Questions regarding precipitous ideals and the ω_{1}-Chang Model

Paul B. Larson

January 3, 2011

The following question is well known.

1 Question. Does Bounded Martin’s Maximum imply that NS_{ω_{1}} is precipitous?

Since BMM is preserved by forcings which do not add a subset of ω_{1}, a negative answer to Question 1 would be given by a positive answer to the following question.

2 Question. If Bounded Martin’s Maximum holds, is it possible to destroy the precipitousness of NS_{ω_{1}} without adding a subset of ω_{1}?

Bounded Martin’s Maximum may not be the most relevant hypothesis for Question 2, but some hypothesis is needed, as the answer is certainly no if NS_{ω_{1}} is ℵ_{1}-dense, or ℵ_{1}-dense in densely many places. So one could modify Question 2 as follows.

3 Question. If there exists A ∈ NS_{ω_{1}}^{+} such that for no stationary B ⊆ A is NS_{ω_{1}}↾B ℵ_{1}-dense, is it possible to destroy the precipitousness of NS_{ω_{1}} without adding a subset of ω_{1}?

Woodin has shown that it is possible to force over the ℙ_{max} extension of L(ℝ) to destroy the saturation of NS_{ω_{1}} without adding an ω_{1}-sequence of ordinals. This argument can be strengthened to destroy presaturation as well. Thus BMM implies neither of these saturation properties. One can also show via an iterated forcing argument that BMM does not imply saturation, and I believe that this argument can be modified to reproduce the result that BMM does not imply presaturation. The following question remains open, as far as I know. Again, the hypothesis can be varied.

4 Question. Does Martin’s Maximum imply that it is possible to destroy the saturation of NS_{ω_{1}} without adding a subset of ω_{1}?

Consider the following game G of length ω_{1} + 1. In each round α, Player I chooses a set A_{α} ∈ NS_{ω_{1}}^{+}, and Player II chooses a set X_{α} ⊆ NS_{ω_{1}}^{+} of size ℵ_{1}. Player II must play to ensure that for all α < β, A_{β} \A_{α} ∈ NS_{ω_{1}}, and, for each B ∈ X_{α}, A_{α} ∩ B ∈ NS_{ω_{1}}^{+} implies A_{β} ∩ B ∈ NS_{ω_{1}}^{+}. Player II wins the run of the game if in any round he cannot play. Say that NS_{ω_{1}} is game complete if Player II does not have a winning strategy in this game. In the ℙ_{max} extension, NS_{ω_{1}} is game complete, and game completeness implies that it is possible to force that NS_{ω_{1}} is not saturated, without adding a subset of ω_{1}. Game completeness also implies precipitousness.

5 Question. Does Martin’s Maximum imply that NS_{ω_{1}} is game complete?

More generally, one could ask for an iterated forcing argument producing a model in which saturation of NS_{ω_{1}} can be destroyed without adding a subset of ω_{1}.

Given a cardinal κ, the κ-Chang Model (κ-CM) is L(Ord^{κ}). Precipitousness of NS_{ω_{1}} is computed in ω_{1}-CM.

6 Question. Is it (always) possible to change the theory of ω_{1}-CM without adding a subset of ω_{1}?

One could ask the same question about arbitrary κ.

7 Question. Given a cardinal κ, is it (always) possible to change the theory of κ-CM without adding a subset of κ?

Again, Questions 6 and 7 can be varied by context. One interesting case is when Bounded Martin’s Maximum holds.

For fun, one can ask the following, and similar questions for arbitrary κ.

8 Question. It is consistent with all large cardinals that ω_{1}-CM is correct about ω_{3}?

In ZFC, precipitousness of NS_{ω_{1}} is a local property, in that it is decided in H((2^{ℵ_{1}})^{+}).

9 Question. Is precipitousness of NS_{ω_{1}} a local property in ω_{1}-CM?

10 Question. Is game completeness of NS_{ω_{1}} computed in ω_{1}-CM?

If NS_{ω_{1}} is saturated, then there exists for each ordinal α a canonical function for α, that is, a function from ω_{1} to the ordinals which is forced to represent α in all V -generic ultrapowers for NS_{ω_{1}}. The existence of canonical functions for each ordinal implies precipitousness.

11 Question. Does Bounded Martin’s Maximum imply that there is a canonical function for each ordinal?

Again, a negative answer to Question 11 would be given by a positive answer to Question 12.

12 Question. If Bounded Martin’s Maximum holds, is it possible to force the nonexistence of a canonical function for some ordinal without adding a subset of ω_{1}?

13 Question. Is the existence of a canonical function for each ordinal a local property?

14 Question. Is the existence of a canonical function for each ordinal a local property in ω_{1}-CM?

Finally, we ask a variation of a question asked by Justin Moore. Recall that Shelah has shown in ZFC that there exists a club guessing sequence at ω_{2}.

15 Question. Is is consistent with all large cardinals that there exists a sequence in ω_{1}-CM which is club guessing at ω_{2} in V?

Tetsuya Ishiu and I have noticed that if MA_{ℵ_{1}} holds then it is possible to add a club subset of ω_{2} not guessed by any ground model sequence, without adding an ω_{1}-sequence of ordinals.