find a sentence $\alpha$ in some language L such that

Question

Let $K=\{k\in\mathbb N : k\mod2\not=0$ and $k\mod3\not=0\}$ find a sentence $\alpha$ in some language L such that $K=${$n\in\mathbb N :$exists a structure $M$ such that $M\models\alpha$ and $|W^M|=n$}

Answer 1

Outline: We are working in the predicate calculus with equality, and with an additional binary predicate symbol $R$. We build the sentence $\alpha$ as follows:

(i) Let $\beta$ be the conjunction of the usual axioms that say that $R$ is an equivalence relation (reflexivity, symmetry, transitivity).

(ii) Let $\gamma$ be the sentence that says the following. There is an object $x$ such that the only object $y$ such that $R(x,y)$ is $x$, and such that any object $z$ not equal to $x$ bears the $R$ relation to precisely $6$ objects.

(iii) Let $\delta$ be the sentence that says the following. There exist $5$ objects which are distinct and congruent to each other, and such that any object $z$ not equal to one of these $5$ bears the $R$ relation to precisely $6$ objects.

If we let $\alpha$ be the sentence $\beta\land(\gamma\lor\delta)$, then $\alpha$ will have the desired property.