Find a subset of $(\mathbb N,\mid)$ isomorphic to $(\mathcal P (\{1, \dots, n \}), \subseteq)$ as a partially ordered set.
I can only think of one isomorphism. Let $S$ be the subset of $\mathbb R$ given by the range of the following mapping:
$$\phi: \mathcal P(\{1, \dots, n\}) \to S$$
where$$\phi(\{n\}) = \mbox{n-th prime number},$$
$$\phi(\{n_1, \dots, n_k \}) = \prod_{i=1}^k \phi(n_i)$$
1. This function is injective because all of its values are unique products of prime numbers. If two values are to be the same, then the input sets must be identical.
2. This function is surjective because its image is the whole set $S$
3. Suppose that $A \subseteq B$. Then there exists such $m$ that $\phi(B) = m\phi(A)$ and so $\phi(A) \mid \phi(B)$
And so this is in fact an isomorphism.
What do you think of my solution? Can it work?