Adding a real with infinite conditions

While Prikry-Silver forcing satisfies Axiom A and hence does not collapse $\omega_{1}$, it is independent of $MA + \neg CH$ whether Prikry-Silver forcing preserves the continuum. Under $MA + \neg CH$, this question is equivalent to whether the additivity of the Silver ideal is the continuum. So, for example, PFA implies that Prikry-Silver forcing does indeed preserve the continuum. For the corresponding question for Sacks forcing, see:

H. Judah, A. W. Miller and S. Shelah, Sacks forcing, Laver forcing, and Martin's axiom. Arch. Math. Logic 31 (1992), 145–161.

This is the forcing to add a Prikry-Silver real, discussed on page 17 of Jech's book Multiple Forcing. 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.