Annotated Content §1. Weak Diamond sufficient condition

a rX iv:mat h /1727v1[mat h.LO]29J u l21WEAK DIAMONDSAHARON SHELAH Abstract.Under some cardinal arithmetic assumptions,we prove that every stationary of λof a right cofinality has the weak diamond.This is a strong negation of uniformization.We then deal with a weaker version of the weak diamond-colouring restrictions.We then deal with semi-saturated (normal)filters.2SAHARON SHELAH Annotated ContentWEAK DIAMOND31.Weak Diamond:sufficient conditionOn the weak diamond see[DvSh65],[Sh:f,Appendix§1],[Sh208],[Sh638]; there will be subsequent work on the middle diamond.Definition1.1.For regular uncountableλ,1.We say S⊆λis small if it is F-small for some function F fromλ≥λto{0,1},which means(∗)F,S for every¯c∈S2there isη∈λλsuch that{λ∈S:F(η↾δ)= cδ}not is stationary.2.Let D wdλ={A⊆λ:λ\A is small},it is a normal deal(the weakdiamond ideal).Claim1.2.Assume(a)λ=λ<λ=2µ(b)Θ={θ:θ=cf(θ)and for everyα<λ,we have|α|<θ><λor just|α|<tr,θ><λ}(see below;so ifλ> ωevery large enoughθ< ωis inΘ)(c)θ∈Θand S⊆{δ<λ:cf(δ)=θ,µωdividesδ}is stationary.Then4SAHARON SHELAH(a)Eεis a club ofλ.(b)ηε∈λ2.(c)ifδ∈Eε∩S then F(ηε↾δ)=1−fε(δ).Now defineη∈δ2byη(µα+ε)=ηε(α)forα<λ,ε<µ.Let E={δ<λ:δis divisible byµωandε<µ⇒δ∈Eεand(∀α<δ)[η↾α∈{ηi:i<δ}]}.Clearly E is a club ofλhence we canfindδ∗∈E∩S.So by the definition of Pδwe haveη↾δ∈Pδand forε<µwe have gη↾δ,ε∈δ2 is equal toηε↾δ(Why?note thatµδ=µasδ∈E and see the definition of gη↾δ,εand ofη,so:α<δ⇒gη↾δ,ε(α)=η(µα+ε)=ηε(α)).Hence hη↾δ∈µ2 is well defined and by the choice ofηwe haveε<µ⇒gη↾δ,ε=ηε↾δso by its definition,hη↾δfor eachε<µsatisfies hη↾δ(ε)=F(gη↾δ,ε)=F=(ηε↾δ). Now by the choice of fεwe have F(ηε↾δ)=fε(δ)=g∗δ(ε)so hη↾δ=g∗δ,but hη↾δ∈Pδwhereas we have chosen g∗δsuch that g∗δ/∈Pδ,a contradiction.1.2We may consider a generalization.Definition1.5. 1.We say¯C is aλ−Wd-parameter if:(a)λis a regular uncountable,(b)S a stationary subsets ofλ,(c)¯C= Cδ:δ∈S ,Cδ⊆δ(1A)We say¯C is a(λ,κ,χ)-Wd-parameter iffor everyα<λwe have λ>|{Cδ∩α:δ∈S}|.Similarly to1.2we haveClaim1.6.AssumeWEAK DIAMOND5(a)¯C is a good(λ,κ,χ)-Wd-parameter.(b)|α| tr,κ <λfor everyα<λ.(c)λ=2µandλ=λ<χ(d)F is a¯C-colouring.Then6SAHARON SHELAHHence F(νη∗↾Cδ,ε)=F(ηε↾Cδ),however asδ∈E∗⊆Eεclearly F(ηε↾Cδ)=ρ∗δ(ε),together F(νη∗↾Cδ,ε)=ρ∗δ(ε).Asη∗↾Cδ∈Pδclearlyρη∗↾cδ∈µ2,moreover for eachε<µwehaveρη∗↾Cδ(ε),see its definition above,is equal to F(νη∗↾Cδ,ε)which bythe previous sentence is equal toρ∗δ(ε).As this holds for everyε<µandρη∗↾Cδ,ρ∗δare members ofµ2,clearly they are equal.Butη∗↾Cδ∈Pδsoρη∗↾Cδ∈{ρη:η∈Pδ}whereasρ∗δhas been chosen outside this set,contradiction.Well,are there good(λ,κ,κ)-parameter?(on I[λ]see[Sh420,§1]). Claim1.7. 1.If S is a stationary subset of the regular cardinalλand S∈I[λ]and(∀δ∈S)cf(δ)=κthenthere is a stationary S∈I[λ]with (∀δ∈S)[cf(δ)=κ].Proof. 1.By the definition of I[λ]2.By[Sh420,§1].We can noteClaim1.8. 1.Assume the assumption of1.6or1.2with Cδ=δand D is aµ+-completefilter onλ,S∈D,and D include the clubfilter.Thenthere is a D−F-Wd-sequence.3.In1.6+1.8(2)we can omit“λregular”.Proof. 1.The same proof.2.Let H∗:λ→2µbe increasing continuous with unbounded rangeand let S∈I[λ]be stationary,such that(∀δ∈S)cf(δ)=κ,and ¯C= Cδ:δ∈S is a good(cf(λ),κ,κ)-Wd-parameter,let S′={h∗(α):α∈S},C′h∗(δ)={h∗(α):α∈Cδ},¯C′= Cβ:β∈S′and repeat the proof usingλ′=2µ,¯C′= C′δ:δ∈S′ insteadλ,¯C.Except that in the choice of the club E we should use E′={δ<λ: for everyα∈δ∩Rang(h∗)we haveη∗(α)<δandδis a limit ordinal andδ′∈S′∧C′=C′δ∩α⇒η∗↾C′∈T<δ}.3.Similarly.WEAK DIAMOND7 This lead to considering the natural related ideal.Definition1.9.Let¯C be a(λ,κ,χ)-parameter.1.For a family F of¯C-colouring and P⊆λ2,let id¯C,F,P be{W⊆λ:for some F∈F for every¯c∈P for someη∈λλthe set {δ∈W∩S:F(η↾Cδ)=cδ}is not stationary}.2.If P is the family of all¯C-colouring we may omit it.If we write Definstead F this mean as in[Sh576,§1].We can strengthen1.6as follows.Definition1.10.We say theλ-colouring F is(S,χ)-good if:(a)S⊆{δ<λ:cf(δ)<χ}is stationary(b)we canfind E and Cδ:δ∈S∩E such that(α)E a club ofλ.(β)Cδis an unbounded subset ofδ,|Cδ|<χ.(γ)ifρ,ρ′∈δδ,δ∈S∩E,andρ′↾Cδ=f↾Cδthen F(ρ′)=F(ρ)(δ)for everyα<λwe haveλ>|{Cδ∩α:δ∈S∩E}|(ǫ)δ∈S⇒|δ| tr,cf(δ) <λor justδ∈S⇒λ>|{C:C⊆δis unbound and for everyα<δfor someγ∈S we have C∩α=Cγ∩α]:Claim1.11.Assume(a)λ=cf(2µ)(b)F is an(S,κ)-goodλ-colouring.Thenthere is a good Cδ:δ∈S(d)forθ,S,¯C as above,if F= Fδ:δ∈S and Fδ(η)depend just onη↾Cδand D is a normal ultrafilter onλ(or less),and lastly S∈D then8SAHARON SHELAH2.On versions of precipitousnessDefinition2.1. 1.We say the D is(P,D˜)-precipituous if(a)D is a normalfilter onλ,a regular uncountable cardinal.(b)P is forcing notion with∅P minimal.(c)D˜a P-name of an ultrafilter of the Boolean Algebra P(λ)(d)letting for p∈PD p,D˜=:{A⊆λ:p A∈D˜}we have:(α)D∅P,D˜=D and(β)D p,D˜is normalfilter onλ(e) P“Vλ/D˜is well founded”.(1A)If D˜is clear from the context(as in part(2))we may omit D˜.2.Forλregular uncountable and D a normalfilter onλlet NOR D={D′:D′a normalfilter onλextending D}ordered by inclusion andD ˜=∪{D′:D′∈G˜NOR D}Woodin[W99]define and was be interested in semi saturation forλ=ℵ2. Definition2.2.Forλregular uncountable cardinal,a normalfilter D onλis called semi saturated when for every forcing notion P and P-name D˜ofa normal(for regressive f∈V)ultrafilter on P(λ)V,we have:D is(P,D˜)-precipitous.Woodin prove Con(Dω2↾S20is semi saturated).He proved that theexistence of suchfilter has large consistency strength by proving2.3below. This is related to[Sh:g,V].Claim2.3.Ifλ=µ+,D a semi saturatedfilter orλ,thenevery f∈λλis<D-smaller than theα-th function for someα<λ+Proof.The point is that(a)if D is a normalfilter onλ, fα:α<λ+ is<D-increasing inλandf∈λλ,α<λ+⇒¬(f≤D fα)thenthe clubfilter onλis not semi saturated.WEAK DIAMOND9 2.Ifλ=µ+≥ ωthenthere is no semi saturated normalfilter D∗onλto which{δ<λ:cf(δ)=κ}belongs.4.In1),2),3),if“D is Nor D-semi saturated”thenIf not then for everyα<λthere isfα∈κλsuch that fα<D f and rk D(f)=αand defineDα=:{A⊆κ:A∈D or1recall Sλ={δ<λ:cf(δ)κ}κ10SAHARON SHELAHForζ<λlet g∗ζ∈κλbe constantlyζ,and let g∗∈λλbe defined by g∗(ζ)=rk J bdκ(g∗ζ)(∗)0g∗∈λλ[why?by an assumption]Forα<λ+we define f∗α∈λλby:f∗α(ε)= rk J bdκ(fα◦gǫ)ifε∈Sλκ0ifε∈λ\SλκNote that fα◦gδis a function fromκtoλ.Now(∗)1f∗α∈λλforα<λ+[Why?as fα◦gδ∈κλ,so by a hypothesis rk J bdκ(fα◦gδ)<λ](∗)2forα<λ∗(∗)2αEα={δ<λ:ifε<λthen f∗α(ε)<δ}is a club ofλ[Why?Obvious](∗)3forα<λ+we haveδ∈Eα⇒f∗α(δ)<g∗(δ),so f∗α<Dλg∗∈λλ[Why?thefirst statement by the definition of Eαand of g∗(δ).The second by thefirst(∗)0.](∗)4ifα<β<λ+thenWEAK DIAMOND11 [Sh208]Saharon Shelah.More on the weak diamond.Annals of Pure and Applied Logic, 28:315–318,1985.[Sh420]Saharon Shelah.Advances in Cardinal Arithmetic.In Finite and Infinite Com-binatorics in Sets and Logic,pages355–383.Kluwer Academic Publishers,1993.N.W.Sauer et al(eds.).[Sh:g]Saharon Shelah.Cardinal Arithmetic,volume29of Oxford Logic Guides.Ox-ford University Press,1994.[Sh:f]Saharon Shelah.Proper and improper forcing.Perspectives in Mathematical Logic.Springer,1998.[Sh589]Saharon Shelah.Applications of PCF theory.Journal of Symbolic Logic, 65:1624–1674,2000.math.LO/9804155[Sh460]Saharon Shelah.The Generalized Continuum Hypothesis revisited.Israel Jour-nal of Mathematics,116:285–321,2000.math.LO/9809200Institute of Mathematics The Hebrew University of Jerusalem Jerusalem 91904,Israel and Department of Mathematics Rutgers University New Brunswick, NJ08854,USAE-mail address:shelah@math.huji.ac.ilURL:/~shelah。