On Countable Stationary Towers
Yo Matsubara
Nagoya University
Joint work withToshimichi Usuba (Waseda University)
1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 RP (the regularityproperties of reals)”.
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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 = 𝜔0 1𝑗 𝜔0 = 𝜆• (2) 3 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 satisifies RP.
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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 ℙ7 in the ground model 𝑉
with cardinality less than λ and a ℙ7 -generic 𝐻 ∈𝑉 𝐺 such that 𝑥 ∈ 𝑉 𝐻 .
With this property, one can show that the reals in
𝑀 (= ℝ1 ; ) are the reals in a generic extension of𝑉 by Levy collapsing our inaccessible λ．Hence, by
Solovay’s theorem, 𝑀 thinks that 𝐿 ℝ satisfies RP．
So, by the elementarity of 𝑗, 𝐿 ℝ satisfies RP in 𝑉.
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Countable Stationary Towers

Definition Let λ be a limit ordinal > 𝜔0 .
Let ℚ? denote the set𝑝 ∈ 𝑉? ∶ 𝑝 𝑖𝑠 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑖𝑛 ℘3G ⋃𝑝.

For 𝑝, 𝑞 ∈ ℚ? , we let𝑝 ≤ℚM 𝑞 if “ ∪ 𝑞 ⊆∪ 𝑝 ∧ ∀𝑥 ∈ 𝑝 𝑥 ∩ ∪ 𝑞 ∈ 𝑞 ”• The p.o. ℚ? , ≤ℚM is known as the countablestationary tower of height λ.
5RemarkIf λ is an inaccessible cardinal, then ℚ? isforcing equivalent to the poset associated to thetower 𝑁𝑆3GT : 𝛼 < 𝜆 where 𝑁𝑆3GT is thenonstationary ideal over ℘3G 𝛼.

6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 “𝑐𝑜𝑓 𝜆 > 𝜔”
If ℚ? = 𝜆 , then presaturation of ℚ? implies that“ 3𝑈𝑙𝑡 𝑉; 𝐺 ⊆ 𝑈𝑙𝑡 𝑉; 𝐺 ”
i.e. 𝑈𝑙𝑡 𝑉; 𝐺 is closed under taking ω sequences in 𝑉 ℚM .
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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; 𝐺with 𝑐𝑟𝑖𝑡 𝑗 = 𝜔0 1 .
Remark If ℚ? is presaturated with ℚ? = 𝜆, then it is precipitous. Furthermore if 𝑗 ∶ 𝑉 → 𝑀 is ageneric elementary embedding associated with suchℚ? , then 𝑗 𝜔0 = 𝜆 and 3 𝑀 ⊆ 𝑀 in 𝑉 ℚM .

8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 RP” .
Even ω many Woodin cardinals cannot imply 𝐿 ℝ satisfies
RP .
Woodin showed the consistency of “ω many Woodin
cardinals + 𝐿 ℝ satisfies AC ”
from the consistency of ZFC +ω many Woodin cardinals.
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If we let 𝛿 denote ℚ? l , then the well-foundednessof 𝑈𝑙𝑡 𝑉m ; 𝐺 is equivalent to the well-foundedness of𝑈𝑙𝑡 𝑉; 𝐺 .
So for an inaccessible λ, the precipitousness of ℚ? isequivalent to “ℚ? forces 𝑈𝑙𝑡 𝑉?n ; 𝐺 is well-founded”.
This motivated the next definition.
Definitionℚ? is semi-precipitous if ℚ? forces“𝑈𝑙𝑡 𝑉? ; 𝐺 is well-founded”
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We are interested in the following questions:
1. Can we prove “𝐿 ℝ satisfies RP” from theproperties of ℚ? ’s such as precipitousness orpresaturation without explicit use of strong large
cardinals like Woodin cardinals ?
2. Can ℚ? be precipitous when λ is not a large
cardinal? For example, can ℚ? be precipitous for asingular cardinal λ? We know that ℚ? cannot bepresaturated if λ is a singular cardinal with ℚ? = 𝜆.
3. Is the semi-precipitousness of ℚ?
a “large cardinal” property?

11Results
Theorem1
Suppose ℚT is precipitous.
Then for each 𝑝 ∈ ℚT , there is a stationary subset 𝑝∗of 𝒫3G 𝑉Tl0 such that , for every limit ordinal 𝛽 > 𝛼,𝑝∗ ≤ℚr 𝑝 and 𝑝∗ ⊩ℚr “𝐺̇ ∩ ℚT 𝑖𝑠 ℚT -generic ”
where 𝐺̇ is the canonical name for a ℚu -generic filter .
Note If ∃𝛿 > 𝛼 (ℚm is semi-precipitous), then theconclusion of the above theorem implies theprecipitousness of ℚT .
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To prove Theorem1, we used the following game ΓT :Players I and II alternately play elements of ℚT suchthat 𝑝x ≥ 𝑞x ≥ 𝑝0 ≥ 𝑞0 ≥ ⋯ where 𝑝{ ʼs are Iʼs moves
and 𝑞{ ʼs are IIʼs moves. Player II wins this game iff
there is some 𝑥 ∈ 𝑉T 3 such that 𝑥 ∩ ⋃𝑝{ ∈ 𝑝{ forevery 𝑛 < 𝜔.
Theorem (Doug Burke)
Let 𝛼 be a limit ordinal > 𝜔. Then the following areequivalent:(1) ℚT is precipitous.
(2)Player I does not have a winning strategy in thegame ΓT .
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We used the next claim to prove Theorem 1:
Claim If ℚT is precipitous, then for every 𝑝 ∈ ℚT ,the following set 𝑝∗ is stationary:𝑝∗ = { 𝑥 ∈ 𝑉Tl0 3 : 𝑥 ∩ ∪ 𝑝 ∈ 𝑝 ∧∀𝐷 ∈ 𝑥 𝐷 𝑖𝑠 𝑑𝑒𝑛𝑠𝑒 𝑖𝑛 ℚT → ∃𝑟 ∈ 𝐷 ∩ 𝑥 𝑥 ∩∪ 𝑟 ∈ 𝑟
We can prove this claim by producing a winning strategy
for Player I in the game ΓT using the assumption of theabove set being non-stationary.
Theorem 1, together with Woodin’s results, imply the nextresult.

14Theorem 2If λ is an inaccessible cardinal such that(i) ℚ? is presaturated and(ii) the set 𝛼 < 𝜆 ∶ ℚT 𝑖𝑠 𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠 isunbounded in λ,then 𝐿 ℝ satisfies RP.
Note: By Theorem 1, (ii) of Theorem 2 implies thatℚ? forces
𝛼 < 𝜆: 𝐺 ∩ ℚT 𝑖𝑠 ℚT 𝑔𝑒𝑛𝑒𝑟𝑖𝑐 to be unbounded in𝜆 where 𝐺 is ℚ? -generic.
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We obtained some results concerning the notion ofsemi-precipitousness.
Theorem 3
Suppose 𝑉? = 𝜆 and 𝜔 < 𝑐𝑜𝑓(𝜆) < 𝜆.
If ℚ? is semi-precipitous, then it is precipitous.
The precipitousness of ℚ? where λ is aninaccessible cardinal implies the existence of many𝛼’s < λ where ℚT is semi-precipitous.
16Theorem 4
If ℚ? is precipitous where λ is inaccessible ,then 𝛼 < 𝜆 ∶ 𝑐𝑜𝑓 𝛼 = 𝜔 ∧ ℚT 𝑖𝑠 𝑠𝑒𝑚𝑖𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠 isstationary in λ.
Furthermore, if λ is a Mahlo cardinal, then𝛼 < 𝜆 ∶ 𝛼 𝑖𝑠 𝑖𝑛𝑎𝑐𝑐𝑒𝑠𝑠𝑖𝑏𝑙𝑒 ∧ ℚT 𝑖𝑠 𝑠𝑒𝑚𝑖𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠is stationary in λ.
The last result indicates that“semi-precipitous” cannot imply “precipitousness” ingeneral.
Otherwise, from λ being a Woodin cardinal, using
Theorem 2 and Theorem 4, we get“𝐿 ℝ satisfies RP”. But, by Woodin’s work, one Woodin
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cardinal cannot imply that conclusion.

For an inaccessible λ, 𝛼 < 𝜆 ∶ ℚT 𝑖𝑠 𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠
being stationary has interesting consequences.
Theorem 5Suppose λ is an inaccessible cardinal.
If 𝛼 < 𝜆 ∶ ℚT 𝑖𝑠 𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠 is stationary, thenℚ? is precipitous. In fact, it is presaturated,
Theorem 6
If ℚ? is precipitous where λ is a weakly compact
cardinal, then 𝛼 < 𝜆 ∶ ℚT 𝑖𝑠 𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑜𝑢𝑠 is stationary,
impling the presaturation of ℚ? .
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Theorem 7
If ℚ? is precipitous where λ is a weakly compact
cardinal, then 𝐿 ℝ satisfies RP.

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Questions
1. Can we derive “𝐿 ℝ satisfies AD” from ourhypotheses concerning the properties of countablestationary towers?
2. We still do not have answers to the following problems;(i) Can ℚ? be precipitous where λ is not a large cardinal?
(ii) Is semi-precipitousness of ℚ? a “large cardinal”
property in general?
We know the following;“ℚ? is semi-precipitous + ℚ? forces 𝑗 𝜔0 = 𝜆” implies∃0# .
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