Are all generalized Scott sets realized as generalized standard systems?

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 $PA$.

The positive answer alluded to above follows from Tanaka's self-embedding theorem, 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 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 $I\Delta_0$ + $Exp$ + $B\Sigma_1$.

The above result appears as Theorem 4.6 in this recent paper 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 $M$.

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 $M$).

Your question has a positive answer: there do exist proper end extensions for countable models with the required coded sets.

The PA requirement can be weakened. Generalizing results of Friedman[1973], the MacDowell-Specker Theorem extensions of Phillips[1974] and Gaifman[1976], and the results of Kirby, MacAloon, and Murawski[1979], Joseph Quinsey proved in his unpublished (and perhaps unreadable) 1980 thesis Lemma 5.9:

Suppose $M$ is a countable model of arithmetic and $\mathcal{X}$ is a countable family of subsets of $M$ such that

$$(M, \mathcal{X})\models exp + \Delta_0^0\text{-Induction} + \Delta_0^0\text{-Arith-Collection} + \text{WKL}$$

where Arith-Collection is the schema (with $q$ not free in $\theta$):

$$\forall m \lt n\, \exists p\,\theta \rightarrow \exists q\,\forall m \lt n \, \exists p \lt q\,\theta \,.$$

Let $\text{T} \in \mathcal{X}$ be such that in $(M, \mathcal{X})$, $\text{T}$ is is the encoding of a consistent set of sentences of arithmetic. Then there exists a proper end-extension $N$ of $M$ which is a model of the standard sentences of $\text{T}$ and is such that $SS_M(N)=\mathcal{X}$.

Moreover, if $(M, \mathcal{X})$ is a model of $\Pi_1^0\text{-overspill}$ (or if $M$ is the standard model), $N$ can be chosen to be recursively saturated.

If additional conditions are placed on $(M, \mathcal{X})$, then using the embedding techniques of Friedman[1973], Wilkie[1977], and Wilmers[1977], we may obtain elementary end extensions. Theorem 5.8 of Quinsey's thesis gives:

Let $k \ge 1$. Suppose $M$ is a countable model of arithmetic and $\mathcal{X}$ is a countable family of subsets of $M$ such that

$$(M, \mathcal{X})\models \Delta_0^0\text{-CA} + \Delta_0^0\text{-Arith-Collection} + \text{WKL} + \text{“the $\bar{k}$th Turing jump of the empty set exists"}$$

Further suppose that $\Pi_1^0\text{-overspill}$ holds in $(M, \mathcal{X})$. Let $c \in M$. Then there exists a structure $(N, \mathcal{Y})$ isomorphic to $(M, \mathcal{X})$ such that

• $N$ is a $(k - 1)$-elementary initial segment of $M$,
• $SS_N(M)=\mathcal{Y}$, and
• the isomorphism fixes $c$.

If in addition we have that $\Sigma_1^1\text{-overspill}$ holds in $(M, \mathcal{X})$ and that

$$(M, \mathcal{X})\models \text{“the $\bar{k}$th Turing jump of the empty set exists"}$$

for each standard integer $k$, then we may require that $N \prec M$.

Since the two structures are isomorphic, we may swap them to get the end-extension. Note in passing that these two results have nothing to do with the main thrust of the thesis, which is to explore applications of Kripke's notion of fulfillability.