The notion of minimality in model theory is related to the existence of a *gap* in the size of definable subsets of a model. Now consider the following generalization:


**Definition 1:** Let $M$ be a $\mathcal{L}$ - structure. Define:


$Def(M):=\{X\subseteq Dom(M)~|~\exists n \in \omega~~~\exists \varphi (x ,y\_1 , ...,y\_n)\in \mathcal{L}-Form~~~$


$\exists b\_1 , ...,b\_n\in Dom(M);~~~~~X=\{a\in Dom(M)~|~M\vDash \varphi(a, b\_1,...,b\_n)\}\}$


**Definition 2:** Let $\kappa >\lambda \geq \aleph\_{0}$ be two cardinals. A model $M$ called $(\kappa , \lambda)$ - minimal if $\forall X\in Def(M);~~~~ |X|\geq \kappa~\vee~~|X|\leq \lambda$.


Assuming consistency of $\text{ZF}$:


**Question 1:** Does $\text{ZF}$ have a $(\kappa , \lambda)$ - minimal model for each $\kappa >\lambda \geq \aleph\_{0}$? 


If the answer of the above question is negative:


**Question 2:** Does $\text{ZF}$ have an extension like $T$ such that $T$ has a $(\kappa , \lambda)$ - minimal model for each $\kappa >\lambda \geq \aleph\_{0}$? 


**Question 3:** For which (probably large) cardinals like $\kappa >\lambda \geq \aleph\_{0}$ does $\text{ZF}$ have a $(\kappa , \lambda)$ - minimal model? 



$(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$
The answer is yes, one can always find models with as large a gap in their definable classes as desired.


**Theorem.** For every $\kappa\gt\lambda\geq\aleph\_0$, and for any consistent theory with an infinite model in a countable language, there is a $(\kappa,\lambda)$-minimal model. 


Proof. Let $M$ be any $\kappa$-saturated model of the theory. It follows that every infinite definable subset of $M$ has size at least $\kappa$, since by saturation we may always find an additional satisfying instance of the definition different than any fewer than $\kappa$ many known instances. Thus, $M$ is $(\kappa,\aleph\_0)$-minimal. QED


Indeed, this argument shows that one may attain the sharper separation property that $|X|\geq\kappa$ or $|X|\lt\lambda$, since in a $\kappa$-saturated model every definable class has size at least $\kappa$ or is finite.


