Fragility of Properness
Yasuo YOSHINOBU
Nagoya University

RIMS Set Theory Meeting
November 18, 2020

The theorem

In this short talk we show just a single theorem.
Theorem
Whenever V ⊊ W are models of ZFC with the same ordinals.
Then there exists a poset P in V such that P is proper in V but
improper in W .

Short background story
Karagila once asked the following question in his website, where helists some open problems in set theory.
Problem (Karagila [1], Problem 3(b))
Is there a cardinal κ such that every proper poset remains proper
after any κ-closed forcing?
This problem was solved negatively by the speaker [2] (2018).
After this, Karagila added another problem somewhat with an‘opposite’ tone.
Problem (Karagila [1], Problem 5)
For any nontrivial poset P, is there a proper poset Q whose
properness is destroyed after forcing over P?
Our theorem solves this problem aﬃrmatively.
The background of these questions is explained more in detail in
Karagila [3].

Proof of the theorem
First let:
κ := the least ordinal s.t. κ Ord ∩ (W \ V ) ̸= ∅.
λ := the least ordinal s.t. κ λ ∩ (W \ V ) ̸= ∅.
Then we have:κ is a regular infinite cardinal both in V and W .
λ is a cardinal in V with λ ≥ 2.
We will prove the theorem by two cases.
Case 1 κ > ω: We use a variation of Shelah’s example ([4], XVII
Observation 2.12, p.826) of two proper posets whose product isimproper.
Case 2 κ = ω: We generalize an argument by Shelah (introducedby Goldstern [5]) to prove some σ-closed forcing turns improper byadding a Cohen real.
Remark These techniques have been mentioned in Karagila [3] as‘examples’.

Case 1: κ > ω (1/3)
Note that ω1V = ω1W . (So we will write them as ω1 .)Work in V .
Let T := the tree <κ λ (ordered by end-extension).
There are θ := λκ branches through T .
Note that by the choice of κ and λ, T has a branch in W \ V .
P

Let P := Add(ω) and Q̇ := Col(ω1 , θ)V .
P ∗ Q̇ adds no new branch through T by the following theorem,and thus in V P∗Q̇ there are ω1 branches through T .
Theorem (Mitchell [6])Let cfγ > ω. Then forcing with Add(ω) ∗ (σ-closed) adds no new
sequence s of ordinals of length γ such that every initial segmentof s is in the ground model.

Case 1: κ > ω (2/3)
Now work in V P∗Q̇ .
We can pick a club subset Ċ ⊆ κ of order type ω1 .
Let T ↾ Ċ := {s ∈ T | length(s) ∈ Ċ}.
Note that branches through T ↾ Ċ are essentially those through T .
So T ↾ Ċ is a tree of size and height ω1 with ω1 branches.
Now let Ṙ be as in the following theorem.
Theorem (Baumgartner [7] §7)
Let T be a tree of size and height ω1 with at most ω1 branches.
Then there is a c.c.c. poset R which specializes T , that is,“T is a countable union of subsets, each of which is⊩R a union of mutually incompatible tails of branches
in the ground model.”Case 1: κ > ω (3/3)

Note that (in V P∗Q̇ ) forcing with Ṙ adds no new branches throughT ↾ Ċ.
Nor does any further extension, unless it collapses ω1 .
Now in V let P̃ := P ∗ Q̇ ∗ Ṙ.
P̃ is of the form (c.c.c.)*(σ-closed)*(c.c.c.) and thus proper in V .
Let G be any P̃-generic over W (thus over V ).
Then W [G] is an extension of V [G] and contains a branch throughT ↾ Ċ which is not in V (and thus not in V [G]).
Therefore ω1 is collapsed in W [G], and therefore P̃ is not proper.

Case 2: κ = ω (1/2)
In this case, we can prove the following:
Lemma
There exists µ > ω1W regular in W such that (Pω1 µ)W \ V isstationary in W .
(Proof of Lemma) If λ = 2 (i.e. W has a real not in V ), theconclusion holds for µ = ω2W , as proved by Gitik ([8], Theorem
1.1).
Otherwise, let f ∈ ω λ ∩ (W \ V ).
Since W ∩ ω 2 = V ∩ ω 2, no countable set in V contains ran(f ).
Now let µ be a W -regular cardinal such that µ ≥ ω2W , λ.
Then{x ∈ (Pω1 µ)W | ran(f ) ⊆ x} ⊆ (Pω1 µ)W \ Vcontains a club subset of (Pω1 µ)W .

Case 2: κ = ω (2/2)

Let µ be as in Lemma and let P := Col(ω1 , µ)V . P is proper in V .
Now work in W . Let θ be a suﬃciently large regular cardinal. ThenY = {M ≺ Hθ | P ∈ M, |M | = ω, M ∩ µ ∈/ V}
is stationary in Pω1 Hθ .
For each M ∈ Y , M ∩ ω1 is an ordinal and so is δ := M ∩ ω1V .
If p ∈ P were (M, P)-generic, by a density argument we would haveran(p ↾ δ) = M ∩ µ ∈/ V,
which is absurd since p ∈ V .
Therefore P is not proper in W .

Question

Question For any two models V ⊊ W of ZFC with the sameordinals, can one find P totally proper in V but not in W ?

References I[1] Asaf Karagila.
Open problems.
Blog entry, http://karagila.org/problems.html, 2018.
[2] Yasuo Yoshinobu.
Properness under closed forcing.
In Advances in Mathematical Logic / Dedicated to the Memory of Professor Gaisi
Takeuti, SAML 2018, Kobe, Japan, September 2018, Springer Proceedings inMathematics and Statistics, to appear.
[3] Asaf Karagila.
Preserving properness.
Blog enrty, http://karagila.org/2018/preserving-properness/, 2018.
[4] Saharon Shelah.
Proper and Improper Forcing.
Perspectives in Mathematical Logic. Springer-Verlag, 1998.
[5] Martin Goldstern.
Answer of MathOverflow question 193522.
MathOverflow, https://mathoverflow.net/a/193522, 2015.

References II[6] William Mitchell.
Aronszajn trees and the independence of the transfer property.
Annals of Mathematical Logic, 5:21–46, 1972.
[7] James E. Baumgartner.
Applications of the Proper Forcing Axiom.
In K. Kunen and J.E. Vaughan, editors, Handbook of set-theoretic topology,
pages 913–959. North-Holland, 1984.
[8] Moti Gitik.
Nonsplitting subset of Pκ (κ+ ).
Journal of Symbolic Logic, 50(4):881–894, 1985.