Adding a surjection $\omega \to \omega$ by Levy forcing I'm trying to understand the Levy collapse, working through Kanamori's 'The Higher Infinite'. He introduces the Levy forcing $\text{Col}(\lambda, S)$ for $S \subseteq \text{On}$ to be the set of all partial functions $p: \lambda \times S \to S$ such that $|p| < \lambda$ and $(\forall \langle \alpha, \xi \rangle \in \text{dom}(p))(p(\alpha,\xi) = 0 \vee p(\alpha, \xi) \in \alpha$).
In the generic extension, we introduce surjections $\lambda \to \alpha$ for all $\alpha \in S$.
However, the example Kanamori gives is $\text{Col}(\omega,\{\omega\})$, which he says is equivalent to adding a Cohen real. I can see how in the generic extension, we add a new surjection $\omega \to \omega$, but I don't see how this gives a new subset of $\omega$.
Thanks for your help.

A new surjection is a subset of $\omega\times\omega$. We have a very nice way to encode $\omega\times\omega$ into $\omega$, so nice it is in the ground model. If the function is a generic subset, so must be its encoded result, otherwise by applying a function in the ground model, on a set in the ground model, we end up with a function... not in the ground model!
The fact this forcing is equivalent to a Cohen one is easy to see by a cardinality argument. It is a nontrivial countable forcing. It has the same Boolean completion as Cohen, and therefore the two are equivalent.
