Overview

Details

Conclusion and Open Problems

Cichoń’s maximum with evasion number
Takashi Yamazoe
Kobe University

RIMS Set Theory Workshop 2023

Overview

Details

Table

1

Overview

2

Details

3

Conclusion and Open Problems

Conclusion and Open Problems

Overview

Details1

Overview

2

Details

3

Conclusion and Open Problems

Conclusion and Open Problems

Overview

Details

Conclusion and Open Problems

History of Cichoń’s maximum
Cichoń’s maximum : a model where all the cardinal invariants in
Cichoń’s diagram are separated (except for cof(M) and add(M)).
The ﬁrst C.M. was constructed in [GKS19] (assuming LCs):·cov(N )non(M)
cof(N )
2ℵ0

ℵ1add(N )bd·cov(M)non(N )

Another type of C.M. was constructed in [KST19] (with LCs).
·cov(N )non(M)
cof(N )
2ℵ0

ℵ1add(N )bd·cov(M)Later, LCs were eliminated in [GKMS22].

non(N )

Overview

Details

Conclusion and Open Problems

Prediction
We study relationships between C.M. and other cardinal invariants
and focus on evasion number e and prediction number pr.
Deﬁnition (prediction)• π = (D, {πn : n ∈ D}) is a predictor:⇔ D ∈ [ω]ω and each πn is a function ω n → ω.
• π predicts f ∈ ω ω:⇔ for all but ﬁnitely many n ∈ D, f (n) = πn (f ↾ n).
• f evades π :⇔ π does not predict f .
ωpredictf↾ nf (n)f... n − 1n...

DOverview

Details

Conclusion and Open Problems

Evasion/prediction number and Cichoń’s diagram
Deﬁnition (evasion number e, prediction number pr)()
Π consists of predictors,• pr := min |Π| :.
∀f ∈ ω ω , ∃π ∈ Π, π predicts f• e := min {|F | : F ⊆ ω ω , ∀predictor π, ∃f ∈ F, f evades π}.
Evasion number e and prediction number pr can be added toCichoń’s diagram as follows.
cov(N )

non(M)·bd·cov(M)cof(N )
pre
ℵ1add(N )non(N )

2ℵ 0

Overview

Details

Conclusion and Open Problems

Cichoń’s maximum with evasion numbere and pr can be added to Cichoń’s maximum.
Theorem (Goldstern, Kellner, Mejía and Shelah, [GKMS21])
There exists a separation model of the following order:·cov(N )non(M)
cof(N )

2ℵ0preℵ1add(N )bd·cov(M)non(N )

It has been unpublished since it will be a part of the results of amonograph on ccc forcings which they are preparing for 2024.
(Kellner gave a talk of this result at Luminy Workshop in 2021 andthe slides [GKMS21] are uploaded to his personal website.)Overview

Details

Conclusion and Open Problems

Main theorem
We obtained the same separation(C.M.+e) with a diﬀerent method.
Moreover, it turned out that the methods can be mixed.
Main Theorem (Y.)
There exists a separation model of the following order:(blue double lines x = y denote equation x = y.)·cov(N )non(M)
cof(N )
non(E)e∗eubdbeℵ1add(N )·prdprubdpr∗

cov(E)

cov(M)

non(N )

(The deﬁnitions of the newly appeared numbers are given later).

2ℵ 0Overview

Details

Conclusion and Open Problems

Time series through methods[GKS19]C.M.
UF-limit

mix

modifyY.
C.M.+eUF-limit
cUF-limitmix
Y.
C.M.+e + non(E)UF-limit
cUF-limit
FAM-limit[GKMS21]
C.M.+eUF-limit
FAM-limit[KST19]another C.M.
FAM-limitOverview

Details1Overview

2

Details

3

Conclusion and Open Problems

Conclusion and Open Problems

Overview

Details

Conclusion and Open Problems

Variants of prediction
First of all, we give the deﬁnitions of the cardinal invariants which
newly appeared in the complicated diagram above.
First, we introduce variants of evasion/prediction number bychanging the deﬁnition of the prediction.
Deﬁnition (bounding-prediction)• predictor π = (D, {πn : n ∈ D}) bounding-predicts f ∈ ω ω:⇔ for all but ﬁnitely many n ∈ D, f (n) ≤ πn (f ↾ n).
• pr∗ and e∗ denote the prediction number and evasion number
for this bounding-prediction, respectively.
Since bounding-prediction is easier than the normal one, pr∗ ≤ pr.
Conversely, evasion is harder and e ≤ e∗ .

Overview

Details

Conclusion and Open Problems

Variants of prediction
Second, for g ∈ (ω \ 2)ω , we consider predictions on

Qn<ω g(n).

Deﬁnition (g-prediction)()
Π consists of predictors,• prg := min |Π| :.
Q∀f ∈ n<ω g(n), ∃π ∈ Π, π predicts f()
QF ⊆ n<ω g(n), ∀predictor π,• eg := min |F | :.
∃f ∈ F, f evades π(Note that larger g makes evasion easier, eg smaller, prediction
harder and prg larger.)• prubd := sup prg : g ∈ (ω \ 2)ω .
• eubd := min {eg : g ∈ (ω \ 2)ω }.

Overview

Details

Conclusion and Open Problems

Variants and Cichoń’s diagram
These variants, non(E) and cov(E) can be added to Cichoń’s
diagram as follows.
(E denotes the σ-ideal generated by closed measure zero sets.)cov(N )

non(M)

non(E)e∗eubdbeℵ1add(N )··cof(N )
prd

prubd

pr∗

cov(E)

cov(M)

non(N )

2ℵ 0

Overview

Details

Conclusion and Open Problems

Separation order
The order of our separation model is as follows (reposted):(blue double lines x = y denote equation x = y.)·cov(N )non(M)
cof(N )
non(E)e∗eubdbeℵ1add(N )

·2ℵ 0

prd

prubd

pr∗

cov(E)

cov(M)

non(N )

We only talk about the separation of the left side since it is known
that once we separate the left side (with “strong witnesses”), we
can also separate the right side (Boolean Ultrapowers or submodel).

Overview

Details

Conclusion and Open Problems

Forcing posets
Before performing an iteration, we introduce (unfamiliar) forcings
which will be used in the iteration.
• Prediction forcing PR and g-prediction forcing PRg denote theposets which generically add predictors and hence increase eand eg , respectively.
They were introduced in [BS03] and [BS96] (We do not give
their deﬁnitions in this talk, but they are naive).
e is a creature forcing which generically adds eventually• E
diﬀerent reals and hence increases non(M).
e was introduced in [KST19] instead of the standardE
eventually diﬀerent forcing E to use the FAM-limit argument(Later we will explain the reason in detail).

Overview

Details

Conclusion and Open Problems

Iteration
Let λ1 < · · · < λ7 be regular cardinals with some suitable cardinalarithmetics (details omitted).
The goal is to separate the diagram as follows:λ2λ6λ5λ4λ5λ3λ4ℵ1λ1··λ7λ7λ7λ7λ7λ7λ7λ7λ7Overview

Details

Conclusion and Open Problems

IterationWe construct P7 , a ﬁnite support iteration of length λ7 + λ7 .
• The ﬁrst λ7 iterands are Cohen forcings C (For “strongwitnesses”).
• Each of the remaining λ7 iterands is a subforcing of someposet of some size according to the table below left.
posetAB
DPR
PRge
Esize
< λ1< λ2< λ3< λ4< λ5< λ6increaseadd(N )
cov(N )be (and e∗ )eg (and non(E))
non(M)

cov(N )

non(M)

non(E)e∗eubdbeℵ1add(N )·(Goal : add(N ) = λ1 , cov(N ) = λ2 , b = λ3 , e = e∗ = λ4 ,eubd = non(E) = λ5 , non(M) = λ6 and cov(M) = 2ℵ0 = λ7 .)Overview

Details

Conclusion and Open Problems

Iteration(Goal : add(N ) = λ1 , cov(N ) = λ2 , b = λ3 , e = e∗ = λ4 ,eubd = non(E) = λ5 , non(M) = λ6 and cov(M) = 2ℵ0 = λ7 .)• By bookkeeping, it is easily seen that each cardinal invariant isforced to be large (e.g. add(N ) ≥ λ1 ).
• By standard preservation arguments (e.g. “cov(N ) ≤ λ2 (withstrong witnesses)” is preserved through fsi of < λ2 poset,σ-centered, e.t.c.), the smallness of add(N ), cov(N ) andnon(M) holds.
→ The smallness of b, e∗ (and e) and non(E) (and eubd ) remains.
• b : UF-limit argument (by [GKS19])• e∗ : cUF-limit argument ← new !
• non(E) : FAM-limit argument (by Cardona and Mejía)

Overview

Details

Conclusion and Open Problems

UF-limit
Deﬁnition (details omitted)
Let UF denote the set of all ultraﬁlters on ω and P be a poset.
• Q ⊆ P is UF-linked if there exists a function
lim : Qω × UF → P with some suitable properties (omitted).
Moreover, if ran(lim) ⊆ Q, we say Q is cUF-linked(closed-UF-linked).
• P is σ-(c)UF-linked if P has a dense subposet of a countable
union of (c)UF-linked subsets.
Lemma (Brendle and Shelah [BS96])
PR is σ-UF-linked and PRg is σ-cUF-linked for any g ∈ (ω \ 2)ω .
Lemma ([GKMS21])
e is σ-cUF-linked.
EOverview

Details

Conclusion and Open Problems

Main Lemma
When arguing “x ≤ λ”
by XY -limit argument, every iterandof size ≥ λ has to be σ-XY -linked.
In the current situation,
this condition is satisﬁed (right table).
Let c̄˙4 := ⟨ċβ ⟩β<λ4
be the ﬁrst λ4 Cohen reals.

poset
PRPRge
EEUF
✓
✓✓
✓

cUF
×
✓✓
✓

FAM××
✓×

Deﬁnitionc̄˙4 is a strong witness of ⊩P7 e∗ ≤ λ4 if P forces that any predictorπ̇ bounding-predicts only bounded many ċβ in c̄˙4 .
Main Lemma(Y.)
P7 forces e∗ ≤ λ4 with a strong witness c̄˙4 .
(More generally, σ-cUF-linked forcings help to keep e∗ small.)Overview

Details

Conclusion and Open Problems

Proof (Part1)
Sketch of Proof. We divide the proof into three parts.
Part1. ∆-system argument
Assume towards contradiction that there exists a predictor whichbounding-predincts unbounded many ċβ .
That is, there exist p and π̇ = (Ḋ, ⟨π̇k : k ∈ Ḋ⟩) such that p forcesthat unbounded many ċβ is bounding-predicted by π̇.
So, for each i < λ4 , there exist pi ≤ p, βi ∈ [i, λ4 ) and ni < ωsuch that:pi ⊩ “ċβi (k) ≤ π̇k (ċβi ↾ k) for all k ∈ Ḋ above ni ”.
Let S + ⊆ λ7 + λ7 be the set of all stages where the iterands Q̇αe (σ-cUF-linked zone) and S − be theare subforcings of PRg or E
rest (size < λ4 zone). By extending and thinning, we may assume
the following items (next slide):Overview

Details

Conclusion and Open Problems

Proof (Part1)(For i < λ4 , pi ⊩ “ċβi (k) ≤ π̇k (ċβi ↾ k) for all k ∈ Ḋ above ni ”.)• βi ∈ dom(pi ) for all i.
• {dom(pi ) : i < λ4 } forms a ∆-system with root ∇.
• βi ∈/ ∇ for all i. Hence all βi are distinct.
• All pi “folow a common guardrail on ∇”. i.e.,• For α ∈ ∇ ∩ S − , identifying Q̇α with some µ < λ4 (since|Q̇α | < λ4 for α ∈ S − ), all pi (α) are the same element of µ.
• For α ∈ ∇ ∩ S + , all pi (α) belong to the same component ofthe σ-cUF-linked Q̇α .

• All ni are the same n∗ < ω and all Cohen conditions pi (βi ) arethe same s ∈ ω <ω of length n∗ (hence all pi force ċβi ↾ n∗ = s).
e (σ-cUF-linked zone)
S + : PRg , E
β i λ4λ7i· · · λ 7 + λ7···ċβi here,
S − : A, B, C, D, PR (size < λ4 zone)
bounding-predicted above n∗ and ċβi ↾ n∗ = s
Cohen forcings

Overview

Details

Conclusion and Open Problems

Proof (Part2)
Part2. General theory of UF-limitωGenerally speaking, for ⟨rm ⟩m<ω ∈ P7 such that their domains
form a ∆-system with root ∇′ and they follow a common guardrailon ∇′ , we can deﬁne r∞ = lim⟨rm ⟩m<ω , the UF-limit condition of⟨rm ⟩m<ω .
Roughly speaking,dom(r∞ ) := ∇′ and for α ∈ ∇′ ∩ S − , r∞ (α) is
the common value all rm (α) and for α ∈ ∇′ ∩ S + ,r∞ (α) is the UF-limit of ⟨rm (α)⟩m<ω inQ̇α (as they are in the same UF-linked component).

S−∇′S+the same UF-limit
elementr0r1r2r∞

Overview

Details

Conclusion and Open Problems

Proof (Part2)
The main principle of the UF-limit is the following:
UF-limit Principle
r∞ forces that ultraﬁlter many rm are in generic ﬁlter.
The following property is direct from the principle.
UF-Limit Preservation Property
Let φ be a forcing formula (without parameter m).
If all rm force φ, then r∞ also forces φ.
Proof of the property. Let G be any generic ﬁlter containing r∞ .
By the principle, in particular, there exists rm ∈ G. Since rm forces
φ, V [G] ⊨ φ and G is arbitrary containing r∞ .
2OverviewDetails

Conclusion and Open Problems

Proof (Part2)
Recall that Q ⊆ P is cUF-linked if UF-limits
are also in Q and that S + is the σ-cUF-lined zone.
The following lemma holds simply because eachiterand Q̇α is σ-cUF-linked for α ∈ S + and hence
the UF-limit is still in the same component.

S−∇′S+r0
r1
r2
r∞still in the samecomponent !

cUF-limit guardrail closedness Lemma
Let ⟨rm ⟩m<ω and r∞ as above.
Then, r∞ still follows the same guardrail on ∇′ .
Thus, we can consider “limit of limits” !
Strategy
Take limits inﬁnitely many times through preserving some suitableproperties and obtain contradiction.

Overview

Details

Conclusion and Open Problems

Proof (Part3)
Part3. Apply cUF-limit properties to the current situation
Back to the current situation and pick the ﬁrst ω-many pi . Fix abijection ω <ω ↔ ω. For each τ ∈ ω <ω and corresponding i < ω,
let qτ be the extension of pi by changing pi (βi ) = s to s⌢ τ .
(Hence qτ forces ċβi ↾ (n∗ + n) = s⌢ τ where n = |τ |.)Note that all qτ follow a common guardrail on ∇ (since βi ∈/ ∇).
(Hence we can consider limits of qτ ’s.)By qτ ≤ pi and the properties of pi in Part1, we have that for eachτ ∈ ω <ω , qτ forces that ċβi (k) ≤ π̇k (ċβi ↾ k) for k ∈ Ḋ above n∗ .
We apply to qτ ’s the UF-limit principle, which states that UF-limit
forces that ultraﬁlter many conditions are in generic ﬁlter.

Overview

Details

Conclusion and Open Problems

Proof (Part3)
We obtain the following lemma as an application of the principle.
Prediction Point Exclusion Lemma
For n < ω and τ ∈ ω n−1 , lim⟨qτ ⌢ m ⟩m<ω forces n∗ + n − 1 ∈/ Ḋ.
Proof of lemma. Each qτ ⌢ m forces :• ċβim ↾ (n∗ + n) = s⌢ τ ⌢ m.
predict
impossible!
• ċβim (k) ≤ π̇k (ċβim ↾ k)∗∗
for k ∈ Ḋ above n .
ċβi ↾ (n + n − 1)⌢
where im corresponds to τ m.
inﬁniteτ
Let G be generic ﬁlter containing
sthe limit lim⟨qτ ⌢ m ⟩m<ω and assumen∗ − 1 n∗ + n − 1n∗ + n − 1 ∈ Ḋ in V [G]. By theprinciple, ultraﬁlter many qτ ⌢ m are inG. Thus, π̇ has to bounding-predict inﬁnitely many values atn∗ + n − 1 from the same initial segment s⌢ τ , which is impossible.
2Overview

Details

Conclusion and Open Problems

Proof (Part3)
This lemma tells us one limit excludes one possible prediction point.
Using the UF-Limit Preservation Property, we can preserve
information of:• which prediction point is excluded.
• the initial segments of cβi (used for the next exclusion).
In fact, we can ultimately obtain qn∞ excluding prediction points“downward”, by taking limits of suitable sequences inﬁnitely manytimes (details omitted, the direction is like τ → ⟨⟩):Lemma
For each n < ω, there exists qn∞ forcing [n∗ , n∗ + n) ∩ Ḋ = ∅.
Finally, we exclude points “upward”. That is, q ∞ := lim⟨qn∞ ⟩n<ω ,the UF-limit of qn∞ , forces [n∗ , n∗ + n) ∩ Ḋ = ∅ for ultraﬁlter many(inﬁnitely many) n < ω, which is a contradiction!
2(For details : [Yam23], slides of Kobe Set Theory seminar)Overview

Details1

Overview

2

Details

3

Conclusion and Open Problems

Conclusion and Open Problems

Overview

Details

Conclusion and Open Problems

Conclusion and Open Problems
Conclusion:• By modefying the UF-limit method (, which was invented tokeep b small), we invented the cUF-limit method, which keepse∗ (and hence e) small.
• Mixing all the three limit methods, we obtained a new
separation model.
Question 1.
Other than e∗ , is there another cardinal invariant which is notincreased by σ-cUF-linked forcings?
Question 2.
What about “cFAM-limit”?

Overview

Details

Conclusion and Open Problems

References I[BS96]Jörg Brendle and Saharon Shelah.
Evasion and prediction II.
Journal of the London Mathematical Society, Vol. 53, No. 1, pp. 19–27,
1996.

[BS03]

Jörg Brendle and Saharon Shelah.
Evasion and prediction.
Archive for Mathematical Logic, Vol. 42, No. 4, pp. 349–360, 2003.

[GKMS21] Martin Goldstern, Jakob Kellner, Diego A. Mejía, and Saharon Shelah.
Adding the evasion number to Cichoń’s maximum.
XVI International Luminy Workshop in Set Theory, 2021.
[GKMS22] Martin Goldstern, Jakob Kellner, Diego A. Mejía, and Saharon Shelah.
Cichoń’s maximum without large cardinals.
J. Eur. Math. Soc. (JEMS), Vol. 24, No. 11, pp. 3951–3967, 2022.
[GKS19]Martin Goldstern, Jakob Kellner, and Saharon Shelah.
Cichoń’s maximum.
Ann. of Math. (2), Vol. 190, No. 1, pp. 113–143, 2019.

Overview

Details

Conclusion and Open Problems

References II[KST19]Jakob Kellner, Saharon Shelah, and Anda R Tanasie.
Another ordering of the ten cardinal characteristics in Cichoń’s diagram.
Comment. Math. Univ. Carolin, Vol. 60, No. 1, pp. 61–95, 2019.

[Yam23]

Takashi Yamazoe.
Cichoń’s maximum with evasion number.
Kobe Set Theory Seminar, 2023.