I have a theory with finitely many relations, and would like to find a model of it with continuum-many 1-types realized, and one 2-type omitted. Is there a version of the Omitting Types Theorem that would help me in this case? The proof in Marker's "Model Theory: An Introduction" explicitly uses countability of the language, so I can't for instance introduce continuum many constant symbols for the types I want to realize.
Specifically, I'm working with the weak monadic second-order theory of 1 successor (treated as a two-typed first order structure, i.e. Henkin semantics). That is, the two-typed structure $(\mathbb{N}, \mathcal{F}(\mathbb{N}), \in, S)$, where $\mathcal{F}(\mathbb{N})$ is the set of finite subsets of $\mathbb{N}$, and $S$ is the successor operation on $\mathbb{N}$. One can describe $\leq$ in this structure in terms of sets which are closed under $S^{-1}$, so it's often more intuitive to think of this as the structure $(\mathbb{N}, \mathcal{F}(\mathbb{N}), \in, \leq)$.
A nonstandard model of this structure (up to isomorphism) consists of a nonstandard model of $(\omega, \leq)$ (call it $A$) for its first-order part, then a collection of subsets of $A$ for its "second-order" part. This collection of subsets $B$ has to, for instance, be closed under union and set-subtraction, and each "second order" element $X \in B$ has to have a $\leq$-maximal element.
I'm curious if there is a nonstandard model whose "first order part" is $\omega + \zeta$ (thus omitting the 2-type of having two points infinitely far from each other and infinitely far from 0), but whose "second order part" $B$ satisfies: $\{X \cap \omega | X \in B\} = \mathcal{P}(\omega)$. That is, for every subset of the natural numbers, there is a "second order element" in the nonstandard model containing exactly those natural numbers. Since every element of the second type must have a $\leq$-maximal element, naturally all of these second order elements will have to also contain nonstandard first order elements.
If one drops the $\omega + \zeta$ requirement, one can prove the existence of a nonstandard model whose second order part "restricts" to $\mathcal{P}(\omega)$ by a simple compactness argument. For each $K \subset \omega$, add in a constant symbol $C_K$ and add $\{S^n(0) \in C_K\}_{n \in K}$ and $\{S^n(0) \notin C_K\}_{n \notin K}$ to the theory. These are the continuum many 1-types I'd like to realize.
I proved countably many such types can be realized at once in an $\omega+\zeta$ model in section 4.6 of my Ph.D. Thesis: https://ecommons.cornell.edu/handle/1813/44393.
