This paper introduces a weak condition that guarantees the existence of Souslin trees at successors of regular cardinals, improving upon previous theorems in the field.
Contribution
It provides an optimal condition for Souslin tree existence, enhancing earlier results by Gregory and the author.
Findings
01
Established a weak sufficient condition for Souslin trees.
02
Improved upon Gregory's theorem.
03
Enhanced previous results with a more general condition.
Abstract
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
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Full text
Souslin trees at successors of regular cardinals
Assaf Rinot
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel.
Abstract.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals.
The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
2010 Mathematics Subject Classification:
Primary 03E05; Secondary 03E65.
This research was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18)
Introduction
In [Gre76], Gregory proved that for every (regular) uncountable cardinal λ=λ<λ,
if 2λ=λ+ and there exists a non-reflecting stationary subset of E<λλ+,
then there exists a λ+-Souslin tree.
A special case of a result from [Rin17] asserts that for every uncountable cardinal λ=λ<λ,
if 2λ=λ+ and □(λ+) holds, then there exists a λ+-Souslin tree.
By results from inner model theory, Gregory’s theorem implies that if GCH holds,
and there are no ℵ2-Souslin trees, then ℵ2 is a Mahlo cardinal in L,
and our theorem implies that if GCH holds,
and there are no ℵ2-Souslin trees, then ℵ2 is a weakly compact cardinal in L.
While the former corollary follows from the latter, the combinatorial theorem of Gregory does not follow from ours.
The purpose of this note is to present a new combinatorial theorem that implies both:
Main Theorem**.**
Suppose that λ=λ<λ is an uncountable cardinal, and 2λ=λ+.
If there exists a □(λ+,λ)-sequence ⟨Cα∣α<λ+⟩
for which {α∈E<λλ+∣∣Cα∣<λ} is stationary,
then there exists a λ+-Souslin tree.
An immediate corollary to the Main Theorem is an optimal improvement to a result from [Rin17] that was promised in [BR19a]:
Corollary**.**
Suppose that λ is a regular uncountable cardinal.
If GCH and □(λ+,<λ) both hold,
then there exists a λ+-Souslin tree.
The corollary is indeed optimal, since GCH implies that □(λ+,λ) holds for every regular cardinal λ (in fact, with a witnessing
sequence ⟨Cα∣α<λ+⟩
satisfying ∣Cα∣=1 for all α∈Eλλ+),
whereas by a recent striking result of Asperó and Golshani [AG18], ZFC+GCH is
consistent with the non-existence of a λ+-Souslin tree for any prescribed value of a regular uncountable λ.
Notation and conventions
Throughout this note,
κ and λ stand for arbitrary regular uncountable cardinals.
Write [κ]<λ for the collection of all subsets of κ of cardinality less than λ.
Denote Eλκ:={α<κ∣cf(α)=λ}
and E<λκ:={α<κ∣cf(α)<λ}.
Suppose that C and D are sets of ordinals.
Write C⊑D iff there exists some ordinal β such that C=D∩β.
Write acc(C):={α∈C∣sup(C∩α)=α>0}, nacc(C):=C∖acc(C),
and acc+(C):={α<sup(C)∣sup(C∩α)=α>0}.
1. Square principles and Souslin trees
Definition 1.1**.**
For any cardinal μ, □(κ,<μ) asserts
the existence of a sequence ⟨Cα∣α<κ⟩ such that:
(1)
For every limit ordinal α<κ:
•
Cα is a nonempty collection of club subsets of α, with ∣Cα∣<μ;
•
for every C∈Cα and αˉ∈acc(C), we have C∩αˉ∈Cαˉ;
2. (2)
For every club D⊆κ, there is α∈acc(D) such that D∩α∈/Cα.
Remark 1.2*.*
(1)
Note that there are no restrictions on otp(C) for C∈Cα.
2. (2)
We write □(κ,μ) for □(κ,<μ+), and write □(κ) for □(κ,1).
To connect Gregory’s theorem with the Main Theorem, let us point out the following.
Proposition 1.3**.**
Suppose that λ<λ=λ and there exists a non-reflecting stationary subset of E<λλ+.
Then there exists a □(λ+,λ)-sequence ⟨Cα∣α<λ+⟩
for which {α∈E<λλ+∣∣Cα∣<λ} is stationary.
Proof.
Fix a subset S⊆E<λλ+ which is stationary and non-reflecting.
We now define C:=⟨Cα∣α<λ+⟩, as follows:
▶ Let C0:={∅}.
▶ For all α<λ+, let Cα+1:={{α}}.
▶ For all α∈S∪Eλλ+, since S is non-reflecting, we may fix a club Cα in α of order-type cf(α) which is disjoint from S.
Now, let Cα:={Cα}.
▶ For all α∈acc(λ+)∖(S∪Eλλ+), let Cα be the collection of all clubs C in α
such that otp(C)<λ and C∩S=∅. As cf(α)<λ and as S is non-reflecting, we know that Cα
is nonempty. As λ<λ=λ, we also know that ∣Cα∣≤λ.
Let us verify that C is as sought:
•
Evidently, {α∈E<λλ+∣∣Cα∣<λ} covers the stationary set S.
•
Fix arbitrary α∈acc(λ+), C∈Cα and αˉ∈acc(C). There are two options:
▶ If α∈S∪Eλλ+, then cf(αˉ)≤otp(C∩αˉ)<otp(C)=cf(α)≤λ.
Also, C=Cα is disjoint from S, so that, altogether, αˉ∈acc(λ+)∖(S∪Eλλ+).
It now follows from the definition of Cαˉ that C∩αˉ∈Cαˉ.
▶ Otherwise, C is a club α such that
cf(αˉ)≤otp(C∩αˉ)<otp(C)<λ and C∩S=∅.
It again follows from the definition of Cαˉ that C∩αˉ∈Cαˉ.
•
Given any club D in λ+, pick α∈D such that otp(D∩α)=λ+ω. Then α∈acc(D) and D∩α∈Cα.∎
As mentioned earlier, even in the presence of GCH, □(κ,<κ) does not imply the existence of a κ-Souslin tree.
For this, Brodsky and the author have introduced the following slight strengthening of □(κ,<κ):
⊠∗(κ) asserts the existence of a sequence ⟨Cα∣α<κ⟩ such that:
(1)
For every limit ordinal α<κ:
•
Cα is a nonempty collection of club subsets of α, with ∣Cα∣<κ;
•
for every C∈Cα and αˉ∈acc(C), we have C∩αˉ∈Cαˉ;
2. (2)
For every cofinal X⊆κ, there is α∈acc(κ) such that sup(nacc(C)∩X)=α for all C∈Cα.
In this paper, we shall not construct Souslin trees (we refer the reader to [BR17a] for background and definitions);
all we need is encapsulated in the following fact.
Suppose that D is a club in κ. Define a function ΦD:P(κ)→P(κ) by letting,
for all x∈P(κ),
[TABLE]
Lemma 2.2**.**
Suppose that:
•
λ<κ;
•
C=⟨Cα∣α<κ⟩* is a □(κ,κ)-sequence;*
•
S* is a stationary subset of {α∈acc(κ)∣∣Cα∣<λ}.*
Then there exists a club D⊆κ
such that for every club E⊆κ,
there exists α∈S such that sup(nacc(ΦD(C))∩E)=α for all C∈Cα.
Remark 2.3*.*
All ingredients for the upcoming proof may already be found in [BR19a].
For completeness, we give here a self-contained proof that avoids various concepts that appear in [BR19a].
Proof of Lemma.
Suppose not.
Then, for every club D⊆κ, we may find a club ED⊆κ such that, for every δ∈S,
there is CδD∈Cδ with
[TABLE]
Define a sequence ⟨Ei∣i≤λ⟩ of clubs in κ, by recursion, as follows:
•
Set E0:=κ;
•
For all i<λ, set Ei+1:=EEi∩Ei;
•
For all i∈acc(λ+1), set Ei:=⋂j<iEj.
Write E:=Eλ.
For each δ∈S,
since {CδEi∣i<λ}⊆Cδ, and
λ=cf(λ)>∣Cδ∣,
we may pick Cδ∈Cδ such that
Iδ:={i<λ∣CδEi=Cδ} is cofinal in λ.
Now, there are three cases to consider, each leading to a contradiction:
Case 1. Suppose that there exists δ∈S∩E>ωκ for which sup(E∩δ∖Cδ)=δ.
Fix such δ and let {in∣n<ω} be the increasing enumeration of some subset of Iδ.
Since ⟨Ei∣i<λ⟩ is a ⊆-decreasing sequence,
for all n<ω, we have in particular that Ein+1⊆Ein+1⊆EEin, so that αn:=sup(nacc(ΦEin(Cδ))∩Ein+1) is <δ. Put α:=supn<ωαn.
As cf(δ)>ω, we have α<δ.
Fix β∈(E∩δ)∖Cδ above α.
Put γ:=min(Cδ∖β).
Then δ>γ>β>α, and for all i<λ, since β∈E⊆Ei,
we infer that sup(Ei∩γ)≥β. So it follows from the definition of ΦEi(Cδ) that min(ΦEi(Cδ)∖β)=sup(Ei∩γ) for all i<λ.
Since ⟨Ein∣n<ω⟩ is an infinite ⊆-decreasing sequence, let us fix some n<ω
such that sup(Ein∩γ)=sup(Ein+1∩γ).
Then min(ΦEin(Cδ)∖β)=min(ΦEin+1(Cδ)∖β),
and in particular, β∗:=min(ΦEin(Cδ)∖β) is in Ein+1∖(α+1). Now, there are two options, each leading to a contradiction:
▶
If β∗∈nacc(ΦEin(Cδ)), then we get a contradiction to the fact that β∗>α≥αn.
▶
If β∗∈acc(ΦEin(Cδ)), then β∗=β and β∗∈acc(Cδ), contradicting the fact that β∈/Cδ.
Case 2. Suppose that there exists δ∈S∩Eωκ for which sup(E∩δ∖Cδ)=δ.
Fix such δ, and note that, for all i∈Iδ, the ordinal αi:=sup(nacc(ΦEi(Cδ))∩Ei+1) is <δ.
So, as cf(δ)=ω1,
let {iν∣ν<ω1} be the increasing enumeration of some subset of Iδ, for which α:=supν<ω1αiν is <δ.
Fix β∈(E∩δ)∖Cδ above α.
Put γ:=min(Cδ∖β).
Then δ>γ>β>α, and min(ΦEi(Cδ)∖β)=sup(Ei∩γ) for all i<λ.
Fix some ν<ω1
such that sup(Eiν∩γ)=sup(Eiν+1∩γ).
Then β∗:=min(ΦEiν(Cδ)∖β) is in Eiν+1∖(α+1), and as in the previous case, each of the two possible options leads to a contradiction.
Case 3. Suppose that sup(E∩δ∖Cδ)<δ for all δ∈S.
Fix ϵ<κ for which S′:={δ∈S∣sup(E∩δ∖Cδ)=ϵ} is stationary.
Put B:=acc(E∖ϵ), and note that, for every δ∈S′, we have B∩δ⊆acc(Cδ).
Let {βα∣α<κ} denote the increasing enumeration of the club {0}∪B.
For all α<κ, put:
[TABLE]
Claim 2.3.1**.**
(⋃α<κTα,⊑)* is a tree whose αth level is Tα,
and ∣Tα∣≤∣Cβα∣ for all α<κ.*
Proof.
We commence by pointing out that Tα⊆Cβα for all α<κ.
Clearly, T0={∅}=C0=Cβ0.
Thus, consider an arbitrary nonzero α<κ along with some t∈Tα.
Fix δ∈S′ above βα such that t=Cδ∩βα.
Then βα∈B∩δ⊆acc(Cδ),
so that Cδ∩βα∈Cβα. That is, t∈Cβα.
This shows that for all t∈⋃α<κTα:
[TABLE]
Next, consider arbitrary α<κ and t∈Tα,
and let t↓:={s∈⋃α′<κTα′∣s⊑t,s=t} be the set of predecessors of t.
Fix δ∈S′ above βα such that t=Cδ∩βα.
We claim that t↓={Cδ∩βα′∣α′<α},
from which it follows that (t↓,⊑)≅(α,∈).
Consider α′<α.
Then βα′<βα<δ,
so that s:=Cδ∩βα′ is in Tα′,
and it is clear that s is a proper initial segment of t. That is, s∈t↓.
Conversely, consider s∈t↓.
Fix α′<κ such that s∈Tα′.
By our earlier observation, sup(s)=βα′,
so that since s⊑t, s=t, and sup(t)=βα, we must have βα′<βα, and therefore α′<α.
Thus, s=t∩βα′=(Cδ∩βα)∩βα′=Cδ∩βα′, as required.
∎
Consider the stationary set S′′:={α∈S′∣α=βα}.
For each α∈S′′, we have ∣Tα∣≤∣Cα∣<λ,
so T:=⋃α∈S′′Tα ordered by ⊑ is a κ-tree each of whose levels has cardinality less than λ.
Now, by a lemma of Kurepa (see [Kan03, Proposition 7.9]), (T,⊑)
admits a cofinal branch, i.e., a chain C⊆T (with respect to ⊑)
that satisfies ∣C∩Tα∣=1 for all α∈S′′.
Put D:=⋃C and note that D is a club in κ.
As C is a □(κ,κ)-sequence,
let us pick β∈acc(D) such that D∩β∈/Cβ.
Now, by definition of D, let us pick t∈C such that D∩β⊑t.
Then, as t∈T, let us pick δ∈S′ above sup(t) such that t⊑Cδ.
So D∩β⊑Cδ.
As β∈acc(D), we have β∈acc(Cδ), and hence D∩β=Cδ∩β∈Cβ,
contradicting the choice of β.
∎
Corollary 2.4**.**
Suppose that C=⟨Cα∣α<λ+⟩ is a □(λ+,λ)-sequence
for which {α∈E<λλ+∣∣Cα∣<λ} is stationary.
Then there exists a □(λ+,λ)-sequence C∙=⟨Cα∙∣α<λ+⟩
such that, for every club E⊆λ+,
there exists α∈acc(λ+)∩E<λλ+ with ∣Cα∙∣<λ such that sup(nacc(y)∩E)=α for all y∈Cα∙.
Proof.
Appeal to Lemma 2.2 with κ:=λ+,
C, and
S:={α∈acc(κ)∩E<λκ∣∣Cα∣<λ},
and let D⊆λ+ be the outcome club.
Define C∙=⟨Cα∙∣α<λ+⟩ as follows:
▶
C0∙:={∅}.
▶
For all α<λ+, Cα+1∙:={{α}}.
▶
For all α∈acc(λ+), let Cα∙:={ΦD(C)∣C∈Cα}.
By [BR19a, Lemma 2.2], for all α∈acc(λ+) and C∈Cα,
ΦD(C) is a club in α
satisfying that, for all αˉ∈acc(ΦD(C)), αˉ∈acc(C) and ΦD(C)∩αˉ=ΦD(C∩αˉ)∈Cαˉ∙.
In addition, by the choice of the club D, we know that for every club E⊆λ+,
there exists α∈acc(λ+)∩E<λλ+ with ∣Cα∙∣≤∣Cα∣<λ such that sup(nacc(y)∩E)=α for all y∈Cα∙.
Finally, given an arbitrary club D′ in λ+, consider the club E:=acc(D′),
and fix α∈acc(λ+) such that sup(nacc(y)∩E)=α for all y∈Cα∙.
It follows that, for all y∈Cα∙, nacc(y)∩acc(D′∩α)=∅, let alone y=D′∩α.
∎
We now arrive at the heart of the matter.
Theorem 2.5**.**
Suppose that λ<λ=λ, 2λ=λ+,
and there exists a □(λ+,λ)-sequence ⟨Cα∣α<λ+⟩
for which {α∈E<λλ+∣∣Cα∣<λ} is stationary.
Then ⊠∗(λ+) holds.
Proof.
Let C∙=⟨Cα∙∣α<λ+⟩ be given by Corollary 2.4.
Fix a bijection π:λ+×λ↔λ+.
Also, for each β<λ+, fix a bijection gβ:λ↔E<λβ+1×λ.
By [Gre76, Lemma 2.1], λ<λ=λ and 2λ=λ+ imply together that ♢∗(E<λλ+) holds.
This means that we may fix a matrix ⟨Zβ,j∣β∈E<λλ+,j<λ⟩ such that, for every Z⊆λ+,
for some club D⊆λ+, we have
[TABLE]
As λ<λ=λ, the main result of [EK65] provides us with a sequence ⟨fi∣i<λ⟩
of functions from λ+ to λ, such that, for every function f:e→λ
with e∈[λ+]<λ, for some i<λ, we have f⊆fi.
Now, let i<λ be arbitrary.
First, define a coloring ci:[λ+]2→λ+ by letting, for all η<β<λ+,
[TABLE]
Then, for every y∈P(λ+), let
[TABLE]
Finally, for every α∈acc(λ+), let Cαi:={yi∣y∈Cα∙}.
Also, let C0i:={∅}, and let Cα+1i:={{α}} for all α<λ+.
Claim 2.5.1**.**
Suppose that α∈acc(λ+) and C∈Cαi. Then:
(1)
C* is a club in α;*
2. (2)
For all αˉ∈acc(C), C∩αˉ∈Cαˉi.
Proof.
Fix a club y∈Cα∙ such that C=yi.
(1) It is easy to see that for any two successive elements η<β of the club y, we have that C∩(η,β] is a singleton.
Consequently, sup(C)=sup(y)=α, and acc+(C)⊆acc(y).
But, by definition of C=yi, we also have acc(y)⊆C, so, C is a club in α.
(2) Let αˉ∈acc(C) be arbitrary. By the above analysis, αˉ∈acc(y), so that y∩αˉ∈Cαˉ.
But C∩αˉ=yi∩αˉ=(y∩αˉ)i, and hence C∩αˉ∈Cαˉi.
∎
Claim 2.5.2**.**
There exists i<λ for which ⟨Cαi∣α<λ+⟩ witnesses ⊠∗(λ+).
Proof.
Suppose not. It follows from Claim 2.5.1 that for each i<λ,
we may pick some cofinal subset Xi⊆λ+ such that, for all α∈acc(λ+),
for some C∈Cαi, we have sup(nacc(C)∩Xi)<α.
Let Z:=π‘‘⋃i<λ(Xi×{i}),
and then fix a club D in λ+ such that for all β∈D:
•
π[β×λ]=β;
•
sup(Xi∩β)=β for all i<λ;
•
if cf(β)<λ, then there exists j<λ with Z∩β=Zβ,j.
Consider the club E:=acc(D).
By the choice of C∙,
we may now pick α∈acc(λ+)∩E<λλ+ with ∣Cα∙∣<λ such that sup(nacc(y)∩E)=α for all y∈Cα∙.
Since cf(α)<λ and ∣Cα∙∣<λ, let us fix some e∈[E∩α]<λ such that sup(nacc(y)∩e)=α for all y∈Cα∙.
Define a function f:e→λ by letting, for all β∈e,
[TABLE]
Fix i<λ such that f⊆fi. By the choice of Xi, let us fix C∈Cαi such that sup(nacc(C)∩Xi)<α.
Find y∈Cα∙ such that C=yi.
Fix a large enough β∈nacc(y)∩e such that η:=sup(y∩β) is greater than sup(nacc(C)∩Xi).
In particular, ci(η,β) must be an element of nacc(C)∖Xi.
Now, there are two cases to consider, each leading to a contradiction:
▶ If cf(β)<λ, then for some j<λ, we have gβ(f(β))=(β,j) and Z∩β=Zβ,j.
But β∈e⊆E⊆D, so that gβ(fi(β))=(β,j),
π[β×λ]=β, and
[TABLE]
As β∈D, we have sup(Xi∩β)=β,
so, ci(η,β)∈Xi∩β. This is a contradiction.
▶ If cf(β)=λ, then let (γ,j):=gβ(f(β)), so that γ∈D and Z∩γ=Zγ,j.
In particular,
{ξ<γ∣π(ξ,i)∈Zgβ(fi(β))}=Xi∩γ and sup(Xi∩γ)=γ.
Since β∈nacc(y),
it also follows that η=sup(y∩β)<γ<β.
Consequently, ci(η,β)∈Xi∩β. This is a contradiction.
∎
This completes the proof.
∎
We are now ready to derive the Main Theorem.
Proof of the Main Theorem.
By [Gre76, Lemma 2.1], λ<λ=λ and 2λ=λ+ imply together that ♢∗(E<λλ+) holds.
In particular, as λ is uncountable, ♢(λ+) holds.
In addition, by Theorem 2.5, ⊠∗(λ+) holds.
So, by Fact 1.5, there exists a λ+-Souslin tree.111In fact, instead of appealing to Fact 1.5, we may appeal to a finer theorem from [BR17b],
thereby getting a λ+-Souslin tree which is moreover λ-complete.
∎
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