Countable Stationary Towers andReflection Principles

Yo Matsubara
Chubu University
Joint work withToshimichi Usuba (Waseda University)1History of stationary tower forcing
In 1985, inspired by the work of Foreman ,Magidor, and Shelah, H.Woodin introduced
stationary tower forcing.
In the Cabal seminar at UCLA, Woodin
circulated handwritten notes which contained
many important results.
One of them is the following theorem:
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Theorem ( H. Woodin )If there exists a supercompact cardinal,
then every set of reals in 𝐿 ℝ is Lebesgue
measurable and has the Baire property andthe perfect set property.
Abbreviation : We denote the conclusion of thistheorem by “𝐿 ℝ satisifies the regularity
properties of reals”.
3Soon after the announcement of Woodin‘sresults, the large cardinal hypothesis of thetheorem was considerably weakened and theconclusions were drastically strengthened.
One of the unique features of Woodin’s proof isthat forcing was used to prove the fact about
the universe V, unlike the usual use of genericextensions implying some propositions
concerning consistency. The key in his proof wasobtaining a suitable generic elementaryembedding satisfying the hypotheses of thefollowing lemma:

4Woodin’s Lemma
Let λ be an inaccessible cardinal.
Suppose there is a p.o. ℙ such that ℙ forces the following:
There is a generic elementary embedding 𝑗 ∶ 𝑉 → 𝑀 with• (1) 𝑐𝑟𝑖𝑡 𝑗 = 𝜔! "𝑗 𝜔! = 𝜆• (2) # 𝑀 ⊆ 𝑀 (i.e 𝑀 is closed under taking 𝜔 sequencesin 𝑉 ℙ )• (3) every real is obtained by a small (size less than λ)forcing
Then 𝐿 ℝ satisfies the regularity properties of reals.
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By “every real is obtained by a small forcing”, wemean that in 𝑉 𝐺 , where 𝐺 is ℙ -generic, for eachreal 𝑥 there are a poset ℙ! in the ground model 𝑉with cardinality less than λ and a ℙ! -generic 𝐻 ∈𝑉 𝐺 such that 𝑥 ∈ 𝑉 𝐻 .
With this property, one can show that the reals in
𝑀 (= ℝ" # ) are the reals in a generic extension of
𝑉 by Levy collapsing our inaccessible λ．Hence, by
Solovay’s theorem, 𝑀 thinks that 𝐿 ℝ satisfies
the regularity properties of reals ．
So, by the elementarity of 𝑗, 𝐿 ℝ satisfies
the regularity properties of reals in 𝑉.
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Countable Stationary Towers

Definition Let λ be a limit ordinal > 𝜔! .
Let ℚ" denote the set𝑝 ∈ 𝑉" ∶ 𝑝 𝑖𝑠 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑖𝑛 ⋃𝑝 #.

For 𝑝, 𝑞 ∈ ℚ" , we let𝑝 ≤ℚ! 𝑞 if “ ∪ 𝑞 ⊆∪ 𝑝 ∧ ∀𝑥 ∈ 𝑝 𝑥 ∩ ∪ 𝑞 ∈ 𝑞 ”• The p.o. ℚ" , ≤ℚ! is known as the countablestationary tower of height λ.
7RemarkIf λ is an inaccessible cardinal, then ℚ! isforcing equivalent to the poset associated to thetower 𝑁𝑆"! # : 𝛼 < 𝜆 where 𝑁𝑆"! # is thenonstationary ideal over 𝒫"! 𝛼.

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If 𝐺 is a ℚ$ -generic filter over 𝑉, just as with thenonstationary ideal, one can take the generic ultrapower
𝑈𝑙𝑡 𝑉; 𝐺 of 𝑉.
In general, there is no guarantee that 𝑈𝑙𝑡 𝑉; 𝐺 is wellfounded.
Definition(i) ℚ$ is precipitous if ℚ$ forces“𝑈𝑙𝑡 𝑉; 𝐺 is well-founded”(ii) ℚ$ is presaturated if ℚ$ forces“ %𝑈𝑙𝑡 𝑉; 𝐺 ⊆ 𝑈𝑙𝑡 𝑉; 𝐺 ”
i.e. 𝑈𝑙𝑡 𝑉; 𝐺 is closed under taking ω sequences in 𝑉 ℚ! .
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If 𝑈𝑙𝑡 𝑉; 𝐺 is well-founded, then, by taking
its transitive collapse, we obtain a genericelementary embedding%𝑗 ∶ 𝑉 → 𝑀 ≅ 𝑈𝑙𝑡 𝑉; 𝐺 with 𝑐𝑟𝑖𝑡 𝑗 = 𝜔$ .
Remark If ℚ! is presaturated, then it isprecipitous.

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Theorem (Woodin)
If 𝜆 is a Woodin cardinal, then ℚ$ is presaturated.
Remark The existence of one Woodin cardinal is notenough to imply “ 𝐿 ℝ satisfies the regularity propertiesof reals” .
Even ω many Woodin cardinals cannot imply 𝐿 ℝ
satisfies the regularity properties of reals . .
Woodin showed the consistency of “ω many Woodin
cardinals + 𝐿 ℝ ⊨ AC ”
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Main Questions:
We are interested in the following questions:
1. Can ℚ$ be precipitous when λ is not a large
cardinal? For example, can ℚ$ be precipitous fora singular cardinal λ?
2.

Can ℚ$ be precipitous without beingpresaturated?

12Remark
We are particularly interested in the question ofwhether ℚ$ can be precipitous where λ is a singularcardinal.
For the nonstationary ideal 𝑁𝑆'$ over 𝒫' 𝜆, it is notknown whether 𝑁𝑆'$ can be precipitous for a singularλ.
In fact, it is known that 𝑁𝑆'$ cannot be precipitous
for any strong limit singular λ.

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The following lemma was known in some form oranother.
Lemma 1Let 𝛾 be a limit ordinal > 𝜔( . The following (1) and(2) are equivalent:(1) ℚ) forces “𝑐𝑜𝑓 𝛾@ > 𝜔”.
(2) ℚ) is presaturated.

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Reflection Principles
We use the following form of the reflection
principle;
Definition: Let 𝜅 and 𝜆 be cardinals withℵ$ < 𝜅 ≤ 𝜆. We say that 𝑅𝑃 𝜅, 𝜆 holds if, for"every stationary 𝑆 ⊆ 𝜆 and 𝛼 < 𝜅, there is some𝐴 ∈ 𝒫& 𝜆 such that 𝑆 ∩ 𝐴 " is stationary in 𝐴 "and 𝛼 ⊆ 𝐴.
We know that the existence of a supercompact
cardinal implies the above reflection principle.

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Lemma 2 If 𝜅 is a supercompact cardinal, then𝑅𝑃 𝜅, 𝜆 holds for every cardinal 𝜆 ≥ 𝜅.
By the definition of 𝑅𝑃 𝜅, 𝜆 , it is easy to seethat the following lemma holds:
Lemma 3 Let 𝜅 be a limit cardinal and 𝜆 be acardinal with 𝜅 < 𝜆.
If 𝜇 < 𝜅 |𝑅𝑃 𝜇, 𝜆 ℎ𝑜𝑙𝑑𝑠 is unbounded in 𝜅,then 𝑅𝑃 𝜅, 𝜆 holds as well.
Remark 4 From the above two lemmas(lemmata?), if 𝜅 is a limit of supercompact
cardinals, then 𝑅𝑃 𝜅, 𝜆 holds for every cardinal 𝜆≥ 𝜅.
16Our reflection principle can imply theprecipitousness of some countable stationary tower.
Theorem 5
Suppose 𝛾 is a limit ordinal > 𝜔 and 𝜅 = 𝑉) .
If 𝑅𝑃 𝜅, 2' holds , then ℚ) is precipitous.
From this theorem, Lemma 1 and above Remark 4,
the next corollary follows immediately:
Corollary 6
If 𝜅 is a limit of supercompact cardinals, then ℚ' isprecipitous. Furthermore if cf 𝜅 = 𝜔, then ℚ' isprecipitous but not presaturated.
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The last corollary gave positive answers toboth questions in Main Questions.
We state a few other results concerning
countable stationary towers.
Proposition 7 If 𝜅 is a supercompact cardinal,
then there is a forcing extension in which 𝜅 is aleast measurable cardinal and ℚ' is precipitous.

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By an Easton support iteration of Levy collapses, one can prove
the next result:

Proposition 8
Suppose there are proper class many supercompactcardinals. Then there is a class forcing extension of 𝑉in which ℚ' is precipitous for every cardinal 𝜅 > 𝜔(with 𝑉' = κ. Furthermore, in that extension, ℚ' ispresaturated for every inaccessible cardinal 𝜅.
The “presaturation” part is due to the following theorem whichappeared in our previous paper.
Old Theorem
Suppose λ is an inaccessible cardinal.
If 𝛼 < 𝜆 ∶ ℚ" 𝑖𝑠 𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠 is stationary, thenℚ# is precipitous. In fact, it is presaturated.

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