Set theory seminar -Forcing axioms and inner models VI

I presented a sketch of a nice proof due to Todorcevic that {sf PFA} implies the P-ideal dichotomy {sf PID}. I then introduced Viale’s covering property {sf CP} and showed that it follows from {sf PID}. Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:

Theorem (Viale). Assume Msubseteq V is an inner model.

  1. If {sf PFA} holds in V and M computes cardinals correctly, then it also computes correctly ordinals of cofinality omega.
  2. If {sf MM} holds in V, kappa is a strong limit cardinal, (kappa^+)^M=kappa^+, and in M we have that kappa is regular, then in V, the cofinality of kappa cannot be omega_1

It follows from this result and the last theorem from last time that if V is a model of {sf MM} and a forcing extension of an inner model M by a cardinal preserving forcing, then {sf ORD}^{omega_1}subset M.

In fact, the argument from last time shows that we can weaken the assumption that V is a forcing extension to the assumption that for all kappa there is a regular cardinal lambdagekappa such that  in M we have a partition S^lambda_omega=sqcup_{alpha<kappa}S_alpha where each S_alpha is stationary in V

It is possible that this assumption actually follows from {sf MM} in V. However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable V cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of V that preserves cardinals, does not add reals, and (for some cardinal kappa) the set of points of countable V-cofinality in lambda is nonstationary for every regular lambdagekappa^+. Obviously, this situation is incompatible with {sf PFA} in V, by Viale’s result.


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