Saharon Shelah- Weak Diamond

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WEAK DIAMOND

SAHARON SHELAH

Abstract. Under some cardinal arithmetic assumptions, we prove thatevery stationary subset of of a right cofinality has weak diamond.This is a strong negation of uniformization. We then deal with a weakerversion of the weak diamond that involves restricting the domain of thecolourings. We then deal with semi- saturated (normal) filters.

Key words and phrases. Set theory, Normal ideals, Weak diamond, precipituous filters,semi saturated filters .

Research supported by the United States-Israel Binational Science Foundation Publi-cation 755. corections after proof reading to the journal.

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Annotated Content1. Weak Diamond: sufficient condition

[We prove that if = 2 =


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1. Weak Diamond: sufficient condition

On the weak diamond see [DvSh 65], [Sh:f, Appendix 1], [Sh 208], [Sh 638];there will be subsequent work on the middle diamond.

Definition 1.1. For regular uncountable ,

(1) We say S is small if it is F-small for some function F from >to {0, 1}, which means

()F,S for every c S2 there is such that { S : F(

) = c} is not stationary.(2) Let Dwd = {A : \ A is small }, it is a normal ideal (the weak

diamond ideal).

Claim 1.2. Assume

(a) =

every large enough regular < is in )

(c) S { < : cf() , and divides } is stationary.

Then S is not in the ideal Dwd of small subsets of .

Definition 1.3. (1) Let = Min{|P| : P [] and every A []

is included in the union of < members of P}.(2) tr = sup{|lim(t)| : t is a tree with nodes and levels }

Remark 1.4. (1) On tr see [Sh 589], on see there and in [Sh 460]

but no real knowledge is assumed here.(2) The interesting case of 1.2 is (weakly) inaccessible; for successorwe know more; but in later results even if 2 is successor we say onit new things.

(3) Actually only F (S

mark. ??

Proof. Let F be a function fromS

to {0, 1}, i.e., F is a colouring, and

we shall find f S as required for it.Let {i : i < } list


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sequence for F, we are done so assume that (for each < ) there are and E such that:

(a) E is a club of .(b)

.(c) if E S then F( ) = 1 f() and

.Now define 2 by ( + ) = () for < , < .

Let E = { < : is divisible by and < E and ( < )[ {i : i < }]}. Clearly E is a club of hence we can find E S. Soby the definition of P we have P and for < we have g,

is equal to (Why? note that = as E and see the definition ofg, and of, so : < g,() = ( + ) = ()). Hence h

2is well defined and by the choice of we have < g, = so byits definition, h for each < satisfies h() = F(g,) = F = ( ).

Now by clause (c) and the choice of f we have F( ) = 1 f() = g ()so h = g

, but h P whereas we have chosen g

such that g

/ P, a

contradiction.

We may consider a generalization.

Definition 1.5. (1) We say C is a Wd-parameter if:(a) is a regular uncountable,(b) S a stationary subsets of ,(c) C = C : S, C

(1A) We say C is a (,,) -Wd-parameter if in addition ( S)[cf() = |C| < ]. We may also say that C is (S,,)-parameter.

(2) We say that F is a C -colouring if: C is a -Wd-parameter and F isa function from > to 2 such that :

if S, 0, 1 and 0 C = 1 C then F(0) = F(1).

(2A) If C = : S we may omit it writing S - colouring(2B) In part (2) we can replace F by F : S where F :

(C) 2such that S F() = F( C). So abusingnotation we may write F( C)

(3) Assume F is a C-clouring, C a -Wd-parameter.We say c S2 (or c 2) is an F-wd-sequence if :(*) for every , the set { S : F( ) = c} is a stationary

subset of .

We also may say c is an (F, S)-Wd-sequence.(3A) We say c S2 is a D F-Wd-sequence if D is a filter on to

which S belongs and(*)for every we have

{ S : F( ) = c} = modD

(4) We say C is a good -Wd-parameter, if for every < we have > |{C : S and C}|.


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WEAK DIAMOND 5

Similarly to 1.2 we have

Claim 1.6. Assume(a) C is a good (,,)-Wd-parameter.

(b) ||tr, < for every < .(c) = 2 and =


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But by our choice of clearly cd()) = (), so

C C,() = () so C, = C,

Hence F(C,) = F( C), however as E E clearly F(

C) = (), together F(C ,) =

().

As C P clearly C 2, moreover for each < we know

that C (), see its definition above, is equal to F(C,) which by theprevious sentence is equal to (). As this holds for every < andC ,

are members of

2, clearly they are equal. But C Pso C { : P} whereas

has been chosen outside this set,

contradiction.

Well, are there good (,,)-parameters? (on I[] see [Sh 420, 1]).

Claim 1.7. (1) If S is a stationary subset of the regular cardinal andS I[] and ( S)cf() = then for some club E of , there isa good (S E,,)-parameter.

(2) If = cf(), + < = cf() then there is a stationary S I[] with( S)[cf() = ].

Proof. (1) By the definition of I[](2) By [Sh 420, 1].

We can note

Claim 1.8. (1) Assume the assumption of 1.6 or 1.2 with C = andD is a +- complete filter on , S D, and D include the club filter.Then we can get that there is a D F-Wd-sequence.

(2) In 1.6, we can weaken the demand = 2 to = cf(2) that is,assume(a) C is a good (,,)-Wd-parameter.

(b) ||tr, < 2 for every < .(c) = cf(2) and 2 = (2)


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and repeat the proof using = 2, C = C : S instead , C.

Except that in the choice of the club E we should use E = { < :

for every Rang (h) we have () < and is a limitordinal and S C = C

C T |{C : S E and C}|

() S |cf()tr ] :

Claim 1.11. Assume

(a) = cf(2)(b) F is an (S, )- good -colouring.

Then there is a (F, S)-Wd-sequence, see Definition 1.5(3).

Remark 1.12. So if = cf(2) and we let =: { = cf() and ( 0 and for every f we have

rkJbd (f) < then there is no semi-saturated normal filter D on to which { < : cf() = } belongs.

(4) In 1), 2), 3), if D is NorD-semi-saturated then the conclusionholds for D.

Remark: We can replace by any strong limit uncountable cardinal.

Proof. (1) Follows by (2)(2) By [Sh 460] for some 0 < , for every regular (0,) we

have: = , see 1.3. Let D = {A : sup( \ A) < }.By part (3) it is enough to prove if f then rkD(f) <

proof of If not then for every < there is

f such that f


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(3) We can find u, : < , < + such that:

(a) u, : < is -increasing continuous such that |u,| < ,

and {u, : < } = .(b) if < < + and u, then u, = u,.

Let f be f() = otp(u,), so it is well known that f/D

is the -th function, in particular < f


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(4) The same proof.

Remark: In the proof of 2.5(2) it is enough that UJbd () = (see [Sh 589]).

[References of the form math.XX/ refer to arXiv.org ]

References

[DvSh 65] Keith J. Devlin and Saharon Shelah. A weak version of which follows from20 < 21 . Israel Journal of Mathematics, 29:239247, 1978.

[Sh 638] Saharon Shelah. More on Weak Diamond. The East-West Journal of Mathe-matics, accepted. math.LO/9807180.

[Sh 208] Saharon Shelah. More on the weak diamond. Annals of Pure and Applied Logic,28:315318, 1985.

[Sh 420] Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and Infinite Com-binatorics in Sets and Logic, pages 355383. Kluwer Academic Publishers, 1993.N.W. Sauer et al (eds.). 0708.1979.

[Sh:g] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Ox-ford University Press, 1994.

[Sh:f] Saharon Shelah. Proper and improper forcing. Perspectives in MathematicalLogic. Springer, 1998.

[Sh 589] Saharon Shelah. Applications of PCF theory. Journal of Symbolic Logic,65:16241674, 2000.

[Sh 460] Saharon Shelah. The Generalized Continuum Hypothesis revisited. Israel Jour-nal of Mathematics, 116:285321, 2000. math.LO/9809200.

[Sh 576] Saharon Shelah. Categoricity of an abstract elementary class in two successivecardinals. Israel Journal of Mathematics, 126:29128, 2001. math.LO/9805146.

[Sh 775] Saharon Shelah. Middle Diamond. Archive for Mathematical Logic, 44:527560,

2005. math.LO/0212249.

Institute of Mathematics The Hebrew University of Jerusalem Jerusalem

91904, Israel and Department of Mathematics Rutgers University New Brunswick,

NJ 08854, USA

E-mail address: [email protected]: http://www.math.rutgers.edu/~shelah

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Saharon Shelah- Weak Diamond