Open AccessArticle

Applying Set Theory

Saharon Shelah

^1,2

Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

Department of Mathematics Hill Center—Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA

Axioms 2021, 10(4), 329; https://doi.org/10.3390/axioms10040329

Submission received: 19 October 2021 / Revised: 19 November 2021 / Accepted: 21 November 2021 / Published: 30 November 2021

(This article belongs to the Collection Mathematical Analysis and Applications)

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Abstract

We prove some results in set theory as applied to general topology and model theory. In particular, we study

ℵ_{1}

-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

Keywords:

set theory; model theory; general topology; monadic logic

MSC:

03C55; 03C85; 03E75; 03E35

1. Introduction

In §2 we prove a result in general topology saying: if

◊_{ℵ_{1}}^{*}

then any normal space is

ℵ_{1} - CWH

(=collectionwise Hausdorff); done independently of and in parallel to Fleisner and Alan D. Taylor.

In §3 we prove the Chang Conjecture for Magidor-Malitz Quantifiers. A recent related piece of work is [1].

In §4 we prove the Monadic Theory of the tree

^{ω >} 2

is complicated under a quite weak set theoretic assumption. Earlier [2] proved this (i.e., the result on the monadic logic) assuming

CH

or at least a consequence of it.

The present note was circulated in the Spring of 1979 in a collection; it include each of the sections (as well as other preprints) but those three were not published. However, Ref. [3]. have results related to Section 3; in particular it was conjectured there (in Remark 2.15) that there are two non-principal ultrafilters of

ω

with no common lower bound in the Rudin Keisler order; a conjecure which had been refuted in [4].

Later, Gurevich-Shelah [5] proved undecidability in

ZFC

, with further developments then more in Shelah [6], still the older proof gives information not covered by them. For more see [4,7,8].

The results are old, still in particular, §2 gives a direct proof of the result compared to others and §4 gives a considerably more transparent easier proof of the later result of [5] but with an extra weak hypothesis.

The author would like to thank Shai Ben David for stimulating discussions on part §3.

We thank the referee for his help, well beyond than the call of duty.

2. A Note in General Topology If $⋄_{ℵ_{1}}^{*}$ Then Any Normal Space Is $ℵ_{1} - CWH$ (=Collectionwise HAUSDORFF)

The normal Moore space problem has been a major theme in general topology, see the recent survey Dow-Tall [9]. In this connection, Fleissner ([10], p. 6) proved:

(V = L)

every normal first countable (topological) space is

CWH

(

CWH

means collectionwise Hausdorff). He used a strengthening of diamond. The author proved Fleissner strengthening (for

ℵ_{1}

) does not follow from

ZFC + ◊_{ℵ_{1}}^{+}

(see [11], Th. 5, p. 31). Here we prove nevertheless

◊_{ℵ_{1}}^{*}

implies every normal first countable space is

ℵ_{1} - CWH

The central idea of the proof in §2 is inspired by one key idea in Fleissner [10]. Fleissner implicitly used a stronger combinatorial principle

◊_{S S}

. In 1979, the author and independently both Fleissner and Alan D. Taylor all saw (as mentioned in [12,13] that a weaker principle,

◊_{ω_{1}}^{*}

, would suffice. Later Smith and Szeptycki [12] derive better results. On more recent results on diamond and strong negation, see [14] and references there.

Definition 1.

Below δ always denotes a limit ordinal

(< ω_{1})

. For transparency, below we refer to the following equivalent form of

◊_{ω_{1}}^{*}

Definition 2.

Let

◊_{ℵ_{1}}^{*}

mean that there exist a sequence

〈 g_{δ} : δ < ω_{1} 〉

where

g_{δ} = 〈 {\bar{g}}^{δ, k} : k < ω 〉

is of the form

{\bar{g}}^{δ, k} = 〈 g_{n}^{δ, k} : n < ω 〉

, where

g_{n}^{δ, k} : δ \to ω

has the property that, for any sequence

\bar{g} = 〈 g_{n} : n < ω 〉

with

g_{n} : {δ : δ < ω_{1}} \to ω

, there is a club (closed unbounded) set

C \subseteq ω_{1}

such that, for each

γ \in C

, there is

k = k (γ) \in ω

with

\bar{g} ↾ γ : = 〈 g_{n} ↾ γ : n < ω 〉 = {\bar{g}}^{γ, k} = 〈 g_{n}^{γ, k} : n < ω 〉 .

Theorem 1.

Assume

◊_{ℵ_{1}}^{*}

. If X is Hausdorff first countable normal and

| X | = ℵ_{1}

then X is CWH.

Proof.

Let

〈 g_{δ} : δ < ω_{1} 〉

be as in Definition 2.

Without loss of generality

X_{*} = {δ : δ < ω_{1}} \subseteq X

and

X_{*}

is closed discrete in the space X. Let

U_{n}^{δ} (n < ω)

be a basis of open neighborhoods of

δ

(for

δ < ω_{1}

). We shall define by induction on

α < ω_{1}

a limit ordinal

γ_{α} < ω_{1}

and

〈 f_{n} (γ) : n < ω, γ < γ_{α} 〉

such that

γ_{α}

is increasing continuous with

α

and

γ_{0} = 0

. For

α = 0

choose

γ_{α} = ω; f_{n} (γ) = 0

. For limit

α

let

γ_{α}

\cup {γ_{β} : β < α}

. For

α = β + 1

γ_{α} > α

then we let

γ_{α} = γ_{β} + ω

and let

f_{n} (γ) = 0

for

γ \in [γ_{β}, γ_{α})

. Finally assume that

α = δ^{*}, γ_{δ^{*}} = δ^{*}

δ^{*} \in X_{*}

We have chosen above the functions

〈 g_{n}^{δ^{*}, k} : n < ω, k < ω 〉

with

g_{n}^{δ^{*}, k} : δ^{*} \to ω

; now for each

n, k < ω

let

A_{ℓ}^{δ^{*}, n, k} = \cup {U_{g_{n}^{δ^{*}, k} (δ)}^{δ} : δ < δ^{*}, f_{n} (δ) = ℓ}

(for

n < ω, ℓ < 2

). Call

k < ω

good for

δ^{*}

when for infinitely many (pairs)

n, ℓ

we have

B_{ℓ}^{δ^{*}, n, k} : = cl (A_{ℓ}^{δ^{*}, n, k}) \cap (X_{*} \ δ^{*}) \neq \emptyset .

We let

γ_{α} = γ_{δ^{*} + 1} = min {δ : δ > δ^{*}

and if

ℓ < 2

and

n, k < ω

and

B_{ℓ}^{δ^{*}, n, k} \neq \emptyset

then

(δ^{*}, δ) \cap B_{ℓ}^{δ^{*}, n, k} \neq \emptyset}

Now we choose

f_{n} ↾ [δ^{*}, γ_{α})

such that for any k good for

δ^{*}

, for some

n, ℓ, δ \geq δ^{*}

we have

f_{n} (δ) = 1 - ℓ (f o r δ \in cl (A_{ℓ}^{δ^{*}, n, k})) .

Then we complete arbitrarily the

f_{n}

so that its domain is

γ_{α}

Thus we have defined

f_{n} (n < ω)

with

f_{n} : ω_{1} \to 2

. For each n the sets

f_{n}^{- 1} {1} \cap X_{*}, f_{n}^{- 1} {0} \cap X_{*}

form a partition of

X_{*}

, both are closed and discrete subsets of X. But X is normal. So there are functions

g_{n} : X_{*} \to ω

for

n < ω

so that letting for

ℓ = 0, 1

A_{ℓ}^{n} = \cup {U_{g_{n} (δ)}^{δ} : δ \in X_{*}, f_{n} (δ) = ℓ}

we have

A_{0}^{n} \cap A_{1}^{n} = \emptyset .

Let

g_{n}^{+}

be any function from

ω_{1}

ω

extending

g_{n}

. For some closed unbounded set

C \subseteq X_{*}

we have:

δ^{*} \in C \Rightarrow (\exists k) (〈 g_{n}^{+} ↾ δ^{*} : n < ω 〉 = 〈 g_{n}^{δ^{*}, k} : n < ω 〉)

. Let the first such k be denoted

k (δ^{*})

. Without loss of generality every

δ^{*} \in C

satisfy

γ_{δ^{*}} = γ

hence if

δ^{*} \in C \land n < ω \land k < ω \land ℓ < 2

and

B_{ℓ}^{δ^{*}, n, k} = cl (A_{ℓ}^{δ^{*}, n, k}) \cap (X_{*} \ δ^{*}) \neq \emptyset

then

min (B_{ℓ}^{δ^{*}, n, k}) < min (C \ (δ^{*} + 1))

For

δ^{*} \in C

now

k (δ^{*})

cannot be good for

δ^{*}

, (by the definition).

Now for at least one n (in fact, for infinitely many n-s) we have

cl (A_{ℓ}^{n} | δ^{*}) \cap (X_{*} \ δ^{*}) = \emptyset

for

ℓ \in {0, 1}

, let

n (δ^{*})

be the first such n.

Define

B_{n} = {δ : for some δ^{*} \in C \cup {0} we have δ^{*} \leq δ < min (C \ δ) and n = max {n (δ^{*}), n (δ)}

Now

⋃_{n} (g_{n} ↾ B_{n})

almost exhibits

X_{*}

has the right sequence of neighborhoods. Now we can deal with each

B_{n}

separately (just choose

U_{n}

by induction on n such that

U_{n}

is open,

U_{n} \cap X_{*} = B_{n}

and

U_{n} \subseteq X \ c ℓ (⋃_{ℓ < n} U_{ℓ})

, possible by normality).

By dealing as follows with each interval

[δ^{*}, min (C \ (δ^{*} + 1))]

for

δ^{*} \in C \cup {0}

we have

U_{g_{n} (δ)}^{δ} (δ \in B_{n})

as required.

That is, for

γ \in C \cup {0}

with

γ^{+}

its successor in C, choose a (countable) family of pairwise disjoint open sets

U_{γ} (β)

for

β \in X_{*} \land γ \leq β < γ^{+}

, with

β \in U_{γ} (β)

, this is possible as in the choice of the

U_{n}

’s.

Now for

β \in X_{*}

we let

W_{β} = U_{n (β)} \cap U_{γ (β)} (β) \cap U_{g_{n (β)} (β)}^{β}

where:

$γ (γ) = max (C \cap (β + 1))$
$m (β) = max {n (δ^{*}), n ({(δ^{*})}^{+}) : δ^{*} = max (C \cap β) \leq β < {(δ^{*})}^{+}}$

Finally

〈 W_{β} : β \in X_{*} 〉

is a sequence of pairwise disjoint open sets of X with

β \in X_{*} \Rightarrow β \in W_{β}

, so we are done. □

Remark 1.

As in [10] it suffices to assume every point in the space has a neighborhood basis of cardinality

ℵ_{1}

3. Chang Conjecture for Magidor-Malitz Quantifiers

Silver (see [15]) had proved the consistency of Chang conjecture, i.e.,

⊕: any model M with universe $ℵ_{2}$ (and countable signature = vocabulary) $τ$ , has an elementary submodel $N, (∥ N ∥ = ℵ_{1}, | N \cap ω_{1} | = ℵ_{0})$

Silver did this by starting with a model

V

with

κ

Ramsey (in fact, something weaker suffices), forcing

MA

and then collapsing

κ

ℵ_{2}

P_{Set}^{κ} = {f : Dom (f) \subseteq {μ : ℵ_{1} < μ < κ, μ a cardinal}

has cardinality

\leq ℵ_{1}

, and for some

α < ω_{1}

(\forall μ \in Dom f) (f (μ)

is a function from

α

μ)}

. See also Koszmider [16] for a topological application.

We can ask whether this submodel N can inherit more properties from M.

Definition 3.

Let us define a (technical variant of) Magidor-Malitz quantifiers.

M ⊨ (Q^{n} \bar{x}) φ (x_{1}, \dots, x_{n})

means that there is a set

A \subseteq M, A

is of cardinality

∥ M ∥

such that

(\forall a_{1} \dots a_{n} \in A) φ (a_{1} \dots a_{n})

The result is that:

Claim 1.

In ⊕ above, we can have N an elementary submodel of M even for the logic

L {(Q^{0}, Q^{1}, \dots)}_{n < ω}

. So e.g., Suslinity of trees is preserved.

For this we need the following.

Definition 4.

Call a forcing

P

suitable when for any sequence

〈 p_{i} : i < ω_{1} 〉

of members of

P

there is a set

U \subseteq ω_{1}

of cardinality

ℵ_{1}

such that: for any finite

u \subseteq U

there is

q \in P

such that

\underset{i \in u}{⋀} q \geq p_{i}

Claim 2.

Forcing by suitable forcing preserves satisfaction of sentences of Magidor-Malitz quantifiers for models of power

ℵ_{1}

See ([17], 1.5–13, p. 34).

Claim 3.

There is a suitable forcing

P, | P | = 2^{ℵ_{1}}

, such that in

V^{P}

: if

Q

is a suitable forcing of power

ℵ_{1}, \underset{~}{M}

Q

-name of a model of power

ℵ_{1}

, in a language

L \in V

, universe

ℵ_{1}

, then there is a directed

G \subseteq P

, which determines

\underset{~}{M}

as M and such that for any sentence

ψ

from the

L (Q^{0}, Q^{1}, \dots)

(the variant of Magidor-Malitz logic from Definition 3)

⊩_{Q} “ \underset{~}{M} ⊨ ψ ” implies M ⊨ ψ .

Proof.

Just iterate the required forcing notions, with direct limit (i.e., finite support) and remember it is known that suitability is preserved under iteration, i.e., Claim 2.

Proof of Main result Claim 1:

Do as Silver, start with

V ⊨ “ κ

Ramsey”, force by

P

from Claim 3, and then use

P_{Set}^{κ}

. The rest is as in his proof.

But we have to choose G as in Definition 3, and notice that more is reflected to the submodel he uses, (just check the definition carefully) and work a little, and remember that

ℵ_{1}

-complete forcing preserves satisfaction of sentences in

L (Q^{0}, \dots)

(and

P_{Set}^{κ}

ℵ_{1}

-complete). □

4. A Remark on the Monadic Theory of Order

In [2] we prove the undecidability of the monadic theory of (the order) R, assuming CH, or the weaker Baire-like hypothesis that

R

is not the union of fewer than continuum sets of first category sets. This condition is weaken below to ‘‘not (

St

) at least for T where a closely related theory is the monadic theory T of

M = (^{ω \geq} 2, ◃)

where

^{ω \geq} 2

is the set of sequences of zeros and ones of length

\leq ω, ◃

is the (partial) order of being initial segment. T is closely related to Rabin’s monadic theory of

(^{ω >} 2, ◃)

which he proved decidable [18]. We prove here that the statement “

\neg (St)

” implies the undecidability of T (and all results on its complexity, see [2] and the paper of Gurevich on the subject) but it was not clear (at that time) whether (St) is consistent with ZFC.

Definition 5.

A Cantor [set] C is a non-empty subset of

^{ω \geq} 2

with the properties

(a): C is closed under initial segments,
(b): if $η$ has length $ω$ then $η \in C \equiv (\forall n) (η ↾ n \in C)$ ,
(c): $η \in C \cap^{ω >} 2$ implies $η ⌢ 〈 0 〉 \in C$ or $η ⌢ 〈 1 〉 \in C$ ,
(d): for every $η \in C \cap^{ω >} 2$ , there is $ν \in C \cap^{ω >} 2$ $η ◃ ν$ and $ν ⌢ 〈 0 〉 \in C, ν ⌢ 〈 1 〉 \in C$ .

Definition 6.

(1) For a Cantor C, the set of its splitting points is

Sp (C) = {η \in^{ω >} 2 : η ⌢ 〈 0 〉 \in C

and

η ⌢ 〈 1 〉 \in C}

(2) For a set

A \subseteq^{ω >} 2, C

is an A-Cantor, if

Sp (C) \subseteq A

(3) For a set

S \subseteq ω, C

is called an S-Cantor, if

Sp (C) \subseteq ⋃_{n \in S}^{n} 2 .

(4) An odd Cantor is one that is an

{2 n + 1 : n < ω}

-Cantor. An even Cantor is one that is an

{2 n : n < ω}

-Cantor.

Now the statement we speak about is

Definition 7.

Let

(St)

mean: the set

^{ω} 2

is the union of less than

2^{ℵ_{0}}

Cantors each of them odd or even.

Problem 1.

(St)

consistent with

ZFC

?; solved in [4].

Claim 4.

Let

{C_{i} : i < α}

be a family of odd and even Cantors,

^{ω \geq} 2 = ⋃_{i < α} C_{i}

. Then

2^{ℵ_{0}} \leq {| α |}^{+}

Proof.

Let for

η, ν \in^{ω} 2, ρ = p (η, ν)

be defined by

ρ (2 n) = η (n), ρ (2 n + 1) = ν (n)

, and then let

η = {pr}_{1} (ρ), ν = {pr}_{2} (ρ)

Now for any even C, and

η

there is at most one

ν

such that

p (η, ν) \in C

; why? if

ν_{0}, ν_{1}

are such

ν

’s,

ρ_{ℓ} = p (η, ν_{ℓ})

, then, by the definition of

p (-, -)

, for some

m < ω, ρ_{0} ↾ m = ρ_{1} ↾ m, ρ_{0} (m) \neq ρ_{1} (m)

. If

m = 2 n

then

ρ_{ℓ} (m) = ρ_{ℓ} (2 n) = η (n)

so they are equal, contradiction. If

m = 2 n + 1

, then (

ρ_{0} (m) \neq ρ_{1} (m)

and)

ρ_{0} ↾ m = ρ_{1} ↾ m

is a splitting point of C, however m is odd and C is an even Cantor, a contradiction. So really there is at most one

ν

, and let

ϱ (η, C)

be the unique

ν

p (η, ν) \in C

if there is one and

η

otherwise.

Similarly if C is odd and

η \in^{ω} 2

, then for at most one

ν, p (ν, η) \in C

and let

ϱ (η, C)

ν

for this

η

, and let

ϱ (η, C) = η

otherwise. Our definition of the function

ϱ

does not contradict, because no Cantor is odd and even.

Let for

η \in^{ω} 2, Dp (η) = {ϱ (η, C_{i}) : i < α}

. So clearly

Dp (η)

is a subset of

^{ω} 2

of cardinality

\leq | α |

Now if

η, ν \in^{ω} 2

, by hypothesis

ρ = p (η, ν)

belongs to some

C_{i}

. If

C_{i}

is odd this implies

ν = ϱ (η, C_{i}) \in Dp (η)

and if

C_{i}

is even this implies

η = ϱ (C_{i}, ν) \in Dp (ν)

{| α |}^{+} < 2^{ℵ_{0}}

we can easily find a counterexample. □

Definition 8.

Assume

\neg (St)

S_{n} \subseteq ω

are infinite pairwise almost disjoint (for

n \in {0, 1, 2}

C_{i} (i < α < 2^{ℵ_{0}})

are Cantors, each an

S_{n}

-Cantor for some n (or just an

S_{n} \cup S_{2}

-Contor for some n), C is a Cantor for every

η \in C \cap^{ω >} 2, ℓ \in {0, 1}

, there is ν, such that

η ◃ ν \in Sp (C), ν \in ⋃_{k \in S_{ℓ}}^{k} 2

Then there is

η \in C \ ⋃_{i < α} C_{i} \^{ω >} 2

(2) Similarly for

S_{n} \subseteq^{ω >} 2

Proof.

(1) We can find a Cantor

C^{'} \subseteq C

, and

0 = k (0) < k (1) < \dots < k (n) < \dots < ω

(∗): if $η \in^{k (n)} 2$ , then there are exactly two $ν \in^{k (n + 1)} 2 \cap C^{'}, η ◃ ν$ , and if they are $ν_{1}, ν_{2}$ and $m : = min {m : ν_{1} (m) \neq ν_{2} (m)}$ then $m \in S_{0} \cup S_{1}$ but $\notin S_{2} \cup (S_{0}, \cap S_{1})$ . Moreover $m \in S_{0}$ if n is even.

Let

A = {η ↾ k (n) : n < ω, η \in C^{'}}

, so

A \subseteq C^{'}

. Clearly there is an isomorphism f, of the models

(^{ω \geq} 2, ◃), (C^{'}, ◃)

Let

C_{i}^{'} = {f (η) : η \in C^{'}, η \in C_{i}}

, it is easy to check that each

C_{i}^{'}

is countable, or the union of a countable set and a Cantor which is odd or is even.

We can find odd Cantor

C_{i}^{'} (α \leq i < α ω)

all countable sets we mentioned are covered by them. Now by—“

\neg (St)

” there is

η \in^{ω \geq} 2

η \notin ⋃_{i < α ω} C_{i}^{'}

(as

α ω < 2^{ℵ_{0}}

) and

f^{- 1} (η)

is the required elements.

(2) Similarly. □

(Now we have added Claim 5 and Definition 9 in 2019).

Claim 5.

Assume

\neg (St)

. (1) The monadic theory T is undecidable.

Proof.

Below let P vary on Cantors and note that we can repeat the proof of [2] with small adaptation (and prove T is undecidable). That is, the change needed is in ([2], (7.4)) which has a set-theoretic hypothesis (CH or the Baire-like hypothesis mentioned above), so we repeat it with the needed changes below. □

Definition 9.

Assume

\neg (St)

and let J be an index-set of cardinality at most

2^{ℵ_{0}}

, (1) Assume the

D_{i} (i \in J)

countable dense subsets of

^{ω >} 2

and

D = ⋃_{i \in J} D_{i}

and

\bar{D} = 〈 D_{i} : i \in J 〉

(The main case is that the

D_{i}

-s are pairwise almost disjoint). Then there is

Q \subseteq^{ω} 2 \ D, Q = Q [\bar{D}]

such that for every Cantor P:

(A): if $P \cap D \subseteq D_{i} (i \in J)$ and $D_{i}$ is dense in P then $| P \cap Q | < 2^{ℵ_{0}}$
(B): if for some $i \in J$ the sets $P \cap D_{i}, P \ D_{i}$ are dense in P then $P \cap Q \neq \emptyset$ .

(2) For some such

\bar{D}

we can strengthen clause (B) above to

(B): if P is a Cantor and for every $i \in J$ the set $D_{i} \cap P$ is nowhere-dense in P then for every, dense subsets $D_{1}^{*}, D_{2}^{*}$ of $P \cap D$ we can find $D_{1}^{•} \subseteq D_{1}^{*}, D_{2}^{•} \subseteq D_{2}^{*}$ satisfying for any P we have: is $P \cap D_{1}^{•}, P \cap D_{2}^{•}$ are dense in P then $P \cap Q \neq \emptyset$ .

Proof.

(1) Let

{P_{α} : 0 < α < 2^{ℵ_{0}}}

be any enumeration of the Cantor sets. We define

x_{α}, α < 2^{ℵ_{0}}

by induction on

α

For

α = 0, x_{α} \in R

is arbitrary.

For any

α > 0

, if

P_{α}

does not satisfy the assumptions of (B) then let

x_{α} = x_{0}

and if P satisfies the assumptions of (B) (hence in particular D is dense in P) let

x_{α} \in P_{α} - ⋃ {P_{β} : β < α, (\exists i \in J) (P_{β} \cap D \subseteq D_{i}

and D is dense in

P_{β})} = D

This is possible; to prove this let

U = {β < α :

there is

i \in J

such that

P_{β} \cap D \subseteq D_{i}}

and for

β \in U

let

i_{β} \in J

be such that

P_{β} \subseteq D_{i_{β}}

Let

i (*) \in J

be such that

P \cap D_{i (*)}, P \ D_{i (*)}

are dense in P. Now we apply Definition 8(2), (or Definition 8(1) if we restrict the

D_{i}

-s, does not matter).

So by

(St)

and the hypothesis

| P_{α} \cap D | < 2^{ℵ_{0}}

there exists such

x_{α}

Now let

Q = {x_{α} : α < 2^{ℵ_{0}}}

. If P satisfies the assumptions of (A), then

P \in {P_{α} : 0 < α < 2^{ℵ_{0}}}

. So for some

α, P = P_{α}

, hence

P \cap D \subseteq {x_{β} : β < α}

, so

| P \cap D | < 2^{ℵ_{0}}

. If

P = P_{α}

satisfies the assumption of (B) then

x_{α} \in P_{α}, x_{α} \in Q

, hence

P_{α} \cap Q \neq \emptyset

(2) Similarly.

So we have proved the lemma. □

Definition 10.

We can interpret the monadic theory of

(R, <)

in T, but the converse was not clear at the time, but looking at it again probably we can carry the proof for

R

Funding

This research received no external funding.

Acknowledgments

The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Shelah, S. Applying Set Theory. Axioms 2021, 10, 329. https://doi.org/10.3390/axioms10040329

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Applying Set Theory

Abstract

1. Introduction

2. A Note in General Topology If $⋄_{ℵ_{1}}^{*}$ Then Any Normal Space Is $ℵ_{1} - CWH$ (=Collectionwise HAUSDORFF)

3. Chang Conjecture for Magidor-Malitz Quantifiers

4. A Remark on the Monadic Theory of Order

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Acknowledgments

Conflicts of Interest

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Applying Set Theory

Abstract

1. Introduction

2. A Note in General Topology If ⋄ ℵ 1 * Then Any Normal Space Is ℵ 1 − CWH (=Collectionwise HAUSDORFF)

3. Chang Conjecture for Magidor-Malitz Quantifiers

4. A Remark on the Monadic Theory of Order

Funding

Acknowledgments

Conflicts of Interest

References

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2. A Note in General Topology If $⋄_{ℵ_{1}}^{*}$ Then Any Normal Space Is $ℵ_{1} - CWH$ (=Collectionwise HAUSDORFF)