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Simon Thomas
Simon Thomas
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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 preservespreserve 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.

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 preserves 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.

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.

Source Link
Simon Thomas
Simon Thomas
  • 8.3k
  • 4
  • 42
  • 57

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 preserves 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.