Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.


Does $\Bbb P$ collapse the continuum?



Adding a real with infinite conditions
This is the forcing to add a Prikry-Silver real, discussed on [page 17 of Jech's book Multiple Forcing](https://books.google.com/books?id=qYqEYg6TMScC&pg=PA17&lpg=PA17&dq=prikry+silver+forcing&source=bl&ots=vts83a-YWY&sig=q4b6vmf78pwOTfN3HWGc2CJwapQ&hl=en&sa=X&ei=qD3GVI_iJuXIsAT2-ILwCw&ved=0CDgQ6AEwAw#v=onepage&q=prikry%20silver%20forcing&f=false). A fusion argument shows that it satisfies the countable cover property (every new countable set of ordinals is covered by a ground model countable set), and so it does not collapse the continuum. A Prikry-Silver real is minimal.


