I am trying to construct an isomorphism between a partial order of positive square-free integers ordered by divisibility - i.e., $(\mathbb{{PSF}}, |)$ - and the partial order given by $(P_{<_w}(\mathbb{N}), \subseteq)$ where $P_{<_w}(\mathbb{N})$ consists of all finite subsets of positive integers.
So far, my approach has been to map every positive square-free integer with a set of its prime factorization (since prime factors are unique by the fundamental theorem of arithmetic, this would be injective). However, this mapping does not satisfy surjectivity. I was wondering if there is any other way I can construct an isomorphism between these two partial orders. I would appreciate any help!
Your idea is correct, with a slight modification.
Let $p:P\to \mathbb{N}$ be a bijection between the set $P$ of all prime numbers and $\mathbb{N}$. Now compose your map with $p$; that is, consider the following map:
$$ p(i_1)\cdots p(i_n)\mapsto \{i_1,\cdots, i_n\}.$$