Coloring ordinals by reals
arXiv:0704.1884v1  [math.LO]  14 Apr 2007
Coloring ordinals by reals
Jörg Brendle and Sakaé Fuchino
March 24, 2007
Abstract
We study combinatorial principles we call Homogeneity Principle HP(κ)
and Injectivity Principle IP(κ, λ) for regular κ > ℵ1 and λ ≤ κ which are
formulated in terms of coloring the ordinals < κ by reals.
These principles are strengthenings of Cs(κ) and Fs(κ) of I. Juhász, L.
Soukup and Z. Szentmiklóssy [13]. Generalizing, their results, we show e.g.
that IP(ℵ2,ℵ1) (hence also IP(ℵ2,ℵ2) as well as HP(ℵ2)) holds in a generic
extension of a model of CH by Cohen forcing and IP(ℵ2,ℵ2) (hence also
HP(ℵ2)) holds in a generic extension by countable support side-by-side prod-
uct of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the
latter result is optimal (Theorem 5.2).
Relations between these principles and their influence on the values of
the variations b↑, bh, b∗, do of the bounding number b are studied.
One of the consequences of HP(κ) besides Cs(κ) is that there is no pro-
jective well-ordering of length κ on any subset of ωω. We construct a model
in which there is no projective well-ordering of length ω2 on any subset of
ωω (do = ℵ1 in our terminology) while b
∗ = ℵ2 (Theorem 6.4).
2000 Mathematical Subject Classification:
03E05 03E17 03E35 03E65
Keywords:
Homogeneity Principle, Injectivity Principle,
bounding number, projective well-ordering,
Cohen forcing, Brendle-LaBerge forcing, Prikry-Silver forcing
0)The first author is partially supported by Grant-in-Aid for Scientific Research (C) 17540116,
Japan Society for the Promotion of Science. The second author is partially supported by Chubu
University grant 16IS55A.
http://arxiv.org/abs/0704.1884v1
1 Introduction
The Cohen model which is obtained by adding at least ℵ2 Cohen reals over a model
of GCH was the first and simplest model for the negation of CH, and it is still one
of the most important. A plethora of statements have been shown to be consistent
with ZFC by adjoining Cohen reals, and it is therefore natural to look for axioms
which hold in the Cohen model and from which many such statements can be
decided, that is, axioms which capture as much as possible of the combinatorial
structure of the Cohen extension. Something similar has been done for the iterated
Sacks model by Ciesielski and Pawlikowski who devised the Covering Property
Axiom CPA [2].
For Cohen models, several such axioms have been proposed in the past. Some
of them are homogeneity type statements, that is, they assert that given at least
ω2 many reals, many of them “look similar”. Examples are the combinatorial
principles Cs(κ), Ĉs(κ), and Fs(κ) introduced by I. Juhász, L. Soukup and Z.
Szentmiklóssy [13] who showed they hold in Cohen models (see Section 2 below for
definitions).
On the other hand, rather different-looking statements have been also investi-
gated in connection with Cohen models, for example, the axiom WFN asserting
that 〈P(ω),⊆〉 has the weak Freese-Nation Property (see [8], [10] and [5]). Here a
partial ordering 〈P,≤〉 has the weak Freese-Nation Property if there is a mapping
f : P → [P ]ℵ0 such that for all p, q ∈ P , p ≤P q holds if and only if there is an
r ∈ f(p) ∩ f(q) such that p ≤P r ≤P q.
In [8], it is shown that WFN holds in a Cohen model for adding ℵn Cohen
reals for any n < ω. If we start e.g. from V = L then WFN holds even after
adding any number of Cohen reals ([10]). In [5], it was shown that WFN implies
many of the known combinatorial properties of Cohen models and so it may be
seen as an axiomatization of the combinatorial structure of the Cohen extension.
Since WFN can be reformulated in terms of elementary submodels, WFN as well
as some closely related statements have come to be known as elementary submodel
type axioms (see [12] for this).
At first glance it seemed that there would be no connection between these two
types of axioms except that they both hold in a Cohen model. Surprisingly enough
though S. Shelah [17] showed that Cs(ℵ2) follows from the combinatorial principle
he called Princ, which is a consequence of WFN. The proof can be easily recast
to show that WFN implies Cs(κ) for all regular κ > ℵ1 (see [6] for more details).
In this paper, we introduce some new principles of the homogeneity type,
namely, the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ, λ)
which are formulated in terms of homogeneity of colorings of ordinals below the
cardinal κ by reals. We establish that these axioms hold in Cohen models and
address the question in which other models these axioms hold as well. It turns
out that, in fact, these principles seem to capture a good deal of the combinatorial
features of models of set theory obtained by forcing by the side-by-side (finite or
countable support) product of copies of a fixed relatively small partial ordering (see
Theorem 4.3 and Corollary 4.8).
Though the relation of these principles to WFN is not yet completely clear,
our principles imply the principles of I. Juhász, L. Soukup and Z. Szentmiklóssy
(Theorem 2.7) and thus can be seen as natural strengthenings of these principles.
Our paper is organized as follows.
In Section 2, we review the principles Cs(κ), Ĉs(κ) and Fs(κ) of I. Juhász, L.
Soukup and Z. Szentmiklóssy, and introduce our principles HP(κ) and IP(κ, λ).
Some basic facts in ZFC concerning these principles are also proved. In particular,
we show that Cs(κ) and Ĉs(κ) follow from HP(κ) (Theorem 2.7), Fs(κ) follows from
IP(κ,ℵ1) (Theorem 2.8) and HP(κ) follows from IP(κ, κ) (Theorem 2.9).
After reviewing some cardinal invariants introduced in [6] which are variants of
the bounding number b and the shrinking number b∗ in [3], we study in Section 3
the effect of the combinatorial principles Cs(κ), Ĉs(κ) and HP(κ) on the values of
these cardinal invariants.
In Section 4 we give a forcing construction of models of IP(κ, λ) (Theorem 4.3)
and its applications (Corollary 4.8). The results in this section improve consistency
results in [13].
As a further application of Theorem 4.3 we show in Section 5 the consistency
of ¬IP(ℵ2,ℵ1) and IP(ℵ2,ℵ2).
One of the consequences of HP(ℵ2) discussed in Section 3 is that there is no
definable well-ordering of length ω2 on any subset of
ωω (or do = ℵ1 in our notation).
Refining a forcing extension of Brendle and LaBerge [1], we prove in Section 6 the
consistency of do = ℵ1 with b
∗ = ℵ2 (Theorem 6.4). We also show that the model
of do = ℵ1 and b
∗ = ℵ2 we construct in this section satisfies a strong negation of
Cs(ℵ2).
Section 7 is devoted to the consistency proof of the combinatorial principle used
in the proof of Theorem 6.4.
In Section 8, we summarize the consistency results obtained in this paper to-
gether with other consistency results established by some previously known con-
structions. We discuss also some open problems at the end of the section.
2 Combinatorial principles formulated in terms
of coloring of ordinals by reals
For any set X , let
(2.1) ((X))
= {~x ∈ Xn : ~x is injective} and
(2.2) ((X))
((X))
Likewise, for any sets X0,...,Xn−1, let
(2.3) ((X0, ...,Xn−1)) = {~x ∈ X0 × · · · ×Xn−1 : ~x is injective}.
For a cardinal κ, the following principle Cs(κ) was introduced by I. Juhász, L.
Soukup and Z. Szentmiklóssy in [13].
Cs(κ): For any matrix 〈aα,n : α ∈ κ, n ∈ ω〉 of subsets of ω and T ⊆
ω>ω, at
least one of the following holds:
(c0) there is a stationary S ⊆ κ such that
n<|t| aαn,t(n) 6= ∅ for all
t ∈ T and 〈α0, ...,α|t|−1〉 ∈ ((S))
(c1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that
n<|t| aαn,t(n) = ∅ for all 〈α0, ...,α|t|−1〉 ∈ ((S0, ...,S|t|−1)).
For any cardinal κ it is easy to see that Cs(κ) holds if and only if Cs(cf κ) holds.
Thus it is enough to consider Cs(κ) for regular κ. The corresponding assertion is
also true for other combinatorial principles we are going to introduce in this section.
Hence, in the rest of this section, we shall assume that κ is a regular cardinal unless
mentioned otherwise.
The combinatorial principle Ĉs(κ), a sort of dual of the principle Cs(κ), is also
considered in [13]:
Ĉs(κ): For any T ⊆ ω<ω and any matrix 〈aα,n : α < κ, n ∈ ω〉 of subsets of
ω, at least one of the following holds:
(ĉ0) there is a stationary S ⊆ κ such that |
n<|t| aαn,t(n) | < ℵ0 for
every t ∈ T and 〈α0, ...,α| t |−1〉 ∈ ((S))
(ĉ1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such
that |
n<|t| aαn,t(n) | = ℵ0 for every 〈α0, ...,α| t |−1〉 ∈
((S0, ...,S|t|−1)).
The following is easily seen:
Lemma 2.1. (I. Juhász, L. Soukup and Z. Szentmiklóssy [13])
(a) Neither of Cs(ℵ1) and Ĉ
s(ℵ1) holds.
(b) Cs(κ) and Ĉs(κ) hold for any regular κ > 2ℵ0.
Let us call a subset A of H(ℵ1) definable if there are a formula ϕ and a ∈ H(ℵ1)
such that A = {x ∈ H(ℵ1) : 〈H(ℵ1),∈〉 |= ϕ(x, a)}. Note that for any n ∈ ω,
A ⊆ Rn is projective if and only if it is definable in our sense. Note also since all
elements of H(ℵ1) can be coded by elements of
ωω we may assume that a as above
is an element of ωω.
In Theorem 2.7, we show that the following Homogeneity Principle HP(κ) im-
plies both of Cs(κ) and Ĉs(κ).
HP(κ): For any f : κ→ P(ω) and any definable A ⊆ ((P(ω)))
, at least one
of the following holds:
(h0) there is a stationary S ⊆ κ such that ((f ′′S))
\ {∅} ⊆ A;
(h1) there are k ∈ ω \ 1 and stationary S0,...,Sk−1 ⊆ κ such that
((f ′′S0, ..., f
′′Sk−1)) ∩ A = ∅.
Note that P(ω) in the definition of HP(κ) above can be replaced by R, ωω, (P(ω))n
or (ωω)n etc. since these spaces can be coded as definable subsets of P(ω) and vice
versa.
As for Cs(κ) (and Ĉs(κ)), it is enough to consider HP(κ) for regular κ. Lemma
2.1 is also true for HP(κ):
Lemma 2.2. (a) HP(ℵ1) does not hold.
(b) HP(κ) holds for any regular κ > 2ℵ0.
Proof. (a): This follows from Lemma 2.1 and Theorem 2.7.
(b): Let κ > 2ℵ0 be a regular cardinal. Suppose that f : κ → P(ω) and A are
as in the definition of HP(κ). Then there is a stationary S ⊆ κ such that f ↾ S
is constant. If (h0) in the definition of HP(κ) does not hold then we must have
((f ′′S))
∩A = ∅ since ((f ′′S))
= ∅ for all n > 1. Hence (h1) holds with n = 1 and
S0 = S. (Lemma 2.2)
The following combinatorial principle Fs(κ) is also introduced in [13]:
Fs(κ): For any T ⊆ ω<ω and any matrix 〈aα,n : α < κ, n ∈ ω〉 of subsets of
ω, at least one of the following holds:
(f 0) there is a stationary S ⊆ κ such that
n<|t|aαn,t(n) : t ∈ T and 〈α0, ...,α| t |−1〉 ∈ ((S))
} | ≤ ℵ0 ;
(f 1) there are t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that
for every 〈α0, ...,α| t |−1〉, 〈β0, ...,β| t |−1〉 ∈ ((S0, ...,S|t|−1)), if
αn 6= βn for all n < |t|, then
n<|t| aαn,t(n) 6=
n<|t| aβn,t(n).
Lemma 2.3. (I. Juhász, L. Soukup and Z. Szentmiklóssy [13])
(a) Fs(ℵ1) does not hold.
(b) Fs(κ) holds for every regular κ > 2ℵ0.
(c) Fs(κ) implies Ĉs(κ).
A combinatorial principle in terms of coloring of ordinals by reals corresponding
naturally to Fs(κ) might be the following Injectivity Principle IP(κ, λ) for cardinals
κ and λ with λ ≤ κ:
IP(κ, λ): For any f : κ→ P(ω) and definable g : ((P(ω)))
→ P(ω), at least
one of the following holds:
(i0) there is a stationary S ⊆ κ such that | g ′′((f ′′S))
| < λ for
every n ∈ ω;
(i1) there are k ∈ ω\1 and stationary S0,...,Sk−1 ⊆ κ such that
for any 〈x0, ...,xk−1〉, 〈y0, ..., yk−1〉 ∈ ((f
′′S0, ..., f
′′Sk−1)),
if xn 6= yn for all n < k, then we have g(x0, ..., xk−1) 6=
g(y0, ..., yk−1).
Again here, we may replace P(ω) in the definition of IP(κ, κ) above by R, ωω,
(P(ω))n or (ωω)n etc.
Lemma 2.4. (a) For λ ≤ λ′ ≤ κ, IP(κ, λ) implies IP(κ, λ′).
(b) IP(ℵ1,ℵ1) does not hold.
Proof. (a): Immediate from the definition.
(b): By Lemma 2.2, (a) and Theorem 2.9. (Lemma 2.4)
IP(κ,ℵ0) for a regular cardinal κ is equivalent to the cardinal inequality 2
ℵ0 < κ.
Proposition 2.5. For a regular cardinal κ the following are equivalent:
(a) IP(κ,ℵ0) holds; (b) 2
ℵ0 < κ; (c) IP(κ, 2) holds.
Proof. (a) ⇒ (b): Suppose that 2ℵ0 ≥ κ. We show that IP(κ,ℵ0) does not hold.
Let f : κ → P(ω) be any injective mapping and g : ((P(ω)))
→ P(ω) be defined
by g(∅) = ∅, g(〈x〉) = ∅ for all x ∈ P(ω) and
g(〈x0, ..., xn−1〉) = min{m ∈ ω : m ∈ x0 6↔ m ∈ x1}
for 〈x0, ..., xn−1〉 ∈ ((P(ω)))
with n ≥ 2. Let S be any stationary subset of κ.
Then | g ′′((f ′′S))
| ≥ ℵ0: Suppose not and let k ∈ ω be such that g
′′((f ′′S))
Since P(k + 1) is finite, there are α, β ∈ S, α 6= β such that f(α) ∩ (k + 1) =
f(β) ∩ (k + 1). But then, by definition of g, it follows that g(〈f(α), f(β)〉) > k.
This is a contradiction.
Thus (i0) does not hold for these f and g. On the other hand, for arbitrary
stationary subsets S0,...,Sn−1 of κ, as there are only countably many values of g,
we can find 〈x0, ..., xn−1〉, 〈y0, ..., yn−1〉 ∈ ((f
′′S0, ..., f
′′Sn−1)) such that xi 6= yi for
all i < n and g(〈x0, ..., xn−1〉) = g(〈y0, ..., yn−1〉). Thus (i1) neither holds.
(b) ⇒ (c): Suppose 2ℵ0 < κ. For f : κ → P(ω) and g : ((P(ω)))
→ P(ω) as
in the definition of IP(κ, 2), there is a stationary S ⊆ κ such that f is constant on
S. This S witnesses that (i0) holds.
(c) ⇒ (a): This follows from Lemma 2.4, (a). (Proposition 2.5)
Corollary 2.6. IP(ℵ2,ℵ0) is equivalent to CH.
IP(ℵ2,ℵ1) and IP(ℵ2,ℵ2) are thus the first two non-trivial instances of IP(κ, λ).
For κ ≥ ℵ2, the principles introduced in this section and some other principles
discussed in [6] can be put together in the following diagram:
IP(κ,ℵ1)
Fs(κ)
HP(κ)
Ĉs(κ) Cs(κ)
Princ(κ, κ)
SEP(κ, κ)
Theorem 2.8
Theorem 2.9
Theorem 2.7
[13] [17] (see also [6])
IP(κ, κ)
Theorem 2.7
fig. 1
In the rest of the section, we shall prove the implications indicated by the thick
arrows in fig.1.
Theorem 2.7. For a regular cardinal κ, HP(κ) implies both Cs(κ) and Ĉs(κ).
Proof. We prove that HP(κ) implies Cs(κ). The other implication can be proved
similarly.
By Lemma 2.1, (b), we may assume that κ ≤ 2ℵ0. Let 〈ti : i ∈ ω〉 be an
enumeration of ω>ω such that
(2.4) | ti | ≤ i for all i < ω
and let ι : P(ω) → P(ω)ω be a definable bijection. For each x ∈ P(ω) and i < ω,
let (x)i denote the i’th component of ι(x).
Suppose that T ⊆ ω>ω and A = 〈aα,n : α < κ, n ∈ ω〉 is a matrix of subsets of
ω. We show that either (c0) or (c1) holds for these A and T .
Let g : κ → P(ω) be a fixed injective mapping which exists by κ ≤ 2ℵ0 . Let
f : κ→ P(ω) be defined by
(2.5) f(α) = ι−1(〈a′α,n : n ∈ ω〉)
where
(2.6) a′α,n =
g(α), if n = 0,
aα,n−1, otherwise.
Note that f is injective by “if n = 0” clause of (2.6). For i < ω, let
(2.7) A∗i =
{〈x0, ..., xi−1〉 ∈ ((P(ω)))
n<| ti |
(xn)ti(n)+1 6= ∅}, if ti ∈ T,
((P(ω)))
, otherwise
(2.8) A =
A∗i .
It is easy to see that A is definable noting that T ∈ H(ℵ1) and hence T can be
used as a parameter in the definition of A. By HP(κ), we have either (h0) or (h1)
for these A and f .
Assume first that (h0) holds. Then there is a stationary S ⊆ κ such that
((f ′′S))
\ {∅} ⊆ A. We show that this S witnesses (c0) for T and A: for t ∈ T ,
let i ∈ ω be such that t = ti. By (2.4), we have | t | ≤ i. For s ∈ ((S))
| t |
, let
s′ ∈ ((S))
be an end-extension of s. Then 〈f(s′(0)), ..., f(s′(i−1))〉 ∈ ((f ′′S))
since
f is injective. Hence 〈f(s′(0)), ..., f(s′(i − 1))〉 ∈ A∗i by the assumption on S. By
(2.7), we have
n<| ti |
(f(s′(n)))ti(n)+1 =
n<| ti |
s′(n),ti(n)+1
n<| t | as(n),t(n).
Thus T and A satisfy (c0).
Assume now that (h1) holds. In this case, there are i ∈ ω and stationary
S0,...,Si−1 ⊆ κ such that
(2.9) ((f ′′S0, ..., f
′′Si−1)) ⊆ ((P(ω)))
\ A∗i .
Let t = ti. Then t ∈ T by (2.9) and “otherwise” clause of (2.7). For s ∈
((S0, ...,S|t|−1)), let s
′ ∈ ((S0, ...,Si−1)) be an end extension of s. Then we have
〈f(s′(0)), ..., f(s′(i− 1))〉 ∈ ((f ′′S0, ..., f
′′Si−1)). It follows that
〈f(s′(0)), ..., f(s′(i− 1))〉 ∈ ((P(ω)))
\ A∗i
by (2.9). Hence, by (2.7), we have
n<| t |(f(s
′(n)))t(n)+1 =
n<| ti |
s′(n),t(n)+1
n<| t | as(n),t(n).
Thus, T and A satisfy (c1) in this case.
The proof of Ĉs(κ) from HP(κ) is exactly like the proof above with (2.7) replaced
(2.7)
A∗i =
{〈x0, ..., xi−1〉 ∈ ((P(ω)))
n<| ti |
(xn)ti(n)+1 | < ℵ0}, if ti ∈ T,
((P(ω)))
, otherwise.
(Theorem 2.7)
HP(κ) also imply other variants of Cs(κ). For example, let
∗Cs(κ): For any matrix 〈aα,n : α ∈ κ, n ∈ ω〉 of subsets of ω and T ⊆
at least one of the following holds:
(∗c0) there is a stationary S ⊆ κ such that
n<|t| aαn,t(n) is infinite
for all t ∈ T and 〈α0, ...,α|t|−1〉 ∈ ((S))
(∗c1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that
n<|t| aαn,t(n) is finite for all 〈α0, ...,α|t|−1〉 ∈ ((S0, ...,S|t|−1)).
It is easy to see by a proof similar to that of Theorem 2.7 that HP(κ) implies ∗Cs(κ)
as well.
The following can also be proved similarly to Theorem 2.7.
Theorem 2.8. IP(κ,ℵ1) implies F
s(κ).
Theorem 2.9. IP(κ, κ) implies HP(κ).
Proof. Suppose that A ⊆ ((P(ω)))
is definable and f : κ → P(ω). If f−1[{x}]
is stationary for some x ∈ P(ω), then, either (h0) holds for S = f−1[{x}] or (h1)
holds for n = 1 and S0 = f
−1[{x}] depending on whether x ∈ A or not. Otherwise
let g : ((P(ω)))
→ P(ω) be defined by
(2.10) g(∅) = ∅;
(2.11) g(〈x〉) = ∅ for all x ∈ P(ω) and
(2.12) g(〈x0, ..., xn−1, x〉) =
∅, if 〈x0, ..., xn−1〉 ∈ A,
x otherwise
for all 〈x0, ..., xn−1, x〉 ∈ ((P(ω)))
. If (i0) holds for this g with S as in (i0), then,
by (2.12), we must have g ′′((f ′′S))
= {∅}. Hence ((f ′′S))
\ {∅} ⊆ A. On the
other hand, if (i1) holds for some n < ω and S0 ,...,Sn−1, then we should have n ≥ 2
by (2.11) and g(〈x0 , ..., xn−2, xn−1〉) = xn−1 for all xi ∈ f
′′Si, i < n by (2.12). It
follows that ((f ′′S0, ..., f
′′Sn−2)) ⊆ ((P(ω)))
\ A by (2.12). (Theorem 2.9)
3 The bounding number and its variations
In this section, we show that the combinatorial principles introduced in the last
section make some of the cardinal invariants from [6] small.
Adopting the notation of [6], we consider the following spectra of cardinal num-
bers in connection with a partial ordering 〈P,≤〉; unbounded spectrum, hereditary
unbounded spectrum and the spectrum of length of P :
S(P ) = {|X | : X ⊆ P,X is unbounded in P, ∀B ∈ [X ]<|X |(B is bounded in P )} ,
h(P ) = {|X | : X ⊆ P, ∀B ⊆ X (B is bounded in P ↔ |B | < |X |)} ,
↑(P ) = {cf(C) : C ⊆ P, C is an unbounded chain} .
Clearly, we have
(3.1) S↑(P ) ⊆ Sh(P ) ⊆ S(P ).
For P = 〈ωω,≤∗〉, we shall simply write S↑, Sh and S in place of S↑(〈ωω,≤∗〉),
h(〈ωω,≤∗〉) and S(〈ωω,≤∗〉), respectively.
Recall that the bounding number b is defined by
b = min{|X | : X ⊆ ωω is unbounded with respect to ≤∗}.
The variant b∗ of b was introduced and studied in [3] and [14] where
∗ = min{κ : ∀X ⊆ ωω
X is unbounded → ∃X ′ ∈ [X ]≤κ(X ′ is unbounded)
b and b∗ can be characterized in terms of S↑, Sh and S as follows:
Lemma 3.1. (a) b = minS↑ = minSh = minS.
(b) b∗ = supS.
In analogy to Lemma 3.1, (b), let
(3.2) b↑ = supS↑, bh = supSh.
Recall also that the dominating number d is defined as:
d = min{|X | : X ⊆ ωω, X dominates ωω}.
By (3.1) and Lemma 3.1, we have
Lemma 3.2. b ≤ b↑ ≤ bh ≤ b∗ ≤ d.
fig. 2
DO = {cf(otp(〈X,R ↾ X〉)) : X ⊆ ωω, R is a definable binary relation
and R ∩X2 well orders X}
do = supDO.
By definition, S↑ ⊆ DO. Hence
Lemma 3.3. b↑ ≤ do.
Lemma 3.2 and Lemma 3.3 may be put together into the following diagram:
b ≤ b↑ ≤ bh ≤ b∗ ≤ d
fig. 3
If S↑ has a maximal element then we have b↑ = maxS↑. In such case we
shall say that b↑ is attained. Also we shall say that b∗, bh or do is attained if the
corresponding set has a maximal element.
In the following, Reg denotes the class of regular cardinals. The following lemma
can be proved similarly to Lemma 3.7, (c).
Lemma 3.4. ([6]) Sh ∩ Reg ⊆ DO.
Corollary 3.5. If Sh ∩ Reg is cofinal in Sh then bh ≤ do.
Note that the condition “Sh ∩ Reg is cofinal in Sh” holds if 2ℵ0 < ℵω or if b
is regular and attained. Under this condition, we can thus improve the diagram in
fig.3 to the following:
b ≤ b↑ ≤ bh ≤ b∗ ≤ d
fig. 4
For an ideal I over a set X , non(I) and cov(I) denote, as usual, the uniformity
and the covering number of I, respectively. More exactly
non(I) = min{|A | : A ∈ P(X) \ I} and
cov(I) = min{|A | : A ⊆ I,
A = X}.
meager and null denote the ideal of meager sets and the ideal of null sets (over R)
respectively.
Lemma 3.6. Suppose that I is an ideal over R with Borel basis. Then we have
min{non(I), cov(I)} ≤ do. In particular, we have min{non(meager), cov(meager)} ≤
do and min{non(null), cov(null)} ≤ do.
Proof. Suppose that I ⊆ P(R) is an ideal with a Borel basis and κ = min{non(I), cov(I)}.
We can construct inductively a sequence 〈〈fα, gα〉 : α < κ〉 such that
(3.3) fα, gα ∈
ωω for all α < κ;
(3.4) gα codes a Borel set Xα ⊆
ωω such that Xα ∈ I and {fβ : β < α} ⊆ Xα ;
(3.5) fα 6∈
Xβ for all α < κ.
Note that (3.4) is possible by κ ≤ non(I) and (3.5) by κ ≤ cov(I).
The sequence 〈〈fα, gα〉 : α < κ〉 is well ordered in order type κ by the definable
ordering:
〈f ′, g′〉 ≤ 〈f, g〉 ⇔ f ′ is an element of the Borel set coded by g.
It follows that κ ≤ do. (Lemma 3.6)
The following lemma shows the relations of cardinal numbers b, b↑, bh, do to
the combinatorial principles introduced in Section 2.
Lemma 3.7. (a) (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) If there is
a ≤∗-chain of length κ then ¬Cs(κ) and ¬Ĉs(κ). In particular, κ ∈ S↑ implies
¬Cs(κ) and ¬Ĉs(κ).
(b) Cs(κ) (or Ĉs(κ)) implies b↑ ≤ κ. If b↑ is attained then Cs(κ) (or Ĉs(κ))
implies b↑ < κ.
(c) If κ ≤ λ for some λ ∈ Sh with cf λ ≥ κ then ¬Cs(κ) and ¬Ĉs(κ).
(d) If Sh ∩ Reg is cofinal in Sh then Cs(κ) (or Ĉs(κ)) implies bh ≤ κ. If bh is
regular and attained then Cs(κ) (or Ĉs(κ)) implies bh < κ.
(e) κ ∈ DO implies ¬HP(κ).
(f) HP(κ) implies do ≤ κ. If do is attained then HP(κ) implies do < κ.
Proof. (a): See [13].
(b): This follows from (a).
(c): Suppose that κ ≤ λ ∈ Sh and κ ≤ cf λ. We show ¬Cs(κ). ¬Ĉs(κ) can be
proved similarly from these assumptions.
Let X ⊆ ωω with |X | = λ be as in the definition of Sh. Then we can find
fα ∈ X and gα ∈
ωω for α < κ such that
(3.6) fα ≤
∗ gβ for all α < β < κ;
(3.7) fβ 6≤
∗ g+α for all α ≤ β < κ where g
α is defined by g
α (k) = gα(k) + 1 for
all k ∈ ω.
Note that (3.7) is possible since cf(|X |) ≥ κ.
For α < κ, let gα,n ∈
ωω, n ∈ ω be such that
(3.8) {gα,n : n ∈ ω} = {g ∈
ωω : g =∗ gα}.
(3.9) aα,0 = {〈k, ℓ〉 ∈ ω
2 : ℓ ≤ fα(k)} and
(3.10) aα,n+1 = {〈k, ℓ〉 ∈ ω
2 : ℓ > gα,n(k)} for all n ∈ ω.
We show that A = 〈aα,n : α ∈ κ, n ∈ ω〉 with T = {〈0, n〉 : n ∈ ω \ 1} is a
counter-example to Cs(κ).
Suppose first that S ⊆ κ is stationary. For any α ∈ S, let β ∈ S be such
that α < β. Then we have fα ≤
∗ gβ by (3.6). Hence there is n ∈ ω such that
fα ≤ gβ,n. By (3.9) and (3.10), it follows that aα,0 ∩ aβ,n+1 = ∅. This shows that
〈A, T 〉 6|= (c0).
Suppose now that S0, S1 ⊆ κ are stationary and 〈0, n〉 ∈ T . By the definition
of T , it follows that n ∈ ω\1. Let α ∈ S0 and β ∈ S1 be such that β < α. Then, by
(3.7), we have fα 6≤
. Thus, by (3.9) and (3.10), it follows that aα,0 ∩ aβ,n 6= ∅.
This shows that 〈A, T 〉 6|= (c1).
(d): This follows easily from (c).
(e): Suppose that κ ∈ DO and let 〈X,R〉 be such that X ⊆ P(ω), R is a
projective binary relation and otp(〈X,R ∩ X2〉) = κ. Let f : κ → P(ω) be the
mapping sending α < κ to the α’th element of X with respect to R. Let
A = R ∪
k∈ω\{2}((P(ω)))
Then it is easily seen that 〈f, A〉 6|= (h0) and 〈f, A〉 6|= (h1).
(f): This follows from (d) since DO is downward closed. (Lemma 3.7)
Corollary 3.8. (a) HP(κ) implies min{non(I), cov(I)} ≤ κ for any ideal I over
R with Borel basis. In particular, it implies
min{non(meager), cov(meager)} ≤ κ and min{non(null), cov(null)} ≤ κ.
(b) If do is attained then HP(κ) implies min{non(I), cov(I)} < κ for all any I
over R with Borel basis. In particular, it implies
min{non(meager), cov(meager)} < κ and min{non(null), cov(null)} < κ.
Proof. By Lemma 3.6 and Lemma 3.7, (f). (Corollary 3.8)
Corollary 3.9. (a) Cs(ℵ2) (or Ĉ
s(ℵ2)) implies b
h = ℵ1.
(b) HP(ℵ2) implies
do = min{non(meager), cov(meager)} = min{non(null), cov(null)} = ℵ1.
Proof. (a): By Lemma 3.7, (d).
(b): By Lemma 3.7, (f) and Corollary 3.8. (Corollary 3.9)
do = ℵ1
HP(ℵ2)
IP(ℵ2,ℵ1)
h = ℵ1b
↑ = ℵ1b = ℵ1 b
∗ = ℵ1
Cs(ℵ2)
Ĉs(ℵ2)
Fs(ℵ2)
Corollary 3.9,(b)
Corollary 3.9,(a)
fig. 5
4 A forcing construction of models of IP(κ, λ)
In this section, we shall prove that IP(κ, λ) holds in a generic extension by a
homogeneous product of copies of a relatively small partial ordering (Theorem
4.3).
Let us begin with definition of some notions needed for precise formulation of
the theorem.
For cardinals κ and µ, κ is said to be µ-inaccessible if κ is regular and λµ < κ
holds for all λ < κ. Similarly, we say that κ is <µ-inaccessible if κ is regular and
λ<µ < κ holds for all λ < κ. Thus, if µ is a successor cardinal, say µ = µ+0 , then κ is
<µ-inaccessible if and only if κ is µ0-inaccessible. In our context, <µ-inaccessibility
is relevant because of the following variant of the ∆-System Lemma of Erdős and
Rado. For cardinals µ < κ, let
Eκ≥µ = {α < κ : cf(α) ≥ µ}
and let Eκµ , E
≤µ etc. be defined analogously.
Theorem 4.1. (P. Erdős and R. Rado, see [13]) Suppose that κ is <µ-inaccessible
and S ⊆ Eκ≥µ is stationary in κ. For any sequence 〈xα : α ∈ S〉 of sets of
cardinality < µ there is a stationary S∗ ⊆ S such that 〈xα : α ∈ S
∗〉 form a
∆-system.
For a sequence Pα, α < δ of posets and an ideal I ⊆ P(δ), we consider the
I-support product
Pα of Pα, α < δ here as being defined as
(4.1)
Pα = {f : f : D →
Pα for some D ∈ I
and f(α) ∈ Pα \ {1lPα} for all α ∈ D }
with the ordering
(4.2) f ≤QI
g ⇔ dom(f) ⊇ dom(g) and
f(α) ≤Pα g(α) for all α ∈ dom(g)
for all f , g ∈
Pα. In particular, 1lQI
= ∅ is the largest element of
with respect to ≤QI
α<δ Pα
Though this definition of product of posets is different from the standard one,
it gives a poset forcing equivalent to the product given by the standard definition.
The present definition is chosen here for the sake of smoother treatment of p ↾ X ,
P ↾ X , G ↾ X etc. (see (4.5), (4.7) etc.)
As usual, the ideal [δ]<ℵ0 is denoted by fin and
Pα is called the finite
support product of Pα, α < δ.
I. Juhász and K. Kunen [12] proved the following theorem for µ = ℵ1 and
I = [δ]<ℵ0 . Their proof also applies to the following slight generalization.
Theorem 4.2. (I. Juhász and K. Kunen [12]) Suppose that P =
Pα for some
ideal I ⊆ P(δ), P satisfies the µ-c.c. and |Pα | ≤ 2
<µ for all α < δ. Then, for all
<µ-inaccessible κ we have ‖– P “C
s(κ) ”.
Suppose that I ⊆ P(δ) is an ideal and P =
Pα is an I-support product of
posets Pα, α < δ. For p ∈ P, the support supp(p) of p is defined by
(4.3) supp(p) = dom(p).
We assume in the following that P-names are constructed just as in [15]. For a
P-name ȧ, the support supp(ȧ) is defined by
(4.4) supp(ȧ) =
{supp(p) : 〈ḃ, p〉 ∈ tcl(ȧ) for some P-name ḃ}.
For X ∈ P(δ) (not necessarily in I), let
(4.5) P ↾ X = {p ↾ X : p ∈ P}.
By (4.1) and since I is an ideal, we have
(4.6) P ↾ X = {p ∈ P : supp(p) ⊆ X}.
In particular,
(4.7) P ↾ X ⊆ P.
Furthermore, it is easy to see that P ↾ X ≤◦ P. Thus, if G is a (V,P)-generic
filter then G ∩ (P ↾ X) is a (V,P ↾ X)-generic filter. We shall denote the generic
filter G ∩ (P ↾ X) by GX . Note that a P-name ȧ is a P ↾ X-name if and only if
supp(ȧ) ⊆ X .
We shall call an I-support product P =
Pα homogeneous if Pα ∼= Pβ for all
α, β < δ and I is translation invariant, that is, I = {j ′′x : x ∈ I} for all bijections
j : δ → δ.
Note that if I is translation invariant then I = [δ] < λ for some λ.
For a homogeneous P =
Pα, we shall always assume that a commutative
system iα,β : Pα
→ Pβ, α, β < δ of isomorphisms is fixed. With such a fixed system
of isomorphisms, every bijection j : δ → δ induces an isomorphism j̃ : P
defined by
(4.8) dom(j̃(p)) = j ′′ dom(p) ;
for α ∈ dom(j̃(p)), j̃(p)(α) = ij−1(α),α ◦ p ◦ j
−1(α)
for all p ∈ P.
For notational simplicity we shall denote the isomorphism on P-names induced
from j̃ also by j̃.
Note that for P and j as above, p ∈ P, P-names ȧ0,..., ȧn−1 and a formula ϕ in
the language of set theory LZF, we have
(4.9) p ‖–P “ϕ(ȧ0, ..., ȧn−1) ” if and only if j̃(p) ‖–P “ϕ(j̃(ȧ0), ..., j̃(ȧn−1)) ”.
We are now ready to formulate the main result of the present section:
Theorem 4.3. Suppose that
(4.10) λ is a regular uncountable cardinal with 2<λ = λ, µ ∈ {λ, λ+} and κ is a
<λ-inaccessible cardinal.
Let P =
Pα be a homogeneous I-support product such that
(4.11) I ⊆ [δ]<λ;
(4.12) |Pα | ≤ λ for all α < δ and P satisfies the µ-c.c.;
(4.13) P is proper.
Then ‖– P “ IP(κ, µ) ” holds.
The proof of Theorem 4.3 will be given after the following Lemmas 4.4 to 4.7.
As in [15], a P-name ẋ of a subset of ω for a poset P is called a nice P-name
if there are antichains Aẋ,n, n ∈ ω in P such that ẋ = {〈ň, p〉 : p ∈ Aẋ,n}. Note
that, for such a name ẋ, we have supp(ẋ) =
n∈ω Aẋ,n. It is easy to see that, for
all P-names ẋ of subsets of ω, there is a nice P-name ẋ′ such that ‖–P “ ẋ = ẋ
We say that a nice P-name of a subset of ω with Aẋ,n, n ∈ ω as above is slim if
Aẋ,n is countable for all n < ω.
The following lemmas are well-known:
Lemma 4.4. Suppose that P is a proper poset and p ∈ P. For any P-name ẋ of a
subset of ω, there are q ≤P p and a slim P-name ẋ
′ such that q ‖–P “ ẋ = ẋ
Proof. By the remark above, we may assume without loss of generality that ẋ
is a nice P-name. Let Aẋ,n, n ∈ ω be as above and ẏ be a P-name such that
‖–P “ ẏ = {s ∈ P : s ∈ (
n∈ω Aẋ,n) ∩ Ġ} ”. Then we have ‖– P “ ẏ is a countable
subset of P ”. As P is proper there exist q ≤P p and countable y ⊆ P such that
q ‖– P “ ẏ ⊆ y ”. Let ẋ
′ = {〈ň, s〉 : n ∈ ω, s ∈ Aẋ,n ∩ y}. These q and ẋ
′ are as
desired. (Lemma 4.4)
Lemma 4.5. Suppose that P =
Pα is a κ-c.c. I-support product for an ideal
I ⊆ [δ]<λ and κ is <λ-inaccessible. If S ⊆ Eκ≥λ is stationary and pα ∈ P for
α ∈ S are such that supp(pα), α ∈ S form a ∆-system with the root R and there
is p∗ ∈ P ↾ R such that pα ↾ R = p
∗ for all α ∈ S, then
p∗ ‖– P “ {α ∈ S : pα ∈ Ġ} is stationary ”
where Ġ denotes the standard P-name of a (V,P)-generic filter.
Proof. By κ-c.c. of P, κ remains a regular cardinal in P-generic extensions. Let
Ṡ be a P-name of {α ∈ S : pα ∈ Ġ}. Suppose that Ċ is a P-name of a club
subset of κ and p ≤P p
∗. It is enough to show that there is a q ≤P p such that
q ‖– P “ Ċ ∩ Ṡ 6= ∅ ”.
Let θ be sufficiently large and let M ≺ H(θ) be such that
(4.14) I, P, κ, Ċ, 〈pα : α ∈ S〉, p ∈M ;
(4.15) |M | < κ ∩M < κ;
(4.16) [M ]<λ ⊆M and
(4.17) α∗ ∈ S where α∗ = κ ∩M .
(4.16) is possible since κ is <λ-inaccessible. (4.17) is possible since S ⊆ Eκ≥λ and
S is stationary in κ.
Claim 4.5.1. ‖–P “α
∗ ∈ Ċ ”.
⊢ Since P satisfies the κ-c.c., we have
H(θ) |= ∀α < κ ∃β ∈ κ \ α ( ‖–P “β ∈ Ċ ”).
By (4.14), and elementarity of M it follows that
M |= ∀α < κ ∃β ∈ κ \ α ( ‖–P “β ∈ Ċ ”).
Thus ‖–P “ Ċ ∩ α
∗ is unbounded in α∗ ”. Since ‖–P “ Ċ is a club in κ ”, it follows
that ‖–P “α
∗ ∈ Ċ ”. ⊣ (Claim 4.5.1)
Claim 4.5.2. supp(pα∗) ∩M = R and supp(p) ∩ supp(pα∗) = R .
⊢ Suppose u = (supp(pα∗)∩M)\R 6= ∅. By (4.16), u ∈M . Hence by elementarity
M |= ∃α < κ (u ⊆ supp(pα)). Let α ∈ κ ∩M be such that u ⊆ supp(pα). Then
α < α∗ and R ∪ u ⊆ pα ∩ pα∗ . This is a contradiction to the assumption that R is
the root of the ∆-system {supp(pα) : α ∈ S}. This shows supp(pα∗) ∩M = R.
By (4.14) and (4.16), supp(p) ∈ M . It follows that supp(p) ∩ sup(pα∗) =
supp(p) ∩ (sup(pα∗) ∩M) = supp(p) ∩ R = R.
⊣ (Claim 4.5.2)
Since p ↾ R ≤P p
∗ ↾ R = p∗ = pα∗ ↾ R, q = p ∪ pα∗ ∈ P. We have q ≤P p.
By pα∗ ‖–P “α
∗ ∈ Ṡ ” and q ≤P pα∗ , we have q ‖–P “α
∗ ∈ Ċ ∩ Ṡ ”. In particular
q ‖– P “ Ċ ∩ Ṡ 6= ∅ ”. (Lemma 4.5)
The arguments of the following two lemmas are also well-known. For Lemma
4.6 see e.g. [12].
Lemma 4.6. Suppose that P =
Pα is an I-support product and G is a (V,P)-
generic filter. For X, Y ⊆ δ, let Z = X ∩Y . Then, in V [G], for any κ ∈ CardV [G],
we have
[On]<κ ∩ (V [GX ] \ V [GZ ]) ∩ (V [GY ] \ V [GZ ]) = ∅.
Lemma 4.7. Suppose that κ ≤ δ and P =
Pα is a κ-c.c. homogeneous I-
support product, p ∈ P, ȧ0, ..., ȧn−1 are P-names with
(4.18) supp(ȧ0), ..., supp(ȧn−1) ⊆ X
for some X ⊆ δ and ϕ = ϕ(x0, ...,xn−1) is a formula in LZF (possibly with some
parameters from V ).
(a) If
(4.19) p ‖–P “ϕ(ȧ0, ..., ȧn−1) ” and
(4.20) δ \X 6∈ I,
then p ↾ X ‖–P “ϕ(ȧ0, ..., ȧn−1) ”.
(b) If
(4.21) p ‖–P “ (∃x ∈
ωω) ϕ(x, ȧ1, ..., ȧn−1) ”,
(4.22) supp(p) ⊆ X and
(4.23) |X \ (supp(p) ∪ supp(ȧ0) ∪ · · · ∪ supp(ȧn−1)) | ≥ κ,
then there is a PX-name ȧ such that p ‖–P “ϕ(ȧ, ȧ1, ..., ȧn−1) ”.
Proof. (a): Suppose that p ↾ X /‖–P “ϕ(ȧ0, ..., ȧn−1) ”. Then there is q ≤P p ↾ X
such that q ‖–P “¬ϕ(ȧ0, ..., ȧn−1) ”. Let j : δ → δ be a bijection such that
(4.24) j ↾ X = idX and
(4.25) (j ′′ supp(q) \X) ∩ supp(p) = ∅.
Note that the last condition is possible by (4.20). By (4.24) and (4.18), we have
(4.26) j̃(q) ↾ X = j̃(q ↾ X) = q ↾ X and
(4.27) j̃(ȧ0) = ȧ0,..., j̃(ȧn−1) = ȧn−1.
By (4.27) and by the choice of q, we have j̃(q) ‖–P “¬ϕ(ȧ0, ..., ȧn−1) ”. On the other
hand, by (4.26) p and j̃(q) are compatible. This is a contradiction to (4.19).
(b): By maximal principle, there is a nice P-name ȧ′ of a real such that
p ‖–P “ϕ(ȧ
′, ȧ1, ..., ȧn−1) ”.
By the κ-c.c. of P, we have | supp(ȧ′) | < κ. By (4.18), (4.22) and (4.23), we can
find a bijection j : δ → δ such that
(4.28) j on supp(p) ∪ supp(ȧ1) ∪ · · · ∪ supp(ȧn−1) is the identity mapping, and
(4.29) j ′′ supp(ȧ′) ⊆ X .
By (4.28), j̃(p) = p and j̃(ȧ1) = ȧ1,..., j̃(ȧn−1) = ȧn−1. Let ȧ = j̃(ȧ
′). Then
p ‖–P “ϕ(ȧ, ȧ1, ..., ȧn−1) ” and supp(ȧ) ⊆ X by (4.29). (Lemma 4.7)
Proof of Theorem 4.3: By Proposition 2.5, we may assume that ‖– P “ κ ≤ 2
ℵ0 ”.
In particular, by (4.11), (4.12) and (4.13), we may assume that δ ≥ κ. By the
µ-c.c. of P, µ and κ remain regular cardinals in the generic extension by P.
Let G be a (V,P)-generic filter. In V [G], let f : κ→ P(ω) and g : ((P(ω)))
P(ω) be definable, say by a formula ϕ. We may assume that ϕ has a real a ∈ V [G]
as its unique parameter. Let ḟ , ȧ and ġ be P-names of f , a and g respectively such
that ‖–P “ ḟ : κ→ P(ω) ”, ‖– P “ ġ : ((P(ω)))
→ P(ω) ” and
(4.30) ‖– P “ ∀x ∈ ((P(ω)))
∀x ∈ P(ω)
ġ(x) = x↔ H(ℵ1) |= ϕ(x, x, ȧ)
Suppose that, for a p ∈ G,
(4.31) p ‖–P “ (i0) for IP(κ, µ) does not hold for ḟ and ġ ”.
In particular, we have
(4.32) p ‖–P “ ∀α < κ ({β ∈ κ : ḟ(β) = ḟ(α)} is non-stationary) ”.
Claim 4.3.1. There is a stationary S ⊆ Eκ≥λ such that
p ‖–P “ ḟ ↾ S is 1-1 ”.
⊢ By the κ-c.c. of P and by (4.32), there are club sets Cα ⊆ κ (in V ) for each
α < κ such that
p ‖–P “Cα ∩ {β ∈ κ : ḟ(β) = ḟ(α)} = ∅ ”.
Then C = ∆α<κCα is club and S = E
≥λ ∩ C has the desired property.
⊣ (Claim 4.3.1)
We show that p forces (i1) for ḟ and ġ. Let p′ ≤P p. It is enough to show that
there is p∗ ≤P p
′ forcing (i1).
By Lemma 4.4, Theorem 4.1, (4.10), (4.11) and (4.13), there are p′′ ≤P p
slim P-name ȧ′ of a real, a stationary S∗ ⊆ S, a sequence 〈ẋ′α : α ∈ S
∗〉 of slim
P-names and a sequence 〈pα : α ∈ S
∗〉 of conditions in P such that
(4.33) (i) p′′ ‖– P “ ȧ = ȧ
(ii) pα ≤P p
′′ and
(iii) pα ‖– P “ ḟ(α) = ẋ
α ” for every α ∈ S
(4.34) dα = supp(pα)∪ supp(ȧ
′)∪ supp(ẋ′α), α ∈ S
∗ are all of the same cardinality
and form a ∆-system with root R;
(4.35) for each α, β ∈ S∗ there is a bijection jα,β : δ → δ such that
(i) jα,β ↾ (δ \ (dα∆dβ)) = idδ\(dα∆dβ) ,
(ii) jα,β
′′dα = dβ, j̃α,β(pα) = pβ and
(iii) j̃α,β(ẋ
α) = ẋ
β for every α, β ∈ S
Note that, by (4.34), we have
(4.36) supp(ȧ′) = supp(ȧ′) ∩ dα ⊆ R for every α ∈ S
By (4.35), pα ↾ R for α ∈ S
∗ are all the same. Let q = pα ↾ R for some/any
α ∈ S∗. q ≤P p
′′ by (4.33), (ii). Let Ṡ be a P-name such that
(4.37) ‖– P “ Ṡ = {α ∈ S
∗ : pα ∈ Ġ} ”.
By Lemma 4.5, q ‖–P “ Ṡ is stationary ”. Hence, by (4.31),
(4.38) q ‖– P “ ∃n ∈ ω ∀α < κ
| ġ ′′((ḟ ′′(Ṡ \ α)))
| ≥ µ
Let q′ ≤P q and n
∗ ∈ ω be such that
(4.39) q′ ‖–P “ ∀α < κ (| ġ
′′((ḟ ′′(Ṡ \ α)))
| ≥ µ) ”.
(4.40) S∗∗ = {α ∈ S∗ : supp(q′) ∩ dα ⊆ R}.
Since | supp(q′) | < λ by (4.11), we have
(4.41) S∗ \ S∗∗ is of cardinality < λ.
In particular S∗∗ is still stationary and
(4.42) q′ ‖–P “ | ġ
′′((ḟ ′′(Ṡ ∩ S∗∗)))
| ≥ µ ”
by (4.39).
Claim 4.3.2. There is 〈α0, ...,αn∗−1〉 ∈ ((S
such that
q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 /‖– P “ ġ(〈ẋ
, ..., ẋ′αn∗−1〉) ∈ V [ĠR] ”.
⊢ Otherwise, we would have
q′ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ‖–P “ ġ(〈ẋ
, ..., ẋ′βn∗−1〉) ∈ V [ĠR] ”
for all 〈β0, ..., βn∗−1〉 ∈ ((S
Fix 〈α0, ...,αn∗−1〉 ∈ ((S
and let
D = {r ∈ P : r ≤P q
′ ∪ pα0 ∪ · · · ∪ pαn∗−1 ,
supp(r) ⊆ R ∪
{dαi : i < n
∗} ∪ supp(q′),
r ‖– P “ ġ(ẋ
, ..., ẋ′αn∗−1) = ẋ ” for some PR-name ẋ }.
Let A be a maximal antichain in D. By the µ-c.c. of P, | A | < µ.
For each r ∈ A, let ẋr be a PR-name such that
r ‖– P “ ġ(ẋ
, ..., ẋ′αn∗−1) = ẋr ”
and Ẋ be a PR-name such that
‖– P “ Ẋ = {ẋr : r ∈ A} ”.
Then, we have ‖–P “ | Ẋ | < µ ”. By Lemma 4.7, (a)
q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 ‖– P “ ġ(〈ẋ
, ..., ẋ′αn∗−1〉) ∈ Ẋ ”.
Hence by (4.35) and (4.9), we have
q′ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ‖–P “ ġ(〈ẋ
, ..., ẋ′βn∗−1〉) ∈ Ẋ ”
for all 〈β0, ..., βn∗−1〉 ∈ ((S
. But this is a contradiction to (4.42). ⊣ (Claim 4.3.2)
Let 〈α0, ...,αn∗−1〉 ∈ ((S
be as in Claim 4.3.2 and
(4.43) q′′ = q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 .
Note that
(4.44) q′′ ‖–P “ f(αi) = ẋ
” for i < n∗ by (4.43).
Let p∗ ≤P q
′′ be such that
(4.45) p∗ ‖–P “ ġ(ẋ
, ..., ẋ′αn∗−1) 6∈ V [ĠR] ”.
By thinning out S∗∗ further, if necessary, we may assume that supp(p∗) ∩
supp(pα) ⊆ R for all α ∈ S
∗∗. For i < n∗, let Ṡi be a P-name such that
(4.46) ‖– P “ Ṡi = {α ∈ S
∗∗ : j̃αi,α(p
∗) ∈ Ġ} ”.
By Lemma 4.5, we have p∗ ‖–P “ Ṡi is a stationary subset of κ ” for all i < n
∗. Note
that we have j̃αi,α(p
∗) ≤P pα by (4.43) and (4.35), (ii).
Claim 4.3.3.
p∗ ‖– P “ ∀β0 · · · ∀βn∗−1
〈β0, ..., βn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1))
→ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) 6∈ V [ĠR]
⊢ Suppose that q ≤P p∗ and q ‖–P “ 〈β0, ..., βn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) ”. Then, by
(4.46), q ‖–P “ j̃αi,βi(p
∗) ∈ Ġ ” for i < n∗. It follows that
(4.47) q ‖– P “ j̃αi,βi(p
∗) ↾ dβi ∈ Ġ ” for i < n
j̃ = j̃α0,β0 ◦ j̃α1,β1 ◦ · · · ◦ j̃αn∗−1,βn∗−1 .
(4.48) j̃(p∗) = p∗ ↾ (δ \
dαi) ∪ j̃α0,β0(p
∗) ↾ dβ0 ∪ · · · ∪ j̃αn∗−1,βn∗−1(p
∗) ↾ dβn∗−1
by (4.35). Hence
(4.49) q ‖– P “ j̃(p
∗) ∈ Ġ ”
by q ≤P p
∗ and (4.47) and (4.48). By definition of j̃ and q′′, and by (4.35), we have
(4.50) j̃(p∗) ≤P j̃(q
′′) ≤P j̃αi,βi(pαi) = pβi for i < n
∗ and
(4.51) j̃(ẋ′αi) = ẋ
for i < n∗ by (4.44).
Hence by (4.45)
q ‖–P “ ġ(〈ẋ
, ..., ẋ′βn∗−1〉) 6∈ V [ĠR] ”.
By (4.33), (4.49) and (4.50), it follows that
q ‖–P “ ḟ(βi) = ẋ
for i < n∗. Hence q ‖–P “ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) 6∈ V [GR] ”. ⊣ (Claim 4.3.3)
To show that p∗ ‖–P “ (i1) holds ”, suppose that q ≤P p
∗ and 〈β0, ..., βn∗−1〉,
〈γ0, ..., γn∗−1〉 ∈ ((S
are such that
(4.52) {β0, ..., βn∗−1} ∩ {γ0, ..., γn∗−1} = ∅ and
(4.53) q ‖– P “ 〈β0, ..., βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) ”.
Note that it is enough to consider 〈β0, ...,βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((S
(4.52) since we can thin out ṠG0 ,..., Ṡ
n∗−1 afterwards if necessary so that they are
pairwise disjoint.
By the remark after (4.46), we may assume that
q ≤P p
∗ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ∪ pγ0 ∪ · · · ∪ pγn∗−1 .
By Lemma 4.7, (b), there are P-names ẏ, ż such that
supp(ẏ) ∩ supp(ż) ⊆ R and
p∗ ∪ pβ0 ∪ ··· ∪ pβn∗−1 ∪ pγ0 ∪ ··· ∪ pγn∗−1 ‖–P “ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) = ẏ
∧ ġ(〈ḟ(γ0), ..., ḟ(γn∗−1)〉) = ż ”.
By Claim 4.3.3 and Lemma 4.6, it follows that
q ≤P p
∗ ∪ pβ0 ∪ ··· ∪ pβn∗−1 ∪ pγ0 ∪ ··· ∪ pγn∗−1 ‖– P “ ġ(〈ḟ(β0〉, ..., ḟ(βn∗−1)) 6=
ġ(〈ḟ(γ0), ..., ḟγn∗−1))〉 ”.
Since q as above may be chosen below arbitrary r ≤P p
∗, it follows that
p∗ ‖–P “ (i1) holds ”.
(Theorem 4.3)
Corollary 4.8. (a) Assume CH and P = Fn(µ, 2) for some cardinal µ. Then
‖–P “ IP(ℵ2,ℵ1) ” holds.
(b) Assume GCH and P = Fn(µ, 2) for some cardinal µ. Then ‖–P “ IP(κ
+,ℵ1) ”
holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ1) ” for
every inaccessible λ.
(c) Assume CH and P is a finite support product of copies of a productively
c.c.c. poset of cardinality ℵ1. Then ‖–P “ IP(ℵ2,ℵ1) ” holds. In particular, we have
‖–P “HP(ℵ2) ”.
(d) Assume GCH and P is a finite support product of copies of a productively
c.c.c. poset of cardinality ℵ1. Then ‖– P “ IP(κ
+,ℵ1) ” holds for every uncountable
κ of uncountable cofinality and ‖–P “ IP(λ,ℵ1) ” for every inaccessible λ.
(e) Assume CH and P is a countable support product of copies of a proper poset
of cardinality ℵ1 such that its product is also proper. Then ‖–P “ IP(ℵ2,ℵ2) ” holds.
In particular, we have ‖–P “HP(ℵ2) ”.
(f) Assume GCH and P is a countable support product of copies of a proper
poset of cardinality ℵ1 such that its product is also proper. Then ‖–P “ IP(κ
+,ℵ2) ”
holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ2) ” for
every inaccessible λ.
Note that countable support products of Sacks or Prikry-Silver forcing are in-
stances of (e) and (f) above.
Proof. Under CH, ω1 = 2
<ω1 and ω2 is <ω1-inaccessible. In (a) and (b), P is forc-
ing equivalent to a finite support product of copies of the countable poset Fn(ω, 2).
Clearly P’s in all of (a) ∼ (f) are homogeneous; P’s in (a) ∼ (d) satisfy the c.c.c.
and hence they are proper. Thus we can apply Theorem 4.3. The second parts of
(c) and (e) follow from Theorem 2.9. (Corollary 4.8)
Results similar to Theorem ?? and Corollary 4.8 also hold for partial orderings
with product-like structure as those considered in [9]. Thus, we can prove e.g. that
IP(ℵ2,ℵ2) together with clubsuit principle is consistent.
In [7] it is shown that, if we start from a model V which is obtained by adding a
dominating real to a model of GCH + Chang’s conjecture for ℵω, i.e. (ℵω+1,ℵω) →→
(ℵ1,ℵ0), then adding more than ℵω+1 Cohen reals forces ¬WFN. Since V satisfies
GCH, IP(κ,ℵ1) is forced for every κ ≥ ℵ2 which is not a successor of a singular
cardinal of cofinality ω by adding any number of Cohen reals by Corollary 4.8. In
particular:
Corollary 4.9. Suppose that Chang’s conjecture for ℵω is consistent. Then so is
IP(ℵ2,ℵ1) ∧ b
∗ = ℵ1 ∧ ¬WFN.
5 Models of IP(ℵ2,ℵ2) ∧ ¬IP(ℵ2,ℵ1)
Recall that Prikry-Silver forcing S is the forcing with partial functions with co-
infinite domain, that is
S = {f : f : D → 2, D ⊆ ω, |ω \D | = ℵ0}
with the ordering
f ≤S g ⇔ f ⊇ g
for f , g ∈ S.
A (V, S)-generic filter G gives rise to the function sG =
G : ω → 2 which is
often called a Prikry-Silver real.
For f ∈ S let codom(f) = ω \ dom(f).
It is easy to check that Prikry-Silver forcing S as well as its countable support
products SI over any index set I satisfy the Axiom A. Hence they are all proper.
Note that, by definition of ≤S, we have:
(5.1) f , g ∈ S are incompatible if | codom(f) ∩ codom(g) | < ℵ0.
(5.2) For any 〈f0, f1〉 ∈ S
2, there is 〈g0, g1〉 ≤S2 〈f0, f1〉 such that | codom(g0) ∩
codom(g1) | < ℵ0.
Lemma 5.1. For any f ∈ S and 〈gn0 , g
1 〉 ∈ S
2, n ∈ ω such that | codom(gn0 ) ∩
codom(gn1 ) | < ℵ0 there is g ≤S f such that 〈g, g〉 is incompatible with all 〈g
0 , g
n ∈ ω.
Proof. Construct in ∈ 2, n ∈ ω and A ⊆ codom(f) recursively so that
| codom(f) ∩
k≤n dom(g
) | = ℵ0 and
|A ∩ codom(gnin) | < ℵ0 for all n ∈ ω.
Then any extension g of f on ω \ A will do. (Lemma 5.1)
Working in V = L, we can construct recursively a maximal antichain {〈gα0 , g
1 〉 :
α < ω1} in S
2 such that
(5.3) | codom(gα0 ) ∩ codom(g
1 ) | < ℵ0 for all α < ω1.
Note that each step of the recursive construction is possible by (5.2) and (5.2).
Furthermore by choosing 〈gα0 , g
1 〉 in each step of the construction according to
the Σ12-well ordering of the reals (which exists because of V = L), we can make
{〈gα0 , g
1 〉 : α < ω1} a Σ
2-set (actually we can even choose such a maximal antichain
as a Π11-set arguing similarly to [16]).
Let ϕ : S2 → ω2 be a Borel bijection and let g : ((ω2))
→ ω2 be defined by
(5.4) g(〈x0, ..., xn−1〉) =
0 , g
1 ) ; if n = 2, there is α < ω2 such that
x0 ⊇ g
0 , x1 ⊇ g
1 and α
∗ is minimal
among such α’s
0 ; otherwise.
It is easy to check that g is a ∆13-set.
Theorem 5.2. Assume V = L. Then we have
‖– Sω2 “ IP(ℵ2,ℵ2) and ¬IP(ℵ2,ℵ1) ”.
Proof. ‖– Sω2 “ IP(ℵ2,ℵ2) ” follows from Corollary 4.8, (e).
To show that ‖– Sω2 “¬IP(ℵ2,ℵ1) ”, let G be a (V, S
ω2)-generic filter. Working in
L[G], let sβ be the β’th Prikry-Silver real added by G for β < ω2. Let f : ω2 →
be defined by
(5.5) f(β) = sβ for β < ω2
and let g : ((ω2))
→ ω2 be the mapping as in (5.4), or more precisely, let g be the
mapping (in L[G]) defined by the ∆13 definition corresponding to (5.4).
We show that f and g build a counter-example to IP(ℵ2,ℵ1).
Since | rng(g) | ≤ ℵ1, (i1) clearly fails for these f and g. Hence we are done by
showing that f and g do not satisfy (i0).
Assume, for a contradiction, that f and g satisfy (i0). Returning to L, let ḟ ,
ġ, ṡβ, β < ω2 etc. be S
ω2-names of f , g, sβ , β < ω2 etc. respectively. In particular,
we can choose ḟ such that
(5.6) ‖– Sω2 “ ḟ(β) = ṡβ ” for all β < ω2.
Since Sω2 is proper, there are p ∈ G, Sω2-name Ṡ and a countable set Z (in L)
such that
(5.7) p ‖– Sω2 “ Ṡ ⊆ ω2 is stationary and ġ
′′((ḟ ′′Ṡ))
⊆ Z ”.
Let U = {β < ω2 : there is p
′ ≤Sω2 p such that p
′ ‖– Sω2 “ β ∈ Ṡ ”}. Then U is a
stationary subset of ω2. For each β ∈ U , let pβ ≤Sω2 p be such that pβ ‖– Sω2 “β ∈ Ṡ ”
and β ∈ supp(pβ).
By ∆-System Lemma and CH, there is U∗ ∈ [U ]ℵ2 such that
(5.8) supp(pβ), β ∈ U
∗ form a ∆-system with root R which is an initial segment
of all of supp(pβ), β ∈ U
(5.9) supR < minU∗;
(5.10) pβ ↾ R, β ∈ U
∗ are all the same; and
(5.11) pβ(β), β ∈ U
∗ are all the same, say h ∈ S.
Note that pβ, β ∈ U
∗ are compatible by (5.8) and (5.10).
(5.12) X = ϕ−1(Z).
By Lemma 5.1, there is a k ≤S h such that 〈k, k〉 is incompatible with all 〈g
0 , g
from the countable set X .
Fix two distinct β, γ ∈ U∗ and let q ≤Sω2 pβ, pγ be defined by dom(q) =
dom(pβ) ∪ dom(pγ) and
(5.13) q(δ) =
pβ(δ) ; if δ ∈ supp(pβ) \ {β}
pγ(δ) ; else if δ ∈ supp(pγ) \ {γ}
k ; else if δ = β or δ = γ
for δ ∈ dom(q).
By q ≤Sω2 pβ, pγ , we have q ‖– Sω2 “β, γ ∈ Ṡ ”. Thus the following claim yields
a contradiction to (5.7):
Claim 5.2.1. q ‖– Sω2 “ ġ(〈ḟ(β), ḟ(γ)〉) 6∈ Z ”.
⊢ By (5.6), we have to show q ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) 6∈ Z ”.
First, we show that q ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) 6= 0 ”. Note that, by the complete
embedding S2 ∋ 〈g0, g1〉 7→ {〈β, g0〉, 〈γ, g1〉} ∈ S
{β,γ} ≤◦ Sω2 we have:
{{〈β, gα0 〉, 〈γ, g
1 〉} : α < ω1} is a maximal antichain in S
For any r ≤Sω2 q, let α
∗ < ω1 be such that r and {〈β, g
0 〉, 〈γ, g
1 〉} are com-
patible. Let s ≤Sω2 r, {〈β, g
0 〉, 〈γ, g
1 〉}. Then we have
s ‖– Sω2 “ ṡβ ⊇ g
0 , ṡγ ⊇ g
Hence, by (5.4), it follows that s ‖–Sω2 “ ġ(〈ṡβ, ṡγ〉) 6= 0 ”.
Now, suppose, for contradiction, that there is r ≤Qω2 q such that
r ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) ∈ Z ”.
Then, by the first part of the proof, there are s ≤Sω1 r and 〈g
0 , g
1 〉 ∈ X such
that s ‖–Sω2 “ ṡβ ⊇ g
0 and ṡγ ⊇ g
1 ”. In particular s(β) and s(γ) are compatible
with gα0 and g
1 , respectively. Since r ≤Sω2 q ≤Sω2 {〈β, k〉, 〈γ, k〉}, it follows that k
is compatible with both of gα0 and g
1 . This is a contradiction to the choice of k.
⊣ (Claim 5.2.1)
(Theorem 5.2)
We can prove a Lemma similar to Lemma 5.1 for omega product of Sacks forcing.
Thus, by a similar argument as above, we can also prove that IP(ℵ2,ℵ1) fails in a
generic extension by countable support side-by-side product of Sacks forcing.
6 The Consistency of b∗ = ℵ2 ∧ do = ℵ1
In the following we shall refer by (A) the assertion that there is a structure
〈(ω2)
2, A,F〉 with the properties (6.1) ∼ (6.5) below. Recall that a mapping
f : X → X is called an involution if it is a bijection exchanging (some) pairs
of elements of X , that is, if f ◦ f = idX holds.
(6.1) ω2 × ω2 ⊇ A ⊇ {〈α, β〉 ∈ ω2 × ω2 : β < α};
(6.2) For any C ∈ [ω2]
ℵ0 there is an X ∈ [ω2]
ℵ2 such that (C ×X) ∩ A = ∅;
(6.3) For all 〈φ, ψ〉 ∈ F , φ and ψ are involutions on ω2;
(6.4) For each 〈φ, ψ〉 ∈ F and for all 〈α, β〉 ∈ ω2 ×ω2, we have 〈α, β〉 ∈ A if and
only if 〈φ(α), ψ(β)〉 ∈ A;
(6.5) For any stationary S ⊆ Eω2ω1 and any Aζ , Bζ ∈ [ω2]
ℵ0 for ζ ∈ S, there is
a stationary T ⊆ S such that, for any n ∈ ω, if ζi, ηi ∈ T for i ∈ n are
pairwise distinct (2n elements) then there is 〈φ, ψ〉 ∈ F such that
φ ′′Aζi = Aηi , ψ
′′Bζi = Bηi ; and
φ ↾ Aζi : Aζi → Aηi , ψ ↾ Bζi : Bζi → Bηi are order isomorphisms
for all i ∈ n.
The consistency of (A) together with CH over ZFC is proved in the next section.
Below, we will prove the consistency of c = b∗ = ℵ2 ∧ do = ℵ1 ∧ ¬C
s(ℵ2) by
constructing a model of this combination of assertions starting from a model of
(A) and CH.
Let us begin with introducing some notation for the forcing construction we use
in the proof.
For a cardinal κ, a sequence f̄ = 〈fξ : ξ < κ〉 in
ωω and X ⊆ κ, let Df̄ ,X be the
canonical poset adding an element of ωω dominating {fξ : ξ ∈ X}. That is
(6.6) Df̄ ,X = {〈s, F 〉 : s ∈
ω>ω, F ∈ [κ]<ℵ0}
and, for 〈s, F 〉, 〈s′, F ′〉 ∈ Df̄ ,X ,
(6.7) 〈s′, F ′〉 ≤Df̄ ,X 〈s, F 〉 ⇔ s
′ ⊇ s, F ′ ⊇ F,
∀α ∈ F ∩X ∀n ∈ dom(s′) \ dom(s) (fα(n) ≤ s
′(n)).
Since any 〈s, F 〉, 〈s′, F ′〉 ∈ Df̄ ,X with s = s
′ are compatible, we have:
Lemma 6.1. Df̄ ,X is σ-centered.
Note that the underlying set of Df̄ ,X does not depend on the sequence f̄ . So
we shall denote this set with DX . Actually DX as a set does not depend on X
either. Nevertheless we shall add the suffix X so that we can distinguish D’s by
their intended function.
Note also that, as a set,
Df̄ ,Xα for any κ-sequence f̄ of reals is the same:
we shall denote this set by
If d ∈ DX and d = 〈s, F 〉 then we shall write s
d and F d to denote these s and
F respectively.
In the following we assume that a sequence X̄ = 〈Xα : α < κ〉 of nonempty
subsets of κ is fixed. Let
(6.8) QX̄ = Cκ ∗
D ˙̄f,Xα
where Cκ = Fn(κ × ω, ω) and
˙̄f denotes the Cκ-name of the sequence of Cohen
reals (∈ ωω) of length κ added by Cκ. Thus, if G is a (V,Cκ)-generic set and cα is
the α’th element of ˙̄fG, then cα(n) = m if and only if there is a condition c ∈ G
such that 〈α, n〉 ∈ dom(c) and c(α, n) = m.
(6.9) Q
= {〈c, d〉 : c ∈ Cκ, d ∈
α∈κDXα ,
ξ∈dom(d) F
d(ξ) × dom(sd(ξ)) ⊆ dom(c)}
For 〈c, d〉, 〈c′, d′〉 ∈ Q
(6.10) 〈c′, d′〉 ≤
〈c, d〉 ⇔
c′ ≤Cκ c, dom(d
′) ⊇ dom(d),
∀α ∈ dom(d)
′(α) ⊇ sd(α) ∧ F d
′(α) ⊇ F d(α)∧
∀ξ ∈ F d(α) ∩Xα ∀n ∈ dom(s
d′(α)) \ dom(sd(α))
c′(ξ, n) ≤ sd
′(α)(n)
The following can be shown easily by standard arguments:
Lemma 6.2. Φ : Q
→ QX̄ ; 〈c, d〉 7→ 〈c, ď〉 is a dense embedding of Q
into QX̄ .
QX̄ and Q
are thus forcing equivalent.
For p ∈ Q
with p = 〈c, d〉, let
supp0(p) = {α < κ : 〈α, n〉 ∈ dom(c) for some n ∈ ω} and
supp1(p) = dom(d).
For a Q
-name ȧ, supp0(ȧ) and supp1(ȧ) are also defined in analogy to (4.4).
In Theorem 6.4, we assume CH + (A) and let, for a structure 〈(ω2)
2, A,F〉 as
in (A), κ = ω2 and X̄ = 〈Xα : α < ω2〉 where Xα = {β ∈ ω2 : 〈α, β〉 ∈ A} for
α < ω2. For such X̄ , the next lemma follows immediately from (6.3) and (6.4).
Lemma 6.3. Suppose that 〈(ω2)
2, A,F〉 and X̄ are as above. If 〈φ, ψ〉 ∈ F , then
the mapping j〈φ,ψ〉 : Q
defined by
j〈φ,ψ〉(〈c, d〉) = 〈c
′, d′〉
for 〈c, d〉 ∈ Q
where c′ and d′ are such that
dom(c′) = {〈φ(α), n〉 : 〈α, n〉 ∈ dom(c)};
c′(〈φ(α), n〉) = c(〈α, n〉) for 〈α, n〉 ∈ dom(c);
dom(d′) = ψ ′′ dom(d);
′(ψ(ξ)) = F d(ξ) and sd
′(ψ(ξ)) = sd(ξ) for ξ ∈ dom(d)
is an automorphism on the poset Q
Similarly to Section 4, we shall also denote with j〈φ,ψ〉 the corresponding map-
ping on Q
-names.
The following theorem together with the consistency result in Section 7 gives
the consistency of the conjunction of the assertions c = b∗ = ℵ2, do = ℵ1 and
¬Cs(ℵ2) over ZFC.
Theorem 6.4. Assume CH and (A). Let 〈(ω2)
2, A,F〉 be a structure satisfying
(6.1) ∼ (6.5) and let X̄ = 〈Xα : α < ω2〉 where Xα = {β ∈ ω2 : 〈α, β〉 ∈ A}.
Then ‖–
“ c = b∗ = ℵ2 ∧ do = ℵ1 ∧ ¬C
s(ℵ2) ”.
Proof. First, we show that ‖–
“ c = b∗ = ℵ2 ”. Let G be a (V,Q
)-generic filter.
Working in V [G], let f̄ = 〈cα : α < ω2〉 be the sequence of Cohen reals added by
the Cω2 part of QX̄ and dα be the Hechler type real added by Df̄ ,Xα for α < ω2.
By (6.1), {cα : α < γ} is bounded by dγ for all γ < ω2. On the other hand,
{cα : α < ω2} is unbounded by (6.2) and the c.c.c. of
Df̄ ,Xα (in V [f̄ ]). This
shows that V [G] |= ℵ2 ≤ b
∗. Since |Q
| = ℵ2 by CH, we have V [G] |= c ≤ ℵ2.
To show that Q
forces do = ℵ1, suppose that ḟα, α < ω2 are Q
-names of
elements of ωω, ϕ(x, y, z) a formula in LZF and ȧ a Q
-name of an element of ωω
such that
(6.11) ‖–
“H(ℵ1) |= ϕ(ḟα, ḟβ, ȧ) ” for all α < β < ω2.
By Maximal Principle, it is enough to show that there are η1 < η0 < ω2 such that
“H(ℵ1) |= ϕ(ḟη0 , ḟη1 , ȧ) ”.
For ξ < ω2, let
Aξ = supp1(ḟξ) ∪ supp1(ȧ) and
Bξ = supp0(ḟξ) ∪ supp0(ȧ).
By CH, ∆-System Lemma and (6.5), we can find a stationary S ⊆ Eω2ω1 such that
(6.12) Aξ, ξ ∈ S form a ∆-system such that its root is an initial segment of each
of Aξ, ξ ∈ S ; Bξ, ξ ∈ S form a ∆-system such that its root is an initial
segment of each of Bξ, ξ ∈ S ;
(6.13) for any distinct ζ0, ζ1, η0, η1 ∈ S, there is 〈φ, ψ〉 ∈ F such that
(6.13a) φ ′′Aζi = Aηi , ψ
′′Bζi = Bηi ; and
(6.13b) φ ↾ Aζi : Aζi → Aηi , ψ ↾ Bζi : Bζi → Bηi are order isomorphisms
for i ∈ 2;
(6.14) j〈φ,ψ〉(ḟζ) = ḟη for any distinct ζ , η ∈ S and 〈φ, ψ〉 ∈ F as in (6.13) with
ζ0 = ζ and η0 = η.
Note that, by (6.12) and (6.13b), we have
(6.15) j〈φ,ψ〉(ȧ) = ȧ
for any 〈φ, ψ〉 as in (6.13).
Now, let ζ0, ζ1, η0, η1 ∈ S be four distinct elements of S such that ζ0 < ζ1 and
η1 < η0. By (6.11), we have
“H(ℵ1) |= ϕ(ḟζ0 , ḟζ1, ȧ) ” .
Hence, by mapping this situation by j〈φ,ψ〉 for 〈φ, ψ〉 ∈ F as in (6.13) for these ζ0,
ζ1, η0, η1 , we obtain
“H(ℵ1) |= ϕ(ḟη0 , ḟη1 , ȧ) ” .
Thus, η0, η1 above are as desired.
Finally, we show that Q
forces the negation of Cs(ℵ2).
Let 〈r0n, s
n〉, n ∈ ω list all quadruples of finite sequences r
0, s0, r1, s1 ∈
ω>ω such that
(6.16) | r0 | = | s0 | = | r1 | = | s1 | and
(6.17) 〈r0, s0〉 6= 〈r1, s1〉 if | r0 | > 0.
We further assume that the enumeration 〈〈r0n, s
n〉 : n ∈ ω〉 is arranged so
(6.18) | r0n | ≤ n for all n ∈ ω.
Now, working in V [G], let aα, α < ω2 be the subsets of ω defined by
n ∈ aα ⇔ one of the following (6.19) and (6.20) holds:
(6.19) r0n ⊆ cα, s
n ⊆ dα+1,
cα(n) = 0, dα+1(n) = 1, cα(n+ 1) = 2 and dα+1(n+ 1) = 3 ;
(6.20) r1n ⊆ cα, s
n ⊆ dα+1,
cα(n) = 2, dα+1(n) = 3, cα(n+ 1) = 0 and dα+1(n+ 1) = 1 .
(6.21) aα,n = aα \ {k : | r
k | < n} for α < ω2 and n ∈ ω.
We show that the matrix 〈aα,n : α < ω2, n ∈ ω〉 together with T =
2ω is a
counter-example to Cs(ℵ2). For this, it is enough to prove the following:
Claim 6.4.1. If S0, S1 are cofinal subsets of ω2, then
(1) there exist n < ω, α ∈ S0 and β ∈ S1 such that aα,n ∩ aβ,n = ∅; and
(2) for any t ∈ 2ω, there are α ∈ S0 and β ∈ S1 such that aα,t(0) ∩ aβ,t(1) 6= ∅.
⊢ Working in the ground model, let Ṡ0 and Ṡ1 beQ†X̄ -names for the cofinal subsets
of ω2.
Let p ∈ Q
. For α < ω2, let pα ∈ Q
and γα, δα ∈ ω2 be such that
(6.22) γα < δα < γβ < δβ for all α < β < ω2 ;
(6.23) pα ≤Q†
p , pα = 〈c
α, dα〉 for all α < ω2 ; and
(6.24) pα ‖–Q†
“ γα ∈ Ṡ0 , δα ∈ Ṡ1 ”.
By ∆-System Lemma, we find a stationary U ⊆ Eω2ω1 and Aα, Bα ∈ [ω2]
<ℵ0 for
α ∈ U such that
(6.25) supp0(pα) ⊆ Aα, supp1(pα) ⊆ Bα ;
(6.26) Aα, α ∈ U form a ∆-system with root A; Bα, α ∈ U form a ∆-system
with root B;
(6.27) γα, γα + 1, δα, δα + 1 ∈ (Aα ∩Bα) \ (A ∪B) .
By thinning out U further, if necessary, we may also assume that there are some
k∗, n∗ ∈ ω such that
(6.28) dom(cα) = supp0(pα)× k
∗ and dom(sd
α(ξ)) = k∗ for all ξ ∈ supp1(pα) ;
(6.29) cα(γα, ·) = r
n∗ , s
dα(γα+1) = s0n∗ ;
(6.30) cα(δα, ·) = r
n∗ , s
dα(δα+1) = s1n∗ .
Without loss of generality, we may also assume that, for some fixed c∗, d∗,
(6.31) cα ↾ A× k∗ = c∗ and 〈sd
α(η) : η ∈ B〉 = d∗ for all α ∈ U .
Note that pα, α ∈ U are compatible by (6.25), (6.26) and (6.31).
Now, since Ṡ0, Ṡ1, p were arbitrary, Claim 6.4.1, (1) is proved by the following
subclaim:
Subclaim 6.4.1.1. For any α, β ∈ U with α < β, there is q ≤
p such that
“ γα ∈ Ṡ0, δβ ∈ Ṡ1, ȧγα,n∗ ∩ ȧδβ ,n∗ = ∅ ”.
⊢ Let q = 〈cq, dq〉 be the common extension of pα and pβ such that
(6.32) γα ∈ F
dq(δβ+1)
(6.33) dom(sd
q(ξ)) = k∗ for all ξ ∈ dom(dq).
Let G be a (V,Q
)-generic filter with q ∈ G. In V [G], we have
(6.34) cγα(m) ≤ dδβ+1(m) for all m ≥ k
by (6.28), (6.32) and (6.33).
Now, toward a contradiction, assume that aγα,n∗ ∩ aδβ ,n∗ 6= ∅ and let m ∈
aγα,n∗ ∩ aδβ ,n∗. By the definition of aα’s it follows that, for some i, j ∈ 2, we have
rim ⊆ cγα , s
m ⊆ dγα+1 ;
rjm ⊆ cδβ , s
m ⊆ dδβ+1.
On the other hand, since q ∈ G, we have pα, pβ ∈ G. It follows that
r0n∗ ⊆ cγα, s
n∗ ⊆ dγα+1 ;
r1n∗ ⊆ cδβ , s
n∗ ⊆ dδβ+1
by (6.29) and (6.30). By the definition (6.21) of aγα,n’s, we have | r
m | ≥ n
∗. Thus
we have, either
r0n∗ ⊆ r
m ⊆ cγα , s
n∗ ⊆ s
m ⊆ dγα+1 ;
r1n∗ ⊆ r
m ⊆ cγβ , s
n∗ ⊆ s
m ⊆ dγβ+1 ;
r0n∗ ⊆ r
m ⊆ cγα , s
n∗ ⊆ s
m ⊆ dγα+1 ;
r1n∗ ⊆ r
m ⊆ cγβ , s
n∗ ⊆ s
m ⊆ dγβ+1 .
In the first case, we must have cγα(m+ 1) = 2 and dδβ+1(m+ 1) = 1 by (6.19) and
(6.20). This is a contradiction to (6.34). Similarly, in the second case, we have
cγα(m) = 2 and dδβ+1(m) = 1. This is again a contradiction to (6.34).
⊣ (Subclaim 6.4.1.1)
(2) of Claim 6.4.1 follows from the next subclaim:
Subclaim 6.4.1.2. For any t ∈ 2ω and α, β ∈ U with α < β, there is q ≤
such that
“ γα ∈ Ṡ0, δβ ∈ Ṡ1, ȧγ(α),t(0) ∩ ȧδ(β),t(1) 6= ∅ ”.
⊢ For each ξ ∈ {α, β}, let p̃ξ ≤Q†
pξ with p̃ξ = 〈c̃
ξ, d̃ξ〉 and m ∈ ω be such that
(6.35) c̃ξ(γξ, ·) = r
m , s
d̃ξ(γξ+1) = s0m ; c̃
ξ(δξ, ·) = r
m , s
d̃ξ(δξ+1) = s1m ;
(6.36) | r0m | ≥ t(0), t(1) ;
(6.37) supp0(p̃ξ) = supp0(pξ) ; supp1(p̃ξ) = supp1(pξ) ;
(6.38) c̃ξ ↾ A× ω = cξ ↾ A× ω and 〈sd̃
ξ(η) : η ∈ B〉 = 〈sd
ξ(η) : η ∈ B〉.
Let q0 = 〈cq
〉 be the maximal (with respect to ≤
) common extension of p̃α
and p̃β which exists because of (6.37) and (6.38). Extend q
0 further to q = 〈cq, dq〉
such that
(6.39) | cq(γα, ·) | = | c
q(δβ , ·) | = | s
dq(γα+1) | = | sd
q(δβ+1) | = m+ 2 ;
(6.40) cq(γα, m) = 0, s
dq(γα+1)(m) = 1, cq(γα, m+ 1) = 2, s
dq(γα+1)(m+ 1) = 3 ;
(6.41) cq(δβ, m) = 2, s
dq(δβ+1)(m) = 3, cq(δβ, m+ 1) = 0, s
dq(δβ+1)(m+ 1) = 1 .
This is possible because γα 6∈ F
(δβ+1) and δβ 6∈ F
(γα+1) by the maximality of
q0 and (6.37).
By (6.35), (6.40), (6.41), by the definition (6.21) of aα,n’s, and since | r
m | ≥ t(0),
t(1), we have q ‖–
“m ∈ ȧγα,t(0) ∩ ȧδβ ,t(1) ”. Since q ≤Q†
pα, pβ, we also have
“ γα ∈ Ṡ0, δβ ∈ Ṡ1 ”.
Thus, q as above is as desired. ⊣ (Subclaim 6.4.1.2)
⊣ (Claim 6.4.1)
(Theorem 6.4)
Note that in the proof of ¬Cs(ℵ2) in Theorem 6.4, we used only (6.1) from the
assumption (A). Note also that this proof actually shows that in the generic exten-
sion the negation of C(ℵ2) from [13] holds which is a weakening of C
s(ℵ2) obtained
by replacing the condition “stationary” in the formulation of Cs(ℵ2) by “cofinal”.
7 Forcing CH + (A)
In this section, we define under CH a σ-closed ℵ2-c.c. poset P0 which forces the
combinatorial assertion (A) of the previous section.
The poset P0 is defined as follows:
p ∈ P0 ⇔ p = 〈X
p, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉
where
(7.1) Xp, Y p ∈ [ω2]
(7.2) Dp ∈ [ω2]
(7.3) for all ξ ∈ Dp, φ
: Xp → Xp and ψ
: Y p → Y p are involutions (that is,
bijections φ such that φ−1 = φ);
(7.4) for all ξ ∈ Dp, α ∈ Xp and β ∈ Y p,
(7.4a) φ
(α) < α + ξ + ω1 and
(7.4b) ψ
ξ (β) < β + ξ + ω1 ;
Note that we have also α < φ
(α) + ξ + ω1 and α < ψ
(β) + ξ + ω1 for all ξ ∈ D
α ∈ Xp and β ∈ Y p since φ
ξ and ψ
ξ are involutions by (7.3).
(7.5) τ p : Xp × Y p → 2 ;
(7.6) for all ξ ∈ Dp, α ∈ Xp and β ∈ Y p, we have τ p(α, β) = τ p(φ
(α), ψ
(β)) ;
(7.7) τ p(α, β) = 1 for all 〈α, β〉 ∈ Xp × Y p with β < α.
The ordering on P0 is defined by the following: For p, q ∈ P0 with
p = 〈Xp, Y p, τ p, 〈φ
ξ : ξ ∈ D
p〉〉 and
q = 〈Xq, Y q, τ q, 〈φ
: ξ ∈ Dq〉〉,
(7.8) p ≤P0 q ⇔
(7.8a) Xp ⊇ Xq, Y p ⊇ Y q ;
(7.8b) Dp ⊇ Dq ;
(7.8c) φ
and ψ
for all ξ ∈ Dq ;
(7.8d) τ p ⊇ τ q and
(7.8e) τ p ↾ (Xp \Xq)× Y q ≡ 1.
For p ∈ P0 with p = 〈X
p, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉, we intend to approximate
the characteristic function of the set A in the assertion (A) by τ p. More precisely,
in a generic extension V [G] for a (V,P0)-generic G, letting
(7.9) τ =
p∈G τ
p ; φξ =
p∈G φ
and ψξ =
p∈G ψ
for ξ ∈ ω2 ;
(7.10) A = τ−1 ′′{1} and F = {〈φξ, ψξ〉 : ξ ∈ ω2},
we are aiming to force 〈(ω2)
2, A,F〉 to satisfy (6.1) ∼ (6.5) in (A).
Of the conditions in the definition of P0, (7.5) and (7.8d) force τ to be a function.
Furthermore, τ : ω2×ω2 → 2 by density argument and the following Lemma 7.1, (a).
(7.3) and (7.8c) make φξ and ψξ mappings for all ξ ∈ ω2; they are forced to be
involutions on ω2 by (7.3) and the following Lemma 7.1, (a). Thus 〈(ω2)
2, A,F〉 is
forced to satisfy (6.3).
By (7.7) (and by the following Lemma 7.1, (a)), 〈(ω2)
2, A,F〉 is forced to satisfy
the second inclusion of (6.1).
By (7.6), 〈(ω2)
2, A,F〉 is forced to satisfy (6.4).
(7.4) and (7.8e) are technical conditions whose role will be clear later in the
course of the proof.
By the definition of P0, it is clear that P0 is σ-closed. Thus, we are done by
showing that P0 satisfies the ℵ2-c.c. and it forces that 〈(ω2)
2, A,F〉 as above satisfies
the conditions (6.2) and (6.5).
The next Lemma follows readily from the definition of P0.
Lemma 7.1. (a) For any α, β < ω2, the set
Dα,β = { p ∈ P0 : p = 〈X
p, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉,
α ∈ Xp and β ∈ Y p }
is dense in P0.
(b) For any C ∈ [ω2]
ℵ0 and any β < ω2,
EC,β = { p ∈ P0 : p = 〈X
p, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉,
C ⊆ Xp and for some δ ∈ Y p with δ ≥ β
τ p(γ, δ) = 0 for all γ ∈ C }
is dense in P0.
In the rest of the section, we are going to work mainly in the ground model
(where CH holds). Let τ̇ , φ̇ξ, ψ̇ξ for ξ ∈ ω2, Ȧ and Ḟ be P0-names of τ , φξ, ψξ for
ξ ∈ ω2, A and F as above, respectively.
Lemma 7.2. ‖– P0 “ 〈(ω2)
2, Ȧ, Ḟ〉 |= (6.2) ”.
Proof. By density argument with Lemma 7.1, (b). (Lemma 7.2)
For ξ < ω2, X , X
′, Y , Y ′ ∈ [ω2]
ℵ0 with X ′ ⊆ X and Y ′ ⊆ Y , τ : X × Y → 2
and involutions φ′ : X ′ → X ′, ψ : Y ′ → Y ′, let us call the quintuple 〈X, Y, τ, φ′, ψ′〉
a ξ-extendable semi-condition if
(7.11) φ′(α) < α + ξ + ω1 and ψ
′(β) < β + ξ + ω1 for all α ∈ X
′ and β ∈ Y ′ ;
(7.12) τ(α, β) = τ(φ(α), ψ(β)) for all α ∈ X ′ and β ∈ Y ′ ;
(7.13) τ(α, β) = 1 for all α ∈ X and β ∈ Y with β < α ;
(7.14) τ ↾ ((X \X ′)× Y ′) ≡ 1.
fig. 6
Note that the sets X ′ and Y ′, though not mentioned explicitly in the definition
of ξ-extendable semi-condition, can be recovered from φ′ and ψ′.
Note also that (7.14) holds vacuously if X = X ′. Hence, if p ∈ P0 with p =
〈Xp, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉, the quintuple 〈Xp, Y p, τ p, φξ, ψξ〉 is a ξ-extendable
semi-condition for all ξ ∈ Dp.
The following two lemmas explain the choice of the naming of ξ-extendable
semi-conditions.
Lemma 7.3. For any ξ < ω2, X, X
′, Y , Y ′ ∈ [ω2]
ℵ0 with X ′ ⊆ X and Y ′ ⊆ Y ,
τ : X × Y → 2 as well as involutions φ′ : X ′ → X ′, ψ′ : Y ′ → Y ′, if 〈X, Y, τ, φ′, ψ′〉
is a ξ-extendable semi-condition then there are X̃ ⊇ X, Ỹ ⊇ Y , τ̃ : X̃ × Ỹ → 2
with τ̃ ⊇ τ and involutions φ̃ : X̃ → X̃, ψ̃ : Ỹ → Ỹ extending φ′ and ψ′ respectively
such that 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 is a ξ-extendable semi-condition and
(7.15) τ̃ ↾ ((X̃ \X)× Y ) ≡ 1.
fig. 7
Proof. Let X0 ∈ [ω2 \X ]
≤ℵ0 be such that
(7.16) X0 is order-isomorphic to X \ X
′ and the order-isomorphism identifies
points of distance less than ω1 (that is, if α ∈ X \ X
′ and α0 ∈ X0 are
identified then we have α < α0 + ω1 and α0 < α+ ω1).
Since X (and hence also X \X ′) is countable, we can easily choose the elements of
X0 recursively in otp(X \ X
′) steps in accordance with (7.16). Put X̃ = X ∪̇ X0
and let φ̃ be the extension of φ which maps X0 order-isomorphically to X \X
′ and
vice versa. Fix θ < ω1 such that
(7.17) φ̃(α) ≤ α + ξ + θ for all α ∈ X̃ .
There is such θ by (7.11), (7.16) and since | X̃ | ≤ ℵ0.
Let Y0 ∈ [ω2 \ Y ]
≤ℵ0 be such that
(7.18) Y0 is order-isomorphic to Y \Y
′ and the order-isomorphism identifies points
of distance less than ξ + ω1; and
(7.19) if β ∈ Y \ Y ′ and β0 ∈ Y0 are identified then β0 > β + ξ + θ.
It is easy to see that the elements of such Y0 can be chosen recursively in otp(Y \Y
steps.
Now let Ỹ = Y ∪̇ Y0 and let ψ̃ be the extension of ψ
′ which maps Y0 order-
isomorphically to Y \ Y ′ and vice versa.
Finally define τ̃ : X̃ × Ỹ → 2 by
(7.20) τ̃(α, β) =
τ(α, β), if 〈α, β〉 ∈ X × Y
τ(φ̃(α), ψ̃(β)), if 〈α, β〉 ∈ (X ′ × Y0) ∪ (X0 × Y
′) ∪ (X0 × Y0)
1, otherwise
for every α ∈ X̃ and β ∈ Ỹ .
X0 = X̃ \X
fig. 8
〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 satisfies (7.11) by (7.17) and (7.18). It satisfies (7.12) by (7.20).
Thus we are done by checking 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 also satisfies (7.15) and (7.13).
For (7.15), suppose that 〈α, β〉 ∈ (X̃ \X)× Y (= X0 × Y ). If 〈α, β〉 ∈ X0 × Y
then τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) by (7.20). But 〈φ̃(α), ψ̃(β)〉 ∈ (X \ X ′) × Y ′ by
definition of φ̃ and ψ̃. Hence, by (7.14), we have τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) = 1. If
〈α, β〉 ∈ X0 × (Y \ Y
′) then τ̃(α, β) = 1 by the “otherwise” clause of (7.20).
For (7.13), it is enough to check that τ̃(α, β) = 1 for all 〈α, β〉 ∈ (X ′∪X0)×Y0
with β < α by (7.20) and (7.15). For such 〈α, β〉, we have τ̃ (α, β) = τ(φ̃(α), ψ̃(β))
by (7.20). Suppose that τ̃ (α, β) = 0. Then, since τ satisfies (7.13), we should have
φ̃(α) ≤ ψ̃(β). By (7.19), we have β > ψ̃(β) + ξ + θ. On the other hand, by (7.17),
we have α = φ̃2(α) ≤ φ̃(α) + ξ + θ. It follows that
α ≤ φ̃(α) + ξ + θ ≤ ψ̃(β) + ξ + θ < β.
This is a contradiction. (Lemma 7.3)
A quartet p = 〈Xp, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉 (not necessarily an element of
P0) with D
p ∈ [ω2]
ℵ0 is said to be an extendable condition if 〈Xp, Y p, τ p, φ
ξ 〉 is
a ξ-extendable semi-condition for all ξ ∈ Dp.
For extendable conditions p, q with
p = 〈Xp, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉, q = 〈Xq, Y q, τ q, 〈φ
: ξ ∈ Dq〉〉,
we denote p ≤1 q if
(7.21) Xp ⊇ Xq, Y p ⊇ Y q, τ p ⊇ τ q, Dp ⊇ Dq, φ
and ψ
for all
ξ ∈ Dp ;
(7.22) τ p ↾ (Xp \Xq)× Y q ≡ 1.
Note that for p, q ∈ P0, we have p ≤1 q if and only if p ≤P0 q.
Lemma 7.4. (Extension Lemma) Suppose that
p = 〈Xp, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉
is an extendable condition for some Dp ∈ [ω2]
ℵ0. Then there is a q ∈ P0 with
q = 〈Xq, Y q, τ q, 〈φ
ξ : ξ ∈ D
q〉〉 such that Dq = Dp and q ≤1 p.
Furthermore , if p0 ∈ P0 is such that p ≤1 p0 then we have q ≤P0 p0.
Proof. The second part of the lemma is clear once the condition q as in the claim
of the lemma is found since (7.8e) holds for such q and p0 since the relation ≤1 is
easily seen to be transitive.
To construct the desired q ∈ P0, let 〈ξn : n ∈ ω〉 be an enumeration of D
p such
that each ξ ∈ Dp appears infinitely often in the enumeration.
First, construct 〈Xn, Yn, τn, 〈φξ,n, ψξ,n : ξ ∈ D
p〉〉, n ∈ ω recursively such that
(7.23) 〈X0, Y0, τ0, 〈φξ,0, ψξ,0 : ξ ∈ D
p〉〉 = p,
(7.24) 〈Xn+1, Yn+1, τn+1, φξn,n+1, ψξn,n+1〉 is the ξn-extendable semi-condition which
is constructed just as in Lemma 7.3 from the ξn-extendable semi-condition
〈Xn, Yn, τn, φξn,n, ψξn,n〉.
(7.25) φξ,n+1 = φξ,n and ψξ,n+1 = ψξ,n for all ξ ∈ D
p with ξ 6= ξn.
Along with the recursive construction above, it can be shown easily that 〈Xn, Yn, τn,
φξ,n, ψξ,n〉 is a ξ-extendable semi-condition for all n ∈ ω and ξ ∈ D
p. Hence the
construction in (7.24) is actually possible at each step.
n∈ωXn , Y
n∈ωXn , τ
n∈ω τn ,
Dq = Dp and φ
n∈ω φξ,n , ψ
n∈ω ψξ,n for all ξ ∈ D
For all ξ ∈ Dq, there are infinitely many n ∈ ω such that ξn = ξ. For such n, φξ,n
is an involution on Xn and ψξ,n is an involution on Yn. It follows that φ
ξ is an
involution on Xq and ψ
is an involution on Y q. Hence
q = 〈Xq, Y q, τ q, 〈φ
: ξ ∈ Dq〉〉
is a condition in P0. Also we have
τ q ↾ (Xq \Xp)× Y p =
n∈ω τ
q ↾ (Xn+1 \Xn)× Y
p ≡ 1.
Thus, this q is as desired. (Lemma 7.4)
Lemma 7.5. (CH) P0 satisfies the ℵ2-c.c.
Proof. Actually we shall show that P0 satisfies a strong form of ℵ2-Knaster prop-
erty.
Suppose that pζ ∈ P0 with p
ζ = 〈Xζ, Y ζ , τ ζ , 〈φ
: ξ ∈ Dζ〉〉 for ζ ∈ ω2. By
the ∆-System Lemma (Theorem 4.1) and the Pigeon Hole Principle, there are a
stationary S ⊆ ω2, X , Y , D ∈ [ω2]
ℵ0 , τ : X ×Y → 2 and φξ : X → X , ψξ : Y → Y
for ξ ∈ D such that
(7.26) Xζ, ζ ∈ S form a ∆-system with root X and Y ζ , ζ ∈ S form a ∆-system
with root Y ;
(7.27) τ ζ ↾ X × Y = τ for all ζ ∈ S ;
(7.28) Dζ, ζ ∈ S form a ∆-system with root D ;
(7.29) φ
↾ X = φξ and ψ
↾ Y = ψξ for all ζ ∈ S and ξ ∈ D ;
(7.30) τ ζ ↾ (Xζ \X)× Y ≡ 1 for all ζ ∈ S .
Note that (7.27) is possible since, by CH, there are at most |X×Y2 | ≤ 2ℵ0 = ℵ1 < ℵ2
many possible values of τ ζ ↾ X × Y . (7.29) is possible since, by CH, (7.4) and
countability of D, there are at most ℵ1 possible values of 〈φ
: ξ ∈ D〉 and
: ξ ∈ D〉. (7.30) is possible by (7.7) and since we can choose S such that
min(Xζ \X) > sup(Y ) for all ζ ∈ S.
Now suppose ζ , η ∈ S with ζ < η. We show that pζ and pη are compatible. Let
Xp = Xζ ∪Xη, Y p = Y ζ ∪ Y η and Dp = Dζ ∪Dη. For ξ ∈ Dp, let φ
ξ : X
p → Xp
and ψ
: Y p → Y p be defined by
(7.31) φ
, if ξ ∈ Dζ \D,
, if ξ ∈ D,
ξ , otherwise
and ψ
, if ξ ∈ Dζ \D,
, if ξ ∈ D,
ξ , otherwise.
Finally let τ p : Xp × Y p → 2 be such that
(7.32) τ p(α, β) =
τ ζ(α, β), if 〈α, β〉 ∈ Xζ × Y ζ
τ η(α, β), else if 〈α, β〉 ∈ Xη × Y η
1, otherwise.
for all 〈α, β〉 ∈ Xp × Y p.
It is easy to see that p = 〈Xp, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉 is an extendable
condition and p ≤1 p
ζ, pη. In particular, (7.22) for p ≤1 p
ζ and p ≤1 p
η holds
because of (7.30) and “otherwise” clause of (7.32).
By Extension Lemma (Lemma 7.4), there is a q ∈ P0 with q ≤1 p. Hence, by
the second half of the lemma, it follows that q ≤P0 p
ζ, pη. (Lemma 7.5)
A modification of the ∆-system argument in the proof of Lemma 7.5 is also
used to prove the following:
Lemma 7.6. (CH) P0 forces (6.5).
Proof. We show that P0 forces the following:
(7.33) For any stationary S ⊆ Eω2ω1 and Aζ , Bζ ∈ [ω2]
ℵ0 for ζ ∈ S, there is a
stationary T ⊆ S such that for any n ∈ ω and pairwise distinct ζi, ηi ∈ T ,
i ∈ n, there is ξ < ω2 such that φ̇ξ
′′Aζi = Aηi and ψ̇ξ
′′Bζi = Bηi for all
i ∈ n.
Note that, by σ-closedness and ℵ2-c.c. of P0 (proved in Lemma 7.5), ω1 and ω2
in generic extensions by P0 remain ω1 and ω2.
Suppose that Ṡ is a P0-name of a stationary subset of E
. Let 〈Ȧζ : ζ ∈ Ṡ〉
and 〈Ḃζ : ζ ∈ Ṡ〉 be P0-names of sequences of countable subsets of ω2. Let
S̃ = {ζ ∈ Eω2ω1 : /‖–P0 “ ζ 6∈ Ṡ ”}.
Then we have ‖–P0 “ Ṡ ⊆ S̃ ” and hence S̃ is a stationary subset of E
Since P0 is σ-closed, we can find pζ ∈ P0 and Aζ, Bζ ∈ [ω2]
ℵ0 such that
(7.34) pζ = 〈X
ζ, Y ζ , τ ζ , 〈φ
: ξ ∈ Dζ〉〉 and
(7.35) pζ ‖– P0 “ ζ ∈ Ṡ , Ȧζ = Aζ and Ḃζ = Bζ ”
for all ζ ∈ S̃.
Without loss of generality, we may assume that
(7.36) Aζ ⊆ X
ζ and Bζ ⊆ Y
By ∆-System Lemma (Theorem 4.1) and the Pigeon Hole Principle, there are a
stationary S̃0 ⊆ S̃, X , Y , D ∈ [ω2]
ℵ0 , τ : X × Y → 2 and φξ : X → X , ψξ : Y → Y
for ξ ∈ D such that
(7.37) Xζ, ζ ∈ S̃0 form a ∆-system with root X and Y
ζ, ζ ∈ S̃0 form a ∆-system
with root Y ;
(7.38) sup(Y ) < min(Xζ \X) for all ζ ∈ S̃0 ;
(7.39) τ ζ ↾ X × Y = τ for all ζ ∈ S̃0 ;
(7.40) Dζ, ζ ∈ S̃0 form a ∆-system with root D ;
(7.41) φ
↾ X = φξ and ψ
↾ Y = ψξ for all ζ ∈ S̃0 and ξ ∈ D ;
(7.42) τ ζ ↾ (Xζ \X)× Y ≡ 1 for all ζ ∈ S̃0 (this follows from (7.38) and (7.7)) ;
(7.43) Xζ, ζ ∈ S̃0 are order-isomorphic and Y
ζ , ζ ∈ S̃0 are order-isomorphic;
Note that the order-isomorphisms of Xζ ’s and Y ζ ’s do not move elements of X and
Y , respectively.
(7.44) the order-isomorphism sending Xζ to Xη sends τ ζ ↾ ((Xζ \ X) × Y ) to
τ η ↾ ((Xη \X)× Y ) while the order-isomorphism sending Y ζ to Y η sends
τ ζ ↾ (X × (Y ζ \ Y )) to τ η ↾ (X × (Y η \ Y )). These order-isomorphisms
together send τ ζ ↾ ((Xζ \X)× (Y ζ \ Y )) to τ η ↾ ((Xη \X)× (Y η \ Y ));
(7.45) the order-isomorphism sending Xζ to Xη sends Aζ to Aη, and the order-
isomorphism sending Y ζ to Y η sends Bζ to Bη.
Note that p̄ = 〈X, Y , τ , 〈φξ, ψξ : ξ ∈ D〉〉 is a condition in P0 and pζ ≤P0 p̄ for all
ζ ∈ S̃0 (the condition (7.8e) for p̄ and pζ holds by (7.42)).
Let Ṫ be a P0-name such that
(7.46) ‖– P0 “ Ṫ = {ζ ∈ S̃0 : pζ ∈ Ġ } ”
where Ġ is the standard P0-name of the generic set.
Claim 7.6.1. p̄ ‖–P0 “ Ṫ is a stationary subset of ω2 ”.
⊢ Since P0 satisfies the ℵ2-c.c. by Lemma 7.5, for any P0-name Ċ of a club subset
of ω2, there is a club subset C of ω2 (in the ground model) such that ‖–P0 “C ⊆ Ċ ”.
Hence it is enough to show the following:
(7.47) For any q ≤P0 p̄ and any club subset C of ω2, there are p ≤P0 q and
ζ ∈ C ∩ S̃0 such that p ≤P0 pζ .
To show (7.47), let q = 〈Xq, Y q, τ q, 〈φ
ξ : ξ ∈ D
q〉〉 and let ζ ∈ C ∩ S̃0 be such
(7.48) (Xζ \X) ∩Xq = ∅ , (Y ζ \ Y ) ∩ Y q = ∅ and (Dζ \D) ∩Dq = ∅.
This is possible by (7.37) and since C ∩ S̃0 is stationary.
Let X∗ = Xq ∪Xζ, Y ∗ = Y q ∪ Y ζ , D∗ = Dq ∪Dζ. For ξ ∈ D∗, let φ∗ξ and ψ
be partial functions from X∗ to X∗ and from Y ∗ to Y ∗ respectively defined by
(7.49) φ∗ξ =
, if ξ ∈ Dq \D,
, if ξ ∈ D,
, otherwise
and ψ∗ξ =
ψq, if ξ ∈ Dq \D,
, if ξ ∈ D,
, otherwise.
Finally, let τ ∗ : X∗ × Y ∗ → 2 be defined by
(7.50) τ ∗(α, β) =
τ q(α, β), if 〈α, β〉 ∈ Xq × Y q,
τ ζ(α, β), else if 〈α, β〉 ∈ Xζ × Y ζ ,
1, otherwise
for 〈α, β〉 ∈ X∗ × Y ∗.
Then p∗ = 〈X∗, Y ∗, τ ∗, 〈φ∗ξ, ψ
ξ : ξ ∈ D
∗〉〉 is an extendable condition and we
have p∗ ≤1 q, pζ : (7.22) for p
∗ and pζ follows from
τ ∗ ↾ ((X∗\Xζ)×Y ζ) = τ ∗ ↾ ((Xq \X)×(Y ζ \Y ))∪τ ∗ ↾ ((Xq \X)×Y ) ≡ 1
where we have τ ∗ ↾ ((Xq \X) × (Y ζ \ Y )) ≡ 1 by the definition (7.50) of τ ∗ and
τ ∗ ↾ ((Xq \X) × Y ) ≡ 1 by q ≤P0 p̄ (in particular, by the condition (7.8e) in the
definition of ≤P0).
By Extension Lemma (Lemma 7.4) it follows that there is p ∈ P0 with p ≤1 p
and hence p ≤P0 q, pζ . ⊣ (Claim 7.6.1)
Claim 7.6.2. p̄ forces that Ṫ is as in (7.33) for 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉.
⊢ By Claim 7.6.1 it is enough to prove the following:
(7.51) For any q ≤P0 p̄ and n ∈ ω, if ζi, ηi ∈ S̃0 are pairwise distinct and
q ‖– P0 “ ζi, ηi ∈ Ṫ for i ∈ n ”, then there is p ≤P0 q with
p = 〈Xp, Y p, τ p, 〈φ
: ξ ∈ Dp〉〉
and ξ0 ∈ D
p such that φ
↾ Xζi : Xζi → Xηi and ψ
↾ Y ζi : Y ζi → Y ηi are
order-isomorphisms for all i < n.
Without loss of generality we may assume that
(7.52) q ≤P0 pζi, pηi for all i < n.
q = 〈Xq, Y q, τ q, 〈φ
ξ : ξ ∈ D
Take ξ0 ∈ ω2 \ (D
q ∪ sup(Xq) ∪ sup(Y q)) and let D∗ = Dq ∪ {ξ0}. Let X
∗ = Xq,
Y ∗ = Y q and τ ∗ = τ q. Let φ∗ξ0 :
Xζi ∪
Xηi →
Xζi ∪
Xηi be
the involution sending Xζi order-isomorphically to Xηi and vice versa for all i < n
and ψ∗ξ0 :
Y ζi ∪
Y ηi →
Y ζi ∪
Y ηi be the involution sending Y ζi
order-isomorphically to Y ηi and vice versa for all i < n. Let φ∗ξ = φ
and ψ∗ξ = ψ
for ξ ∈ Dq.
Then p∗ = 〈X∗, Y ∗, τ ∗, 〈φ∗ξ, ψ
ξ : ξ ∈ D
∗〉〉 is an extendable condition with
p∗ ≤1 q. To see this, we have to check 〈X
∗, Y ∗, τ ∗, φ∗ξ0, ψ
〉 satisfies (7.12) and
(7.14). But this follows from (7.42), (7.52) and (7.45).
By Extension Lemma (Lemma 7.4) there is p ∈ P0 with p ≤1 p
∗. Clearly p forces
that ξ0 as above satisfies (7.33) together with Ṫ , 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉.
⊣ (Claim 7.6.2)
Since the argument above can be repeated below arbitrary element of P0, it
follows that P0 forces (7.33). (Lemma 7.6)
8 A summary of consistency results and some
open problems
The following is a summary of consistency results in connection with the combina-
torial principles in fig. 5 where (1) ∼ (7) below correspond to the separation lines
(1) ∼ (7) drawn in fig. 9.
(1): By adding random reals. More precisely, start from a model V of CH and
force with (the positive elements of) the measure algebra B of, say, Maharam type
ℵ2. B can be seen as a (measure theoretic) product of random forcing and inherits
thus some of the homogeneity property of finite support product. This is used to
prove do = ℵ1 in the generic extension. It is also well-known that the ground model
functions from ω to ω dominate the functions in a generic extension by a measure
algebra. Hence we have d = ℵ1 in the model. K. Kunen proved that there is a
κ-Lusin gap for an uncountable κ in such a model. On the other hand, I. Juhász,
L. Soukup and Z. Szentmiklóssy proved in [13] that there is no ℵ2-Lusin gap under
Cs(ℵ2). This proves that C
s(ℵ2) does not hold in the generic extension.
This observation may be also interpreted as pinning down of the difference in
the extent of homogeneity of product forcing and the forcing by measure theoretic
products in terms of whether the principle Cs(ℵ2) holds.
(2): A model constructed by J. Brendle and T. LaBerge in [1] realizes this
separation.
(3): By the model in Theorem 3.8 of I. Juhász and K. Kunen [12] in which
Cs(ℵ2) and do > ℵ1 hold. The model is obtained by a finite support product of ℵ2
posets of cardinality ℵ1 starting from a model of CH. From this, it follows easily
that b∗ = ℵ1 and d = ℵ2.
(4): By adding Cohen reals. More exactly, start from a model V of CH and
then add, say, ℵ2 Cohen reals (by Fn(ℵ2, 2)). Then by Corollary 4.8, (c) we have
IP(ℵ2,ℵ1) in the generic extension. Just as in (3), we have d = ℵ2 in such a generic
extension and it is shown in S. Fuchino, S. Koppelberg and S. Shelah [8] that WFN
holds there.
(5): By a model of Hechler.
(6): By Theorem 6.4.
(7): By Corollary 4.9.
(8): By Theorem 5.2. See [11] for the proof of ‖– Sκ “¬WFN”.
do = ℵ1
HP(ℵ2)
h = ℵ1b
↑ = ℵ1b = ℵ1 b
∗ = ℵ1
Cs(ℵ2)
d = ℵ1
(4)(5)
IP(ℵ2,ℵ2)
IP(ℵ2,ℵ1)
fig. 9
Finally, we shall mention some open problems.
In [5] it is shown that a = ℵ1 follows from WFN where a is the almost disjoint
number. In [4], it is then shown that, under some additional assumptions, a = ℵ1
already follows from SEP which is a weakening of WFN. Therefore, it seems natural
to ask the following question:
Problem 1. Does a = ℵ1 follow from HP(ℵ2) or IP(ℵ2, λ) for λ = ℵ1,ℵ2 ?
Problem 2. Does WFN imply HP(ℵ2) or do = ℵ1 ?
The model of b∗ = ℵ2 and do = ℵ1 satisfies a strong form of negation of C
s(ℵ2).
This suggests the following problem:
Problem 3. Does HP(ℵ2) (or even C
s(ℵ2)) imply b
∗ = ℵ1 ?
At the moment, we do not have any model separating HP(κ) and IP(κ, κ) for
κ > κ1.
Problem 4. Is HP(κ) + ¬IP(κ, κ) consistent for some (or any) κ > ℵ1 ?
In Corollary 4.9 which realizes the separation (7) in fig.9, a very strong large
cardinal property is assumed.
Problem 5. Can we construct a model realizing (7) in fig.9 starting from ZFC
without any large cardinal?
The property (A) used in the proof of Theorem 6.4 and proved to be consistent
with CH in Section 7 seems to be of its own interest.
Problem 6. Is ¬(A) consistent with ZFC + CH (or even with ZFC + GCH) ?
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Authors’ addresses
Jörg Brendle
Graduate School of Science and Technology
Kobe University
Rokko-dai 1-1,
Nada KOBE 657-8501 Japan
brendle@kurt.scitec.kobe-u.ac.jp
Sakaé Fuchino
Dept. of Natural Science and Mathematics
College of Engineering, Chubu University,
Kasugai AICHI 487-8501 Japan.
fuchino@isc.chubu.ac.jp
