Is the set of submonoids of $(\Bbb N,+)$ countable? Having the monoid $(\Bbb N,+)$, I wonder if there are countable many submonoids. There are obviously infinitely many since $S_n = \{kn \mid k \in \Bbb N\}$ is a submonoid for any $n \in \Bbb N$.
My conjecture is that the set of all submonoids is countable, because I think the following statements (which I failed to prove so far) hold for any submonoid $S$ of $(\Bbb N,+)$:


*

*there is an odd element in $S$ $\Rightarrow \exists e \forall f: (f \ge e \rightarrow f \in S)$ $\Rightarrow \Bbb N \setminus S$ is finite

*all elements of $S$ are even $\Rightarrow \exists e \forall f: (2f \ge e \rightarrow 2f \in S)$ $\Rightarrow \Bbb N \setminus (S \cup \{1,3,5,...\})$ is finite
In both cases we can identify the submonoid by a finite set of numbers which are not elements of the submonoid. Therefore we have only countable many possibilities.
Can you complete this approach or provide a better one?

All numerical monoids have finitely many generators (called the embedding dimension), and the set  of finite subsets of $\mathbb{N}$ is countable.
