A set $M$ of natural numbers is called a [i]spectrum[/i] if there is a first-order language $L$ and a sentence $\phi$ over $L$ such that: $$M = \{ n \mid \phi \text{ has a model containing exactly $n$ elements}\}$$ For example, consider a sentence $\phi = \exists e . (\forall x. x = e)$ in a first order language with no relation symbol, no function symbol, and no constant symbol. The formula $\phi$ only admits a model containing exactly 1 element. Therefore, the set $\{1\}$ is a spectrum.\\ Show that: [list=1] [*] Every finite subset of $\mathbb{N} \setminus \{0\} $ is a spectrum [/*] [*] The set of even numbers, i.e., $\{2k \mid k \in \mathbb{N}\}$ is a spectrum [/*] [*] For any fixed $m \geq 1$, the set of numbers greater than $0$ that are divisible by $m$, i.e., $\{m\cdot k \mid k \in \mathbb{N}\}$ is a spectrum[/*] [/list]