WEAK PREDICTION PRINCIPLES
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
Abstract. We prove the consistency of the failure of the weak diamond
Φλ at strongly inaccessible cardinals. On the other hand we show that
the very weak diamond Ψλ is equivalent to the statement 2<λ < 2λ and
hence holds at every strongly inaccessible cardinal.
2010 Mathematics Subject Classification. 03E05.
Key words and phrases. Weak diamond, very weak diamond, Radin forcing.
1
2
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
0. introduction
The prediction principle ♦λ (diamond on λ) was discovered by Jensen,
[5], who proved that it holds over any regular cardinal λ in the constructible
universe. This principle says that there exists a sequence hAα : α < λi of
sets, Aα ⊆ α for every α < λ, such that for every A ⊆ λ the set {α < λ :
A ∩ α = Aα } is stationary.
In the ancient era, the main focus was the case of λ = ℵ1 . It is immediate
that ♦ℵ1 ⇒ 2ℵ0 = ℵ1 , but consistent that 2ℵ0 = ℵ1 along with ¬♦ℵ1 . Motivated by algebraic constructions, Devlin and Shelah [2] introduced a weak
form of the diamond principle which follows from the continuum hypothesis:
Definition 0.1. The Devlin-Shelah weak diamond.
Let λ be a regular uncountable cardinal.
The weak diamond on λ (denoted by Φλ ) is the following principle:
For every function c : <λ 2 → 2 there exists a function g ∈ λ 2 such that
{α ∈ λ : c(f α) = g(α)} is a stationary subset of λ whenever f ∈ λ 2.
The idea is that we replace the prediction of the initial segments of a set
(or a function) by predicting only their color. The function c is a coloring,
and the function g is the weak diamond function which gives stationarily
many guesses for the c-color of the initial segments of every function f . It
is easy to see that the real diamond implies the weak diamond.
Concerning cardinal arithmetic, not only that Φℵ1 follows from the continuum hypothesis, but actually 2ℵ0 < 2ℵ1 implies Φℵ1 as proved in [2]. On
the other hand, Φℵ1 implies 2ℵ0 < 2ℵ1 (as noted by Uri Abraham) so both
assertions are equivalent.
The diamond and the weak diamond are prediction principles, while 2ℵ0 <
2ℵ1 and 2ℵ0 = ℵ1 belong to cardinal arithmetic. The above statements show
that a simple connection may exist. Actually, if λ > ℵ1 and λ = κ+ then
the situation becomes even simpler. For the diamond, 2κ = κ+ ⇔ ♦κ+
+
(see [8]), and for the weak diamond 2κ < 2κ ⇔ Φκ+ (a straightforward
generalization of [2]). The negation of ♦ℵ1 with the affirmation of 2ℵ0 = ℵ1
has been recognized as a peculiarity of the first uncountable cardinal.
However, the situation is totally different if λ = cf(λ) is a limit cardinal.
One direction is still easy and has a general nature. If ♦λ then 2<λ = λ and
if Φλ then 2<λ < 2λ (see Claim 1.1 below). Notice that these formulations
coincide with the above description of the successor case λ = κ+ . But can
we prove the opposite implication?
If λ = cf(λ) is a limit cardinal then λ is a large cardinal (in the philosophical sense; its existence cannot be established in ZFC). Now if λ is large
enough then ♦λ holds. Measurability suffices, and actually much less. If λ
is ineffable (a property which may live happily with V = L) or even subtle
then ♦λ and hence also Φλ .
A striking (and unpublished) result of Woodin shows that small large
cardinals manifest much more interesting demeanor. Woodin proved the
WEAK PREDICTION PRINCIPLES
3
consistency of a strongly inaccessible cardinal λ for which ♦λ fails. Moreover,
the construction can be strengthened to strongly Mahlo. In the current
paper we prove the same assertion, upon replacing the diamond by the
weak diamond. For both the diamond and the weak diamond we do not
know what happens if λ is weakly compact.
We conclude that the cardinal arithmetic assumption 2<λ < 2λ is strictly
weaker than the prediction principle Φλ . It is still tempting to look for
a prediction principle which is characterized by 2<λ < 2λ . We define the
following:
Definition 0.2. The very weak diamond.
Let λ be an uncountable cardinal.
The very weak diamond on λ (denoted by Ψλ ) is the following principle:
For every function c : <λ 2 → 2 there exists a function g ∈ λ 2 such that
{α ∈ λ : c(f α) = g(α)} is an unbounded subset of λ whenever f ∈ λ 2.
As we shall see, Ψλ ⇔ 2<λ < 2λ whenever λ = cf(λ) > ℵ0 . Let us mention
Galvin’s property which follows from 2<λ < 2λ but consistent with 2<λ =
2λ (see [3]). Clumping all these principles together we have a systematic
hierarchy of weak prediction principles, each of which implied by the stronger
one but strictly weaker from the next stage.
The basic tool for proving ¬Φλ over small large cardinals is Radin forcing,
[6]. Since there are several ways to introduce this forcing notion we indicate
that our approach is taken from [4], and in particular we use the Jerusalem
notation i.e. p ≤R q means that q is stronger than p.
4
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
1. Weak diamond and Radin forcing
We commence with the easy direction about the connection between Φλ
and cardinal arithmetic. Actually, the claim below applies to Ψλ as well. A
parallel assertion can be proved easily for ♦λ .
Claim 1.1. The basic claim.
Let λ be an uncountable cardinal.
If Φλ (or even Ψλ ) holds then 2<λ < 2λ .
Proof.
Assume towards contradiction that Φλ holds and 2<λ = 2λ . Let b : 2<λ → 2λ
be a surjection, such that for every α < λ and every t ∈ α 2, if there is an
ordinal δ < α such that ε ∈ [δ, α) ⇒ t(ε) = 0 then b(t) = b(t δ). We are
trying to describe a coloring F : <λ 2 → 2, which exemplifies the failure of
the weak diamond.
Assume α < λ and η ∈ α 2. Set F (η) = [b(η)](α). By Φλ we can choose a
function g ∈ λ 2 which predicts F . Let h ∈ λ 2 be the opposite function, i.e.
h(α) = 1 − g(α) for every α < λ. Let t ∈ <λ 2 be any mapping for which
b(t) = h. Let δ < λ be such that t ∈ δ 2. We define f ∈ λ 2 as an extension
of t as follows. If α < δ then f (α) = t(α) and if α ≥ δ then f (α) = 0.
Observe that b(f α) = b(f δ) for every α ∈ (δ, λ). Consequently,
F (f α) = [b(f α)](α) = [b(f δ)](α) = [b(t)](α) = h(α) 6= g(α), so g fails
to predict F (f α) on an end-segment of λ, a contradiction.
Notice that the above argument shows that even the very weak diamond
Ψλ implies 2<λ < 2λ , so the proof is accomplished.
1.1
It follows from the above claim that the weak diamond and the very weak
diamond are equivalent in the successor case. We comment that the same
holds for the common diamond over successor cardinals, if one replaces the
requirement of stationary set of guesses by an unbounded set.
Corollary 1.2. Φλ and Ψλ .
If λ = κ+ then Φλ ⇔ Ψλ . The same holds true if λ is weakly inaccessible
and 2<λ = 2κ for some κ < λ.
Proof.
The implication Φλ ⇒ Ψλ results from the definition. For the opposite
dierction, if Ψλ then 2<λ < 2λ by the above claim, and hence Φλ by [2].
1.2
Claim 1.1 gives one direction. As we shall see in Theorem 1.6 below,
the other direction cannot be proved. Indeed, every strongly inaccessible
cardinal satisfies 2<λ < 2λ , but the negation of Φλ can be forced over some
strongly inaccessible cardinal. However, we can prove that 2<λ < 2λ is
equivalent to the very weak diamond Ψλ . We shall use the fact that if λ is
weakly inaccessible and 2<λ = 2κ < 2λ for some κ < λ then Φλ . This fact
appears already in [2], without explicit proof. Since it plays a key-role in
the theorem below, we spell out the proof:
WEAK PREDICTION PRINCIPLES
5
Claim 1.3. Weak diamond out of nowhere.
If λ is weakly inaccessible, λ < 2<λ < 2λ and 2κ = 2<λ for some κ < λ,
then Φλ .
Proof.
Assume towards contradiction that ¬(Φλ ), and choose a coloring F : <λ 2 →
2 which exemplifies it. For every g ∈ λ 2 we can find f ∈ λ 2 so that {α <
λ : F (f α) = g(α)} ∈ Dλ . Indeed, given any g ∈ λ 2 we choose f ∈ λ 2
for which N = {α < λ : F (f α) = 1 − g(α)} is not a stationary subset of
λ. This can be done by the assumption ¬(Φλ ) with respect to the function
1 − g. However, in this case λ \ N ∈ Dλ and for every α ∈ λ \ N we have
F (f α) 6= 1 − g(α) and hence F (f α) = g(α).
Let S be the collection of all the sequences of the form (α, . . . , gν , fν , . . .)ν<β
when α, β < λ and gν , fν ∈ α 2 for every ν < β. Observe that |S| = 2<λ ,
which equals 2κ by the assumption of the theorem. Let h be a one-to-one
mapping between S and κ 2.
Assume g ∈ λ 2 is any function. We describe a process of defining κ · κmany functions gν , fν ∈ λ 2 such that every pair of functions is a pair of
many guesses. We also choose a club subset of λ which is contained in the
set of these guesses. This is done by induction on η < κ, when we choose in
the η-th stage κ · η many functions and one additional pair of functions at
η = 0.
For η = 0 let g0 = g, f0 = f for some f which is guessed by gTon a club
Cζ . For
C. We set C0 = C. If η is a limit ordinal then we let Cη =
ζ<η
the functions, we just take all the functions that were defined in previous
stages. The interesting case is η + 1, assuming that the stage of η has been
accomplished.
For each α < λ let βα,η be the first member of Cη greater than α. We
define simultaneously the sequence of functions hgκ·η+δ : δ < κi. For this
end, we have to determine the value of all these functions for every α < λ,
so hgκ·η+δ (α) : δ < κi = h(βα,η , . . . , gν βα,η , fν βα,η , . . .)ν<κ·η .
Having the functions gκ·η+δ at hand for every δ < κ, we choose for each
one of them a function fκ·η+δ ∈ λ 2 such that Aκ·η+δ = {α < λ : F (fκ·η+δ α) = gκ·η+δ
T (α)} ∈ Dλ . Finally, we choose a club Cη+1 of λ such that
Cη+1 ⊆
Aκ·η+δ ∩ Cη . The above process can be rendered for every
δ<κ
function g ∈ λ 2. Let hgν : ν < κ · κi enumerate all the functions derived
from g. We shall add g as a superscript to all the functions and club subsets
which emerge out of g, e.g. gνg and so forth.
We define an equivalence relation E on the members of λ 2 by gEg 0 iff:
T
T
0
(a) γ = min( {Cηg : η < κ}) = min( {Cηg : η < κ}).
0
0
(b) gνg γ = gνg γ, fνg γ = fνg γ for every ν < κ · κ.
The number of equivalence classes of E is bounded by λ · 2<λ = 2<λ < 2λ .
Consequently, we can choose two functions g, g 0 ∈ λ 2 such that g 6= g 0 and
6
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
gEg 0 . For the rest of the proof we focus on these two functions, hence
0
we write Cη instead of Cηg and Cη0 instead of CηgT. The same convention
is applied
to the derived functions. Let C be {Cη : η < κ} and let
T
C 0 = {Cη0 : η < κ}. Enumerate the members of C in an increasing order
by {γξ : ξ < λ}. Similarly, {γξ0 : ξ < λ} would be the increasing enumeration
of C 0 . Notice that γ0 = γ00 = γ.
We try to prove by induction on ξ < λ that γξ = γξ0 (hence C = C 0 ) and
that all the functions coincide (i.e., requirement (b) holds for every γξ ). If
we succeed, the proof will be accomplished. Indeed, considering ν = 0 we
shall have g γξ = g0 γξ = g00 γξ = g 0 γξ for every ξ < λ and hence
g = g 0 , a contradiction.
The case ξ = 0 is actually the content of the definition of E, as γ0 = γ.
Part (b) of the definition of E is responsible to the equality of the functions
restricted to γ, so we can start the induction. The case of a limit ordinal
ξ is also simple. By the induction hypothesis, γζ = γζ0 for every ζ < ξ.
S
S 0
Now γξ =
γζ =
γζ = γξ0 (recall that both C, C 0 are closed). For the
ζ<ξ
ζ<ξ
S
functions, if ν < λ then gν γξ =
gν γζ , and the same holds for gν0 , so
ζ<ξ
the induction hypothesis applies. A similar argument shows that the fν -s
coincide.
The remaining case is ξ + 1. For every α < λ and every η < κ let sα,η be
the sequence (βα,η , . . . , gν βα,η , fν βα,η , . . .)ν<κ·η . We define s0α,η in the
same manner, with respect to gν0 , fν0 . By the induction hypothesis:
h(sγξ ,η ) = hgκ·η+δ (γξ ) : δ < κi =
0
γξ ) : δ < κi
hF (fκ·η+δ γξ ) : δ < κi = hF (fκ·η+δ
0
(γξ ) : δ < κi = h(s0γξ ,η ).
= hgκ·η+δ
However, the function h is one-to-one, and hence sγξ ,η = s0γξ ,η . In par0
. The equality
ticular, βγξ ,η = βγ0 ξ ,η for every η < κ and hence γξ+1 = γξ+1
0
between sγξ ,η and sγξ ,η says that the derived functions coincide as well, so
we are done.
1.3
Remark 1.4. Another way to phrase the idea in the above proof is by noticing
that if 2<λ = 2κ then ¬Φλ codes a one-to-one mapping from 2λ into 2<λ ,
and hence 2<λ = 2λ .
1.4
Now we can prove the following:
Theorem 1.5. Very weak diamond and cardinal arithmetic.
For every regular uncountable cardinal λ we have 2<λ < 2λ iff Ψλ .
Proof.
By Corollary 1.2 we may assume that λ is a limit cardinal. If Ψλ holds
WEAK PREDICTION PRINCIPLES
7
then 2<λ < 2λ holds by Claim 1.1. For the opposite direction, assume that
2<λ < 2λ . If 2<λ = 2κ = 2λ for some κ < λ then Φλ holds by Claim 1.3 and
hence also Ψλ . So assume that this is not the situation (as always happens
in the case of a strongly inaccessible cardinal).
We claim that Φα holds for unbounded set of α-s below λ. For proving
this assertion, let us define an increasing continuous sequence of cardinals
hαε : ε < λi as follows. For ε = 0 let θ be the first cardinal for which
2ℵ0 < 2θ . Such a cardinal must be below
S λ by our assumption, and we let
αζ . Finally, if ε = ζ + 1 let θ < λ
α0 = θ. For a limit ordinal ε let αε be
2ℵζ
ζ<ε
θ
2 . Define
be the first cardinal for which
<
αε = θ.
We claim that Φαε holds whenever ε is a successor ordinal (and actually
also when ε = 0). Indeed, if ε = ζ + 1 and αε = γ + for some γ < λ then
+
2γ < 2γ = 2αε and then Φαε holds as indicated in the introduction. If
αε is a limit cardinal then it cannot be singular (by the Bukovsky-Hechler
theorem). It cannot be strongly inaccessible (since it has been chosen as
the first) and hence it must be weakly inaccessible. Notice, however, that
it satisfies the assumptions of Claim 1.3, since the sequence h2γ : γ < αε i is
constant from some point below αε onwards. By claim 1.3 we have Φαε .
We claim now that Ψλ holds. For this, let c : <λ → 2 be a coloring. For
every ζ < λ let ε = ζ + 1 and let cε be the restriction c αζ+1 . Choose a
function gε which exemplifies Φαε with respcet to cε for every ε = ζ + 1 < λ.
Define h : λ → λ as follows. For every β < λ let h(β) be the first ordinal
ε < λ so that αε ≤ β < αε+1 .
We define g ∈ λ 2 as follows. Given β < λ, if h(β) = ε is a limit ordinal
then g(β) = 0. If h(β) = ε is a sucessor ordinal then g(β) = gε (β). Let us
show that g exemplifies Ψλ .
Assume f ∈ λ 2. For every successor oridnal ε = ζ + 1 < λ let fε = f αε .
By Φαε we can choose an ordinal βε ∈ [αζ , αε ) for which gε (βε ) = cε (fε βε ).
By the above definitions it folows that g(βε ) = c(f βε ). Since we have
unboundedly many βε of this form, we are done.
1.5
Our next goal is to demonstrate the fact that Ψλ is strictly weaker than
~ ), when U
~ = hκi_ hUτ | τ < κ+ i is a
Φλ . We shall use Radin forcing R(U
measure sequence for some measurable cardinal κ with o(κ) ≥ κ+ . Recall
~ ) are finite sequences p = hdi | i ≤ ki satisfying the
that conditions in R(U
following conditions.
(ℵ) d~ = hdi | i < ki is a finite sequence. For every i ≤ k, di is either
of the form hκi i where κi < κ is an ordinal, or of the form di =
hµ~i , ai i where µ~i is a measure sequence on a measurable cardinal
κi = κ(µ~i ) ≤ κ and ai ∈ ∩µ~i .
(i) hκi | i < ki is increasing.
~ , Ai.
(‫ )ג‬dk = hU
8
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
For each i ≤ k we denote κi by κ(di ) and ai by a(di ). Given a condition
~ , Ai from the other
p = hdi | i ≤ ki as aobve, we will frequently separate hU
_
~
~
~
components and write p = d hU , Ai where d = hdi | i < ki.
A condition p∗ = hd∗i | i ≤ k ∗ i is a direct extension of p = hdi | i ≤ ki
if k ∗ = k and a(d∗i ) ⊂ a(di ) whenever a(di ) exists. A condition p0 is a one
point extension of p if there exists j ≤ k and a measure sequence ~ν ∈ a(dj )
with κ(~ν ) > κ(dj−1 ) and p0 = p_ h~ν i is either hdi | i < ji_ h~ν i_ hdi | i ≥ ji if
~ν = α is an ordinal, or hdi | i < ji_ h~ν , a(dj ) ∩ Vκ(~ν ) ii_ hdi | i ≥ ji if ~ν is a
nontrivial measure sequence. A condition q extends p if it is obtained from
p by a finite sequence of one point extensions and direct extensions.
~ be a measure sequence. It is shown in [4] that R(U
~ ) is a Prikry
Let U
~
type forcing notion, preserves all cardinals and satisfies κ(U )+ − cc. It is
also proved that κ remains strongly inaccessible in a generic extension by
~ ). Suppose G ⊂ R(U
~ ) is a generic filter. Let M SG ⊂ M S be the set
R(U
~ , Ai
of all ~u ∈ M S for which there exists some p ∈ G of the form p = d~_ hU
such that d~ = hu0 , a0 i, . . . , huk−1 , ak−1 i and ~u = ui for some i < k. Define
CG = {κ(~u) | ~u ∈ M SG }. CG is the Radin club associated with G.
Theorem 1.6. Strong inaccessibility and ¬Φκ .
Assuming the existence of a measurable cardinal κ such that o(κ) ≥ κ+ and
+
2κ = 2κ , it is consistent that there is an inaccessible cardinal κ such that
¬Φκ .
Proof.
~ = hκi_ hUτ | τ < κ+ i is a measure sequence of a measurable
Suppose that U
cardinal κ derived from an elementary embedding j : V → M satisfying
+
~ τ ∈ j(X)} for all
M |= 2κ = 2κ . Therefore, Uτ = {X ⊂ Vκ | U
+
+
τ < κ+ . Clearly, M |= 2κ = 2κ = |([Vκ ]<ω × P(Vκ ))κ×κ |. In V , let
+
H : κ → Vκ be a partial function, dom(H) = {α < κ | 2α = 2α }. For
+
every α ∈ dom(H), Hα : 2α ↔ ([Vα ]<ω × P(Vα ))α×α is a bijection. Let
+
Hκ = j(H)(κ) : 2κ ↔ ([Vκ ]<ω × P(Vκ ))κ×κ .
~ ) be the Radin forcing associated with U
~ . Choose a generic set
Let R(U
~ ). We claim there is no weak diamond on κ in V [G]. To show
G ⊆ R(U
~ , Ai ∈ R(U
~)
this, it will be convenient to identify conditions p = d~_ hU
~ Ai ∈ V <ω × P(Vκ ) is
~ Ai ∈ V <ω × P(Vκ ). We say that hd,
with pairs hd,
κ
κ
~ ) satisfies κ+ .c.c we can represent
a simple representation of p. Since R(U
~ ) using elements in (Vκ<ω × P(Vκ ))κ . We use the phrase
antichians in R(U
simple representation in this context as well.
~ ) name for a function from κ to 2
We first work in V . Let g be an R(U
˜
_
~ , Ai a condition. For every µ
and p = d~ hU
~ ∈ A, consider the one point
_
_
_
~
~
~ )/(p_ h~
extension p h~
µi = d h~
µ, A ∩ Vκ(~µ) i hU , Ai. The forcing R(U
µi)
~
~
factors into a product R(~
µ) × R(U )/(hU , A \ Vκ(~µ) + 1i). It follows that
_
~ , Aµ~ i forcing
p µ
~ has a direct extension of the form d~_ h~ν , A ∩ Vκ(~ν ) i_ hU
WEAK PREDICTION PRINCIPLES
9
g (κ̌(~
µ)) = σ(~
µ), where σ(~
µ) is a R(~
µ) name for an ordinal in {0, 1}. Let
˜A∗ = ∆
~ , A∗ i.
ν ∈ Vκ | ~ν ∈ Aµ~ if κ(~
µ) < κ(~ν )} and p∗ = d~_ hU
µ
~ ∈A Aµ
~ = {~
∗
∗
_
∗
p ≥ p and p h~
µi g (κ̌(~
µ)) = σ(~
µ) for all µ
~ ∈A .
+
<ω
˜
In M , we define a function h : κ → (Vκ × P(Vκ ))κ . For every τ < κ+ ,
let h(τ ) ∈ (Vκ<ω ×P(Vκ ))κ be a simple representation of a maximal antichain
~ τ ) of conditions q ∈ R(U
~ τ ) which force j(σ)(U
~ τ ) = 0̌. We
Aτ ⊂ R(U
+
<ω
κ×κ
can identify h with an element in (Vκ ×P(Vκ ))
. Let f = Hκ−1 (h) : κ →
0
<κ
2. Back to V , define a function F : 2 → Vκ as follows. F 0 (w) is a function
whose domain is the collection of all measure sequences ~ν with κ(~ν ) = α.
+
Note that for every α < κ and w ∈ 2α , Hα (w) ∈ (Vα<ω × P(Vα ))α×α .
We set F 0 (w)(~ν ) = {q ∈ R(~ν ) | q is simply represented by an element of
Hα (w)(len(~ν ))}, where len(~ν ) is the length of the sequence. By our choice
of f = Hκ−1 (h) we see that B = {~ν ∈ Vκ | F 0 (f κ(~ν ))(~ν ) is a maximal
T~
antichain of R(~ν ) of conditions q σ(~ν ) = 0̌} ∈ U
.
<κ
Finally, we define F : 2 → 2 in V [G]. For every ~ν ∈ M SG , let G(~ν )
denote the R(~ν ) generic filter induced by G. For each w ∈ 2κ(~ν ) we set
(
0 if F 0 (w)(~ν ) ∩ G(~ν ) 6= ∅
F (w) =
1 otherwise.
Define F (w) = 0 for every other w ∈ 2α .
~ , A∗ i as above. Let
We may assume G contains a condition p∗ = d~_ hU
T
~ . The claim follows as
X = {κ(~ν ) | ~ν ∈ A∗ ∩ B}. CG ⊂∗ X since X ∈ U
F (f α) = g(α) for every α ∈ X ∩ CG .
1.6
To round out the picture, we are left with the case of weakly but not
strongly inaccessible cardinal. We distinguish three cases. If 2<λ = 2λ
then we already know that ¬Φλ and if 2<λ = 2κ < 2λ for some κ < λ
then Φλ . The remaining case is when the sequence h2θ : θ < λi is not
eventually constant. In which case 2<λ < 2λ , and we do not know if the
weak diamond holds (see [7], Question 1.28) though we have seen that the
very weak diamond holds. It is possible to force Φλ in such cases:
Claim 1.7. It is consistent that λ is weakly inaccessible, h2θ : θ < λi is not
eventually constant, λ is not strongly inaccessible and Φλ holds.
Proof.
We begin with a strong limit cardinal λ in the ground model, aiming to blow
up 2θ for every regular uncountable θ < λ. We Shall add Cohen subsets to
any such θ, and also to λ itself. So let P be the Easton support product of
Add(θ, λ+θ+1 ) for every regular uncountable θ ≤ λ. Notice that P neither
collapses cardinals, nor changes cofinalities.
The forcing P is λ+ -cc and ℵ1 -complete. It follows that h2θ : θ < λi =
hλ+θ+1 : θ < λi in the generic extension, so this sequence is not eventually
constant. Although λ is not strongly inaccessible any more, it is still weakly
inaccessible. It remains to show that Φλ holds in V P .
10
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
Let c be a name of a coloring function from <λ 2 into 2. Observe that
< ˜2λ = λ+λ+1 in V P . Hence, by the chain condition, there exists a
bounded subset B of λ+λ+1 so that P = P B×P (λ+λ+1 \B) and c ∈ V PB .
Denote the lower part P B by Q and the upper part P (λ+λ+1 \˜B) by R.
We may assume that R is ℵ1 -complete and λ-distributive. Let g be a new
λ-Cohen set, added by R.
Working in V Q , Let f be a name of a function from λ into 2, and let D
˜
˜ of λ. For every condition r ∈ R we can define by
be a name of a club subset
induction on ω a sequence hrn : n ∈ ωi of conditions in R such that:
(a) r0 = r and rn ≤ rn+1 .
(b) rn+1 f αn = ǧn for some gn ∈ V Q .
˜ ) = αn+1 .
(c) dom(rn+1
(d) rn+1 ∃β̌n , β̌n ∈ D and α̌n < β̌n < α̌n+1 .
S
˜
rn . By the ℵ1 -completeness of R, p is a condition in R. Denote
Let p be
n∈ω
S
S
αn by α. Since rn+1 ≤ p for every n ∈ ω we see that p f α =
gn .
n∈ω
n∈ω
˜S
Likewise, p α̌ ∈ D since D is forced to be closed and α =
βn .
˜
˜
n∈ω
Our goal is to show that the fixed g chosen above can serve as a weak
diamond function from λ into 2. For this, we shall prove that the condition
p can be extended to force g(α) = c(f α̌). As c ∈ V Q , the value of c(f α̌)
˜
˜
˜ not
is determined by the condition p. ˜However,
functions
from λ into 2˜are
determined in a bounded stage. In particular, we can extend p to a condition
q which forces g(α) = c(f α̌). It follows that for every f : λ̌ → 2 and every
˜
˜ = c(f α), so we
club D there exists an˜ ordinal
α̌ ∈ D for which P g(α)
˜
˜
are done.
1.7
The following diagram summarizes the relationship between the various
prediction principles considered in this paper:
2<λ
f
Diamond
\
Weak
diamonde
\
Very weak
diamond
9
\
Galvin’s s
property
8
\
ZFCp
The downward positive implications ♦λ ⇒ Φλ ⇒ Ψλ are trivial. The fact
that Ψλ implies Galvin’s property appears in [3], as well as the negative
WEAK PREDICTION PRINCIPLES
11
direction upwards (i.e. Galvin’s property does not imply Ψλ ). The consistency of Ψλ with ¬Φλ can be exemplified by a strongly inaccessible cardinal
for which ¬Φλ is forced. The consistency of Φλ with ¬♦λ can be forced by
simple cardinal arithmetic considerations. Finally, it is shown in [1] that
Galvin’s property is not a theorem of ZFC, as its negation can be forced.
Finally, we mention the question of Shelah from [7], which seems to be
the last open case:
Question 1.8. Assume λ is weakly inaccessible and h2θ : θ < λi is not
eventually constant. Is it consistent that ¬Φλ holds?
12
OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT
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(2011m:03084)
Department of Mathematics, University of California, Los Angeles
E-mail address: [email protected]
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem
91904, Israel
E-mail address: [email protected]
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem
91904, Israel
E-mail address: [email protected]