementary end extension $N$* ($N$ is a conservative extension of $M$ here means that the intersection of every parametrically definable subset of $N$ with $M$ is parameterically definable in $M$).


