Power set at $\aleph_\omega$: On a theorem of Woodin

0.1 MB·

by  Mohammad Golshani

POWER SET AT ℵ : ON A THEOREM OF WOODIN
ω
6 MOHAMMADGOLSHANI
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0
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n Abstract. WegiveWoodin’soriginalproofthatifthereexistsa(κ+2)−strongcardinal
a
J κ,then thereisagenericextension of theuniverseinwhichκ=ℵω,GCH holds below

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1 ℵω and2ℵω =ℵω+2.

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O
L
1. introduction
.
h
t
a One of the central topics in set theory since Cantor has been the study of the power set
m
[ function κ 7→ 2κ, and despite many results which are obtained about it, determining its

1 behavior is far from being answered completely. In this paper we consider the very special
v
1 caseofdetermining thepowerof2ℵω, whenℵ isastronglimit cardinal,andsojustdiscuss
ω
4
1
a little about what is known for this case.
4
0
The first important results were obtained by Magidor,who proved the consistency of ℵ
. ω
1
0 is strong limit and 2ℵω =ℵ , where 1<k ≤ω, from a supercompact cardinal [8], and
ω+k+1
6
1 the consistency of 2ℵω = ℵ while GCH holding below ℵ from a huge cardinal and a
ω+2 ω
:
v
i supercompactcardinalbelow it [9]. In [11], ShelahimprovedMagidor’stheoremfrom[8] by
X
r showing the consistency of ℵω is strong limit and 2ℵω =ℵω+α+1, where α is any countable
a
ordinal, still starting from a supercompact cardinal.
In1980th, Woodinwas ableto reduce the largecardinalassumptions usedby Magidorto
the level of strong cardinals, and in particular he proved the following theorem.
Theorem 1.1. (Woodin). Suppose GCH holds and κ is a (κ+2)−strong cardinal. Then
there is a generic extension of the universe in which κ = ℵ , GCH holds below ℵ and
ω ω
2ℵω =ℵ .
ω+2
Theauthor’sresearchwasinpartsupportedbyagrantfromIPM(No. 91030417). HealsothanksRadek
Honzikformanyvaluablecomments.
1
2 M.GOLSHANI.
In [5], Gitik and Magidor introduced a new method of forcing, called extender based
Prikryforcing,andusingit,theywereabletoreducethelargecardinalassumptionsusedby
Magidor and Shelah to the level of strong cardinal. In particular they provedthe following.
Theorem 1.2. Assume κ is a (κ+α+1)-strong cardinal, where α < ω . Then there is a
1
generic extension κ=ℵ , GCH holds below ℵ and 2ℵω =ℵ .
ω ω ω+α+1
In[7],GitikandMerimovichshowedthatifweallowfinitegapatℵ ,thenthecontinuum
ω
function below it can behave arbitrary, in the sense that given any finite m > 1 and any
function φ : ω → ω such that φ is increasing and φ(n) > n, there is a model of ZFC in
which2ℵω =ℵ andfor alln<ω,2ℵn =ℵ (startingfroma(κ+m)-strongcardinal).
ω+m φ(n)
Cummings [1] has given a strengthening of Woodin’s theorem, by producing a model of
ZFC in whichGCH holds at allsuccessorcardinalsbut fails atalllimit cardinals,however
the proof, which uses Radin forcing is quite complicated, while for Woodin’s theoremjust a
variant of Prikry forcing is sufficient. As there is no explicit proof of Woodin’s theorem, in
this paper we will sketch a proof of it, which is based on ideas from [1]. We have avoided
all the details, as they all can be found in [1] or [2].
Let us also mention that by results of Gitik and Mitchel, it is known that some large
cardinalsatthe levelofstrongcardinalsarerequiredfortheresultsstatedabove(see[3],[6]
and [10]). On the other hand by a famous result of Shelah, if ℵ is a strong limit cardinal,
ω
then 2ℵω <min{ℵ(2ℵ0)+,ℵω4}.
2. Woodin’s result
In this section we briefly review Woodin’s original proof of Theorem 1.1. The proof we
present here is based on ideas from [1], and we refer to it for details. Thus assume GCH
holds and let κ be a (κ+2)-strong cardinal. Let E be a (κ,κ++)-extender witnessing this,
and let j : V → M ≃ Ult(V,E) ⊇ V be the corresponding elementary embedding with
κ+2
crit(j)=κ. The proof is in several steps.
2.1. STEP 1. Factor j through the canonical ultrapower to get the diagram
ON A THEOREM OF WOODIN 3
V j ✲ M ≃Ult(V,E)
✟✟✯
✟
i ✟
❄ ✟✟ k
N ≃Ult(V,E )
κ
Let
P1 =hhP1 :τ ≤κ+1i,hQ1 :τ ≤κii
τ ∼τ
be the reverse Easton iteration for adding τ++-many Cohen subsets of τ+, for each inac-
cessible τ ≤ κ. So for each τ ≤ κ,Q1 is the trivial forcing, except τ ≤ κ is inaccessible, in
τ
which case (cid:13)P1τ“Q∼1τ =A∼dd(τ+,τ++).
Let
G1 =hhG1 :τ ≤κ+1i,hH1 :τ ≤κii
τ τ
beP1-genericoverV andV1 =V[G1].Thenbystandardarguments,therearegenericfilters
G1 ,G1 ∈V1 so that the diagram lifts to the following
M N
V1 j1 ✲ M1 =M[G1 ]
M
✑✑✸
✑
i1 ✑
✑ k1
❄ ✑
N1 =N[G1 ]
N
The next lemma is essentially proved in [1].
Lemma2.1. Thereexistsafilterg¯∈V1 whichisi1(Add(κ,κ++) ×Col(κ,κ+) )-generic
V1 V1
over N1.
2.2. STEP 2. Work in V1. Note that
N1 =the transitive collapse of {j1(f)(κ):f :κ→V1},
M1 =the transitive collapse of {j1(f)(a):f :[κ]<ω →V1,a∈[κ++]<ω}.
Let
P2 =hhP2 :τ ≤κi,hQ2 :τ <κii
τ ∼τ
bethereverseEastoniterationforaddingτ++-manyCohensubsetsofτ,foreachinaccessible
τ ≤ κ. So for each τ < λ,Q2 is the trivial forcing, except τ is inaccessible, in which case
τ
(cid:13)P2τ“Q∼2τ =A∼dd(τ,τ++).
4 M.GOLSHANI.
Let
G2 =hhG2 :τ ≤κi,hH2 :τ <κii
τ τ
be P2-generic over V1 and V2 = V1[G2]. Then for some suitable generic filters in V2, we
can lift the diagram one further step and get the following:
V2 j2 ✲ M2 =M1[G2 ]
M
✑✸
✑
✑
i2 ✑
✑ k2
❄ ✑
N2 =N1[G2 ]
N
Let E2 denote the (κ,κ++)-extender derived from j2. We have the following (see [1]).
Lemma 2.2. There exists F ∈V2 such that F is Col(κ+3,<i(κ)) ×Col(i(κ),i(κ+)) -
N2 N2
generic over N2. Further F ∈M2.
2.3. STEP 3. In this section we define the main forcing construction. Work in V2. Let
i2 :N2 →N2 denote the standardembedding of the αth ultrapowerinto the βth one, and
α,β α β
set κ =i2 (κ). Define
α 0,α
P=[Col(κ+3,<κ )×Col(κ ,κ+)] ,
1 1 1 N22
P∗ ={f ∈N2 :dom(f)∈j2(E2):∀β ∈dom(f), f(β)∈[Col(κ+3,<β)×Col(β,β+)] }.
1 κ N12
It is not difficult to see thatF is P-genericoverN2, which givesrise to some F∗ whichis
1
P∗-generic over N2. For α<β set
1
P(α,β)=Col(α+3,<β)×Col(β,β+).
We now define the main forcing construction.
Definition 2.3. A condition in P3 is a finite sequence of the form
hδ ,P ,δ ,...,P ,δ ,H,hi
0 1 1 n n
where
(1) δ <δ <···<δ <κ,
0 1 n
(2) Each Pk ∈P(δk−1,δk),k ≤n,
(3) dom(h)∈E2,dom(h)⊆κ\(δ +1),
κ n
ON A THEOREM OF WOODIN 5
(4) h(β)∈P(δ ,β),
n
(5) dom(H)=[dom(h)]2,
(6) H(α,β)∈P(α,β),
(7) i2 (H)(κ,j2(κ))∈F.
0,2
The order relation on P3 is defined as follows.
Definition2.4. Letp=hδ ,...,P ,δ ,H,hiandq =hξ ,...,Q ,ξ ,I,iibetwoconditions
0 n n 0 m n
in P3. Then p≤q if and only if
(1) n≥m,
(2) ∀k ≤m,δ =ξ and P ≤Q ,
k k k k
(3) ∀k >m, δ ∈dom(i),
k
(4) dom(h)⊆dom(i),
(5) ∀(α,β)∈dom(H),H(α,β)≤I(α,β),
(6) n=m⇒h(β)≤i(β),
(7) If n>m then
• P ≤i(δ ),
m+1 m+1
• m+1<k ≤n⇒ Pk ≤I(δk−1,δk),
• h(β)≤I(δ ,β).
n
Let G3 be P3-generic over V2 and let V3 =V2[G3]. Also let
hδ :n<ωi
n
be the Prikry sequence added by G3. Let us summarize the main properties of the generic
extension. The proof is essentially the same as (and in fact simpler than) the proofs given
in [1], where we also refer to it for more details.
Lemma 2.5. (1) V3 |=“ κ=δ+ω”,
0
(2) V2 and V3 have the same cardinals ≥κ,
(3) In V3, GCH holds in the interval (δ ,κ) and 2κ =(κ++)V.
0
6 M.GOLSHANI.
2.4. STEP 4. ForceoverV3 with P4 =Col(ω,δ+), andletG4 be P4-genericoverV3. Also
0
let V4 =V3[G4]. It is evident that
V4 |= “ℵ =κ,GCH holds below ℵ and 2ℵω =ℵ .
ω ω ω+2
This completes the proof of Theorem 1.1.
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