The first relevant theorem is the following classical result:

**Theorem A.** (Mostowski, Keisler) *If $M$ is a countable model of Kelley-Morse + Choice Scheme, then there is an elementary extension $M^{*}$ of $M$ such that $\mathrm{Set}^{M}= \mathrm{Set}^{M^{*}}$ and $M^{*}$ has cardinality $\aleph_1$.* 

In the above, the Choice Scheme consists of universal generalizations of statements of the following form, where the parameters in $\phi$ are suppressed:

$\forall x \exists Y \phi (x,Y) \rightarrow \exists Z \forall x  \phi(x,(Z)_x)$,

where $x$  range over sets, capital letters range over classes, and $(Z)_x$ [read as: the $x$-th slice of $Z$] denotes $\{t: (x,t)\in Z\}$).

The theorem holds for models of Second Order Arithmetic, with the same proof that works for Kelley-Morse. The proof uses omitting types and elementary chains; see, e.g., Chapter 28 of Keisler's monograph *Model Theory for Infinitary Logic*. It is well-known that Kelley-Morse plus the Choice Scheme is bi-interpretable with with the extension of $ZFC^-$ ($ZFC$ without powerset, where $ZF$ is formulated using the collection and separation schemes) obtained by adding "the last cardinal exists, and it is inaccessible".

On the other hand;

**Theorem B.** (Jensen's Gap-1 Theorem). *Assuming $V=L$, every first order theory that admits a two-cardinal model of the form $(\aleph_{1}, \aleph_{0})$ admits a two-cardinal model of the form $(\kappa^{+},\kappa)$ for any prescribed infinite cardinal $\kappa$.*

By putting Theorems A & B together with the Loewenheim-Skolem theorem,  we obtain the following corollary; in what follows, the "two-cardinal type" of a model $M$ of Kelley-Morse, is the order pair of cardinals $(|M|, |\mathrm{Set}^{M}|)$.

**Corollary.** *It is consistent with $ZFC$ that the following holds: for every consistent extension $T$ of Kelley-Morse + Choice Scheme (in the same language),  the "two-cardinal" types of models of $T$ are precisely of the form $(\kappa, \kappa)$ or $(2^{\kappa}, \kappa)$, where $\kappa$ ranges over infinite cardinals.*