moti gitik and saharon shelah- cardinal preserving ideals

8/3/2019 Moti Gitik and Saharon Shelah- Cardinal Preserving Ideals

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We consider the notion of a -presaturated ideal which was basically introduced by

Baumgartner and Taylor [B-T]. It is a weakening of presaturation. It turns out that this

notion can be preserved under forcing like the Levy collapse. So in order to obtain such an

ideal over a small cardinal it is enough to construct it over an inaccessible and then justto use the Levy collapse.

The paper is organized as follows. In Section 1 the notions are introduced and various

conditions on forcing notions for the preservation of-presaturation are presented. Models

with N S cardinal preserving are constructed in Section 2. The reading of this section

requires some knowledge of [G1,2,4].

The results of the first section are due to the second author and second section to the

first.

Notation. N S denotes the nonstationary ideal over a regular cardinal > 0, NS

denotes the N S restricted to cofinality , i.e. {X |X { < |cf = } N S}, D

denotes a normal filter over a regular cardinal = (D) > 0, D+ = {A |A = mod

D, i.e. A D}. D

Q for a forcing notion Q, such that Q

(D) is regular is the

Q-name of the normal filter on (D) generated by D. By forcing with D+ we mean the

forcing with D-positive sets ordered by inclusion.

1. -Presaturation and Preservation Conditions

The definitions and the facts 1.1-1.5 below are basically due to Baumgartner-Taylor

[B-T].

Definition 1.1. A normal filter (or ideal) D over is -presaturated if D+

every

set of ordinals of cardinality < can be covered by a set of V of cardinality

.

Let us formulate an equivalent definition which is much easier to use.

Definition 1.2. A normal filter D on is -presaturated, if for every 0 < , and

maximal antichains I = {Ai : i < i}, i.e.

Ai D+, [i = j Ai A

j D

+], (A D+)(i < i)A Ai D

+)

(for < 0) for every B D+ there is A D+ such that A B and ( < 0)|{i . For < , < 0 let () be the least s.t. B if such exists and -1

otherwise. Define f : by f() =


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Proof: (b) (c) since |D+| = +. The direction (c) (a), (b), (d), (e), (f) are trivial.

If (b) holds, then, as in [B-T], also (a) holds. (a) implies (d), (e) and (f).

We are now going to formulate various preservation conditions.

Lemma 1.6. Suppose that < are regular cardinals, D a normal filter on = (D),

Q a forcing notion Q

regular and D

Q denotes the normal filter on in VQ which

D generates.

If D is -presaturated (in V) then D

Q is -presaturated in VQ provided that for someK

()1D,K,Q, K is a structure with universe |K|, partial order =K unitary function T =

TK , and partial unitary functions pi = pKi such that

(a) for t K, T(t) D+;

(b) K|= t s implies T(s) T(t)mod D;

(c) pi(t) is defined iff i T(t);

(d) pi(t) Q and [K|= t s, i T(s) pi(t) pi(s)];

(e) t

=

K

t= {i T(t) : pi(t) G

Q} is not forced to be mod D

Q;

(f) if q Q

T

(D

Q)+

for some q Q then there is s, s K, pi(s) q and

Q

s

T

mod D

Q

;

(g) if() < , for < () Y K is such that {T(t) : t Y} is a maximal antichain

of D+ such that [t Y, s Y , > (C D)(i C T(t) T(s))pi(s)

pi(t)], and T D+ then there is s, s K, T(s) T and there are y Y, |y|

for < () such that [ < (), x Y\y T(x) T(s) = mod D] and

< ()(x y)(C D)(i C T(x) T(s))[pi(x) pi(s)].

Proof: Let () < , and I

= {A

i: i < i

} (for < ()) be Q-names of maximal

antichains of (DQ

)+ (as in Definition 1.2) (i.e. Q

I is a maximal antichain.

We define by induction on < () Yh such that

(i) Y K, {T(t) : t Y} a maximal antichain of D+;

(ii) for every < , t Y, s Y for some C D i C T(s) T(t) pi(t) pi(s));

(iii) h : Y ordinals, and

t A

h(t)for t Y.

Using (g) pick s and y| < (). Then s

will be as required in Definition 1.2.

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Definition 1.7.

1) Gm(D , , a) (where a ) is a game which lasts moves, in the ith move if i a

player I, and if i a player II, choose a set Ai D+, Ai Aj mod D for j < i. If at

some stage there is no legal move, player I wins, otherwise player II wins.2) If we omit a, it means a = {1 + 2i : 1 + 2i < }.

3) Gm+(D , , a) is defined similarly, but for player II to win, Ai = has to hold as

well.

By Galvin-Jech-Magidor [G-J-M], the following holds.

Proposition 1.8. D is precipitous if player I has no winning strategy in Gm+(D, ).

Lemma 1.9.

1) Suppose that < are regular cardinals, D a normal filter on = (D), Q a forcing

notion Q

regular, = (D).

If player II wins the game Gm(D , , a) in V, then it wins in Gm(D , , a) in VQ

provided that for some, , the following principle holds:

()2D,K,Q:

K is a structure as in Lemma 1.6,

(a) fort K, T(t) D+;

(b) K|= t s implies T(s) T(t)mod D;

(c) pi(t) is defined iff i T(t);

(d) pi(t) Q and [K|= t s, i T(s) pi(t) pi(s)];

(e) t

= {i T(t) : pi(t) GQ

} and pi(t) Q

t

(D

Q)+

for i T(t);

(f) if, q Q

T

(D

Q)+

for some q Q then there is s, s K, pi(s) q and

Q

sT mod DQ

;

(g) K is -complete, i.e. if ti : i < < is increasing and T(ti) D+ then there

is t K, ti t;

(h) Q is -complete and preserves stationarity of all A D+.

2) The same holds with Player I having no winning strategy.

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Proof:

1) Let us provide a winning strategy for player II. Let G Q be generic over V without

loss of generality the players choose Q-names for their moves. We will now describe

the strategy of player II in V[G].Player II also chooses Ti, tj(j a, i < ), according to the moves. Player II preserves

the following (for a fixed winning strategy F of player II in Gm(D , , a) in V).

(*) the plays Ai

: i < and Ti : i < so far satisfy

(a) Ti : i < is a beginning of a play ofGm(D , , a) (in V) in which player II uses

the strategy F;

(b) for i a, Ai

=ti

, T(ti) = Ti, ti K;

(c) for j < i < , tj ti;(d) V[G]|=A

i[G] D+

.

Proposition 1.10. Suppose < are regular cardinals, < [


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If ()3D,K,Q holds then also the following statements are true:

(1) If D is -presaturated in V, { : cf (in V)} D, then D

Q is -presaturated in

VQ.

(2) If player II wins in Gm(,D,) in V ( ), Q -complete then he wins in Gm(,D,)in VQ provided that

()4D,K,Q :(a) p K pi

Q

p

= mod D

Q for i Tp

(b) ()3D,K,Q

(3) The same holds if we replace The player II wins by Player I does not have winning

strategy or the game Gm(,D,) is replaced by Gm+(,D,).

(4) In (2) we can replace()3D,K,Q by:

D1 a normal filter over in V, K1 like K,

player II wins also in Gm(,D,) and

()5D,K,D1,K1,Q :

() ()4D1,K1,Q() ()4D,K,Q() for every p = pi : i Tp K there are

q = q : S K1 p = pi : i T K

T Tp pairwise disjoint[i T pi p

i ], q

p

= mod D

Q .

Proof: Let us prove (1). The proof of (2) is similar. Let () < , I

= {A

i: i < i

}

( < ()) be () Q-names of maximal antichains of D+ as mentioned in the definition

of -presaturitivity (i.e. it is forced that they are like that).

We define by induction on (), Y, j() and p( Y) such that

(i) Y is a set of sequence of ordinals of length ;

(ii) p K;

(iii) < , Y implies | Y, p| p (i.e. T T|), [i T p|i p ];

(iv) Y0 = {}, p = pi : i < , p

i = Q (the minimal element);

(v) for limit, Y = { : a sequence of ordinals g() = , (i < )|i Yi};

(vi) {T : Y} is a maximal antichain;

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(vii) for limit, Y, T =


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We define by induction on () Y and the function j|Y : Y ordinals and

p(v Y) s.t.

(i) Y a set;

(ii) pv

K let pv

= pvi : i Tv, Tv D+;

(iii) for every v Y and i Tv q0 pvi ;

(iv) Tv : v Y} is a maximal antichain of D+;

(v) for v Y, i Tv, pvi

i A

j(v);

(vi) if Y, < , v Y, and Tv T D+ then for some Cv, D, (i

Tv T Cv,) [pi p

vi ];

(vii) < Y Y = ;

(viii) for < , v Y |{ Y : T Tv D+}| .

If , v


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(v). Now / p (D

+)Q so for some r Q r p (D

+)Q. Hence for some r1,

r r1 Q, and j the following holds:

() 1 p A

j D+.

For each i T choose if possible p

i , pi p

i Q p

i

i A

j . Set Tb = {i : p

i

is defined}. It is necessarily in D+ (otherwise this contradicts ()). Applying (d) of

()6K,Q,D we get a final p contradicting Y maximal but {Tv : v Y} is not.

For = 0, use (b) with our q0.

Proposition 1.12. Let D, D1 are normal filters over a regular cardinal . Suppose that

the following holds

CD,D1 : for every T D+

, F(T) = { < : T is stationary} D+1 .

Let Q be a -closed forcing for a regular < . Then in VQ CD,D1 is not forced to fail

provided that

()7D,K,D1,Q ()()6D,K,Q or ()

3D,K,Q and

(1) for p K

{ < :p

is forced to be stationary} D+1

or (2) for p K

{ < :p

is not forced to be nonstationary} D+1

and Q satisfies c.c .

Proof: Suppose otherwise. Let

S

(D

Q)+ and F(S

) (D

Q

1)+

. Set T = {i is an inaccessibleQ = Col(, is an inaccessible, Q is the

Levy collapse Col(, < ) and D, D1 are normal filters over . Then

1) ()3D,K,Q holds if { < |cf } D and we let K = {pi|i T|T D+, pi Q};

2) ()6

D,K,Q holds if we let K = {pi|i T C|T D+

, pi Q, C is as in Lemma 1.13};3) ()4D,K,Q holds if { < |cf } D and K is as in 2);

4) ()7D,K,D1,Q holds if { < |cf < } D, { < | is an inaccessible } D1 and K

is as in 2).

Proof: 1), 2), 3) easy checking. 4). Suppose that in V CD,D1 holds. Let us show the

condition (2) of ()7D,K,D1,Q. Let p = pi|i T K and C D1. We should find some

C and q Q q p

is stationary.

Using CD,D1 find an inaccessible C such that C, T is stationary and for

every i < pi Col(, < ). Now, as in Lemma 1.13, there is a stationary S T such

that every pi (i S) forces in Col (, < ) that p

is stationary. IfD concentrates on

ordinals of cofinality < , then without loss of generality all elements ofT are of some fixed

cofinality < . The forcing Col(, < ) is -closed, so it would preserve the stationarity of

(p )Col(,


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Corollary 1.15. Let be an inaccessible, < be a regular cardinal and Q = Col(, 0 be ordinals and be a regular cardinal. Then

(a) is an up-repeat point for F if for every A F(, ) there is , < < such that

A F(, );

(b) is a (, )-repeat point for F if (a1) cf = and (a2) for every A {F(, )|

< + } there are unboundedly many s in such that A {F(, )| 2 + GCH or;

(2) N S0+

is -preserving for a regular > 2 + GCH or;

(3) N S+ is 2-preserving for a regular > 2 + GCH or;

(4) NSr+

is 2-preserving for a regular > 2, < + GCH

is the existence of an up-repeat point.

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Proof: Use the model of [G4] with a presaturated N S over an inaccessible and the

Levy collapse Col(,< ). By the results of Section1, N S will be -preserving in the

generic extension.

The rest of the section will be devoted to the construction of NS 1-preserving froman (, + + 1)-repeat point. The following theorem will then follow.

Theorem 2.2.

(1) the strength of N S0 is1-preserving + is an inaccessible is an (, + + 1)-repeat

point;

(2) the existence of an (, + + 1)-repeat point is sufficient for N S is 1-preserving +

is an inaccessible +GCH;

(3) the strength of N S0+ is 1-preserving +GCH for a regular > 2 is (, )-repeat

point;

(4) the existence of an (, + 1)-repeat point is sufficient for N S+ is 1-preserving

+GCH, where is a regular cardinal.

Suppose that U is a coherent sequence of ultrafilters with an (, + + 1)-repeat point

at . Define the iteration P for in the closure of {|( = ) or ( < and is an

inaccessible or = + 1 and is an inaccessible)}.

On the limit stages use the limit of [G1]. For the benefit of the reader let us give a

precise definition.

Let A be a set consisting of s such that < and > 0() > 0. Denote by A the

closure of the set { + 1 | A} A. For every A define by induction P to be the

set of all elements p of the form p

| g where

(1) g is a subset of A.

(2) g has an Easton support, i.e. for every inaccessible , > | dom g |;

(3) for every dom g p = p

| g P and p P

p

Q

.

Let p = p

| g , q = q

| f be elements of P. Then p q (p is stronger

than q) if the following holds:

(1) g f

(2) for every f p P

p

q

in the forcing Q

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(3) there exists a finite subset b of f so that for every f\b, p p

q

in

Q

.

If b = , then let p q.

Suppose that is an inaccessible and P is defined. Define P+1. Let C(+) be theforcing for adding +-Cohen subsets to , i.e. {f VP |f is a partial function from +

into , |f|VP

< and for every < +{|(, ) dom f} is an ordinal}. P+1 will be

P C(+) P(, O

U()), where P(, OU()) is the forcing of [G1,2] with the slight change

described below.

The change is in the definition of U( , , t) the ultrafilter extending U(, ) for } p

j(A

) .

Note that the set D = {q|q||( j(A

)} is E-dense and it belongs to N. So it appears in

the list . Hence some p

is in D.

Let G be a generic subset of P C(+). Recall that by [G2] the sequence is commutative Rudin-Keisler increasing sequence of ultrafilters,

for every .

Now let j : V[G] M = V[G]/U(,, ). Then M = M[j(G)] for a model M

of ZF C which is contained in M. We would like to have the exact description M.

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The next definition is based on the Mitchel notion of complete iteration see [Mi 1,2,3].

Definition 2.3. Suppose that N is a model of set theory, V a coherent sequence in N,

is a cardinal. The complete iteration j : N M of V at will be the direct limit of

j : N M , < for some ordinal , where , j, M are defined as follows. Set

M0 = N, j0 = id, V0 = V, C0(, ) = for all and . If OV() = 0, then = 1.

Suppose otherwise:

Case 1. OV() is a limit ordinal.

If j, M , V , C are defined then set

j+1 : M M+1 = M/V( , ) ,

where is the minimal ordinal so that

(i) ;

(ii) is less than the first > with OV() > 0;

(iii) for some < OV () C(, ) is bounded in ;

and is the minimal satisfying (iii) for .

If there is not such an then set = , j = j , M = M , C = C . Define

C+1|(, ) = C |( , ), +1 = j+1()

C+1(+1, j+1()) =

C(, ), if = C(, ) {}, if =

Case 2. OV() = + 1.

Define j , M, V and C as in Case 1, only in (ii) let be less than the image of

under the embedding of N by V(, ).

Theorem 2.4. Let j : V[G] M be

(a) the ultrapower of V[G] by the ultrafilter U( ,, )

or

(b) the direct limit of the ultrapowers with the Rudin-Kiesler increasing sequence of the

ultrafilters < U(, , )| < >.

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Then M is a generic extension ofM, whereM is the complete iteration of U|(, +1)

at , if (a) holds, and of U|(, ) at , if (b) holds.

Main Lemma. Let M be as in the theorem and let i be the canonical embedding of V

into M. Then there is G i(P C(+)) so that G V[G] and G is M[G] generic.

Proof: Let us prove the lemma by induction on the pairs (, ) ordered lexicographically.

Suppose that it holds for all (, ) < (, ). Let us prove the lemma for (, ).

Case 1. = + 1.

Let N be the ultrapower of V by U(, ) and j : V N the canonical embedding.

We just simplify the previous notations, where N = N , j = j . Then M is the complete

iteration of j( U)| + 1 at in N if is a limit ordinal, or for successor , the above

iteration should be performed -times in order to obtain M. Let us concentrate on the

first case; similar and slightly simpler arguments work for the second one.

Denote by k : N M the above iteration. Then the following diagram is commutative

Nj

V k

i

M

By the inductive assumption there exists G N[G] M-generic subset of Pk()

C((k())+).

If = 0 then k() = 0i(U)(k()) = 0 and C(k()+) is only the forcing used over

k(). Suppose now that > 0. Then k() > 0 and the forcing over k() is C(k()+)

followed by P(k(), k()). Let us define in N[G] a M[G]-generic subset ofP(k(), k()).

Recall that a generic subset ofP(k(), k()) can be reconstructed from a generic sequence

bk() to k(), where bk() is a combination of a cofinal in k() sequences so that b1k()({n})

is a sequence appropriate for the ultrafilters i(U)(k(), ) with on the depth n. We refer

to [G2] for detailed definitions.

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Let us use the indiscernibles of the complete iteration C in order to define bk().

Namely, only < C(k(), )| < k() > will be used.

Set bk() = {< , n, > |n < , < k()+, < k(), for some < k() coded by

< n, > in sense of [G2] C(k(), ))}.Let G

be the subset of P(k(), k()) generated by bk(). Let us show that G

is

M[G]-generic subset. Let D M[G] be a dense open subset of P(k(), k()) and D

its

canonical name. Since M is the direct limit for some less than the length of the complete

iteration, for some D

M, D

is the image of D

. Let us work in M . Denote by G

the appropriate part of G and by t a coherent sequence which generates bk()| . Let

k : M M be the part of the complete iteration k on the step .

Case 1.1. cfM(k()) < k(

).

Then k() changes its cofinality to cfM (k(

)) after the forcing with P(k(), k()).

For every T so that < t, T > P(k(), k()), there exists i < cfM (k()) and

T so that < t, T > is a condition stronger than < t ,T > and < t, T > forces some

< t, T > D with t on the level i is in the generic set. In order to find such T just

use the k()-completeness of the ultrafilters involved in the forcing P(k(), k(

)) and

the Prikry property. Now for every t T which is appropriate for i or some j i there

exists T such that < t, T > D . The same property remains true for k(T) for every

< length of the iteration k. Pick to be large enough in order to contain

elements of bk() appropriate for i. Let t be a coherent sequence generating bk()| . It

is possible to pick such t in k(T). But then for some T < t, T k(T

) > k (D).

The image of < t, T k(T) > under the rest of the iteration will be in G

.

Case 1.2. cfM (k()) = k(

).

Then k() changes its cofinality to .

For every T so that < t, T > P(k(), k()) there exist n < and T so that

< t, T > is a condition stronger than < t, T > and < t, T > forces some < t, T > D ,

with t containing the first n elements of the canonical -sequence to k(), is in the generic

set.

Pick in order to first reach n elements of the canonical -sequence to k().

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Let us check that G is M[G ]-generic subset of C(+). Suppose that D M[G ] is

a dense open subset of C(+). Let D

be a canonical P-name of D. By the assumption

on there are indiscernibles < 1 < < n < such that the support of D

is a

subset of {, 1 , . . . , n}. Since the forcing C(+

) is -closed D can be replaced by itdense open subset with support alone. Let us assume that D

is already such a subset.

Then D

= k(D

) for dense open D

N subset of C(()+). By the choice of the master

condition sequence for some < +

p | p(

)|C(()+) D

.

But this implies

k(p)()|C(+

) D .

So G D = .

Suppose now that 0k(U)() > 0. We need to define G

a M[G G]-generic subset of

P(, 0k(U)()). In this case is a limit of indiscernibles, i.e. C(, ) is unbounded in for

every < 0k(U)()). So we are in the situation considered above. The only difference is that

some ps may contain information about the generic sequence b to . In order to preserve

k(p)| + 1 in the generic set, we need to start b according to k(p)(). Notice, that

further elements of the master condition sequence do not increase the coherent sequence

given by p.

Case B. is a limit ordinal.

Set G = {p P| for every < p| G}. Let us show that G is M-generic

subset of P. Consider two cases.

Case B.1. There are unboundedly many in indiscernibles for ordinals .

Since = k(), is a limit of indiscernibles for . Then is measurable in M and thedirect limit is used on the stage . Let < i|i < > be a cofinal sequence of indiscernibles.

Let D M be a dense open subset of P. Then { < |D H() is a dense open

subset of

is almost contained

in every club of M so there is i0 s.t. D P0 is dense. But then G0 D = . Hence

G D = .

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Case B.2. There are only boundedly many in indiscernibles for ordinals .

Then, since = k(), there is no indiscernibles for ordinals below . So there

are unboundedly many in ordinals which are in the range ofk.

Let D be a dense open subset of P in M. Define D = {p P| for some k(p0)|

G for every < . So

k(p0)| G.

So G is an M-generic subset of P. It completes Case B and hence also Case 1 of

the lemma.

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Case 2. is a limit ordinal.

Use the inductive assumption and the definitions of the generic sets of Case 1. Define

a generic subset of Pk() as in Case B.1.

of the lemma.

Let G V[G] be the M-generic subset of Pi() C(i()+) defined in the Main

Lemma. Then G P C(+) = G and for every < + i(p) G. Define the

elementary embedding i : V[G] M[G] by i(a

[G]) = (i(a

))[G]. Then i|V = i,

i(G) and, if U = {A | i(A)} then U = U(,, ). So the following diagram is

commutative

M[G]i

V[G]

M = V[G]/U

where ([f]U) = i(f)().

It remains to show that = id. Notice, that it is enough to prove that every indis-

cernible in C is of the form i(f)() for some f V[G]. Examining the construction of

G, it is not hard to see that every indiscernible is an element of the generic sequence b

for some k

(N).

So indiscernibles are interpretations of forcing terms with parameters in k

(N). But

such elements can easily be represented by functions on in V[G]. Let C and

= (t())[G] for = k(), where t is a term. Let = [f]U(,) for some f : ,

f V. Define a function g V[G] on as follows: g() = t(f())[G]. Then i(g)() =

t(t(f)())[G] = (t())[G] = .

Let us now turn to the construction of 1-preserving ideal. Suppose that is an

(, + + 1)-repeat point for U in V. Let < + + be an ordinal. Denote by M

the complete iteration ofj (U)|((j (), (j

(

+ +)) in N . Let i, k be the canonical

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elementary embeddings making the diagram

Njk

V

ki

M

commutative.

As in the Main Lemma find in V[G] a P(, )-name of a M-generic subset G

of

Pi

() C(i

(

+

))/P+1 so that each k

(p

) G

for <

+

, where < p

| < +

> is themaster condition sequence for U(,, ). Let G

be obtained from G

by removing all the

information on the generic subset of C(i(+)) except i

(G C(+)).

We shall define a presaturated filter U(, ) on in V[G] extending U(, ) so that

(i) all generic ultrapowers of U(, ) are generic extensions of M;

(ii) every set U(,, ) is U(, )-positive.

Property (i) will insure that the forcing for shooting clubs over i() will be -closed

forcing. Property (ii) is needed for the iteration of forcings for shooting clubs over in

V[G].

Denote G P by G and G C(+) by G. Let G() denote the -th function of G

for < +. Let j be the embedding of V[G] into the ultrapower of V[G] by U(,, ).

Let < | < + > be the list of all the indiscernibles for i() of the complete iteration

used to define M. For every < + extend G

to G

by adding to G

conditions

< + + 1, , > for every < +, i.e. the + + 1 generic function moves to .

Define a filter U(, ) on in V[G] as follows:

A belongs to U(, ) iff for some r G some r

s.t. r

G

, for every

p

C

(j (+))(0) s.t. in N [j

(G)] p j(G(0)), and for some r

s.t. r

G

(p

)

in M

r 1P(,) p

r

Pi()+1

i(A

)

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where (p) < + is the minimal s.t. p| j(G(0)), for the (p) member of some fixed

enumeration of j ().

Claim 1. U(,, ) U(, ).

Proof: Let A U(,, ), then for some < + some r, < , T > P(, ), in N

r {< , T >} p

j (A

) .

Let us split p

into three parts p1

=p

| Pj , p2

=p

C(j (+))(0) and p

3

= the rest of

p

. Then, using k, in M

r {< , T >} k

(p) i

(A) .

Extend p2

to some p

2

, still remaining in the master condition sequence, in order to make

its domain above all the indices of the generic sequences listed in p3

. Extend p2 to some

p

2 C(j (

+))(0) which is incompatible with a member of the master condition sequence.

Then, using k, in M

r {< , T >} k(p1

) k(p

2

) k(p3

) i(A

) .

By the definition of U(, ), it implies that A is a U(, )-positive set.

So every member of U(,, ) is U(, )-positive. The fact that U(,, ) is an

ultrafilter completes the proof of the claim.

Claim 2. U(, ) is normal precipitous cardinal preserving filter and its generic ultra-

power is a generic extension ofM.

Proof: Let U = U(, )V[G, G(0)]. The arguments of Theorem 1 show that a generic

ultrapower by U is isomorphic to a generic extension of the complete iteration of N with

j (U)|j () above . Actually the forcing with U is isomorphic to P(, ) followed by

C(j (+))(0) (in the sense of this iteration) over V[G, G(0)]. Clearly, the generic function

form j (+) into j (

+) produced by this forcing is incompatible with j(G(0)) which

belongs to V[G, G(0)].

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Now the forcing with U(, ) over V[G] does the following: First it picks p

C

(j (+))(0) incompatible with j(G(0)) and then G(p

) is added. G(p

) insures that a

generic ultrapower is a generic extension ofM. So the forcing with U(, ) is isomorphic

to P(, ) followed by a portion of C(i(+)), which is -closed.

Force over V[G] with the forcing Q which is the Backward-Easton iteration of the

forcings adding +-Cohen subsets to every regular < with 0U() > 0. Fix a generic

subset H of Q. All the filters U(, ) extend in the obvious fashion in V[G, H]. Let us

use the same notations for the extended filters.

Set F = {U(, )| + +}. Then F is a -preserving filter in V[G, H]

and forcing with it is isomorphic to P(, ) (for some , + +) followed

by -closed forcing. Let us shoot clubs through elements of F, then through the sets of

generic points and so on, as was done in [G 3,4]. Denote this forcing by B. Let R be its

generic subset over V[G, H]. We shall show that N S is 1-preserving ideal in V[G,H,R].

The forcing with N S consists of two parts:

(a) embedding of B into P(, );

(b) (-closed forcing) (i(B)/i(R)).

By the choice of i(), the forcing i(B) is a shooting club through sets containing a

club (the club of indiscernibles for i()). So it is -closed forcing and part (b) does not

cause any problem.

Let us examine part (a) and show that this forcing preserves 1. Recall that B is the

direct limit of < B| < + > where each B is of cardinality and for a limit , B =

the direct limit of B( < ), if cf = and B = the inverse limit of B(

< )

otherwise. The forcing of (a) is

P = { V[G, H]| for some < +

is an embedding of B into P(, )} .

For 1, 2 P let 1 2 if 1| dom 2 = 2.

Claim. N S is an 1-preserving ideal in V[G,H,R].

Proof: Suppose otherwise. Then for some + + some condition in the

forcing P(, ) (the forcing isomorphic to the adding of+-Cohen subsets of+) P over

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V[G, H] forces 1 to collapse. Let us assume that the empty condition already forces this.

Consider the case when P(,)cf = . The remaining cases are simpler.

Let S be a V[G, H]-generic subset of P(, ) (+-Cohen subsets of +). Denote

V[G,H,S] by V. Let f be a P-name in V of a function from to V

1 . Pick an elementarysubmodel N of < H(),,B, P, , f

>, for big enough, satisfying the following three

conditions:

(1) |N| = ;

(2) N HV();

(3) N + = for some s.t. cfV[G,H] = .

Then B N = B. Also B is a direct limit of < B | < >. Pick in V[G, H]

a cofinal sequence < | < > to and in V a cofinal sequence < n|n < > to .

Consider the subsets of , s.t. B+1/B is the forcing for shooting club

into A

. Assume for simplicity that all As are in V. Let A = 0. Denote n0 by 0. Let

C0 be the generic club through A0 defined by 0 and S. As in Lemma 3.6 [G4], it is

possible to find an element of the generic sequence to , (0) A n0 such that C is a

V[G|(0), H|(0)]-generic club through A . Choosing (0) more carefully, it is possible

to also satisfy the following A (0) =


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= + for a regular > 1. It is possible to use the model constructed above, collapse

to + and apply the results of Section 1. But an (, + + 1)-repeat point was used

in the construction of the model. It turns out that an (, + 1)-repeat point suffices

for N S+ and an (, )-repeat point for N S+ |{ < +

|cf < }. On the other hand,precipitousness of N S0

+implies an (, )-repeat point, by [G4].

Let us preserve the notations used above. Assume that < is a regular cardinal and

some < 0U() is an (, + 1)-repeat point. Let G be a generic subset of P C(+).

Over V[G] instead of the forcing Q, in the previous construction, use Col(, ) the Levy

collapse of all the cardinals , < < on . Let H be a generic subset of Col(, ).

Denote by H() the generic function from on where (, ).

Now the forcing for shooting clubs should come. In order to prevent collapsing

cardinals by this forcing, j() was made a limit of + indiscernibles for the measures

{U(, )| < + +}. But now we have only measures. So the best we can do

is to make j() a limit of indiscernibles and then its cofinality in V will be < . It

looks slightly paradoxical since usually cf j() = +, but it is possible by [G5]. In order

to explain the idea of [G5] which will be used here let us give an example of a precipitous

ideal I on 1 so that:

() I+

cfV(j(1)) = .

Example. Suppose that is a measurable cardinal. Let U be a normal measure on

and j : V N = V/U the canonical embedding. Let P be the Backward-Easton

iteration of the forcings C(+) for all regular < . Let G H be a generic subset ofV.

Collapse to 1 by the Levy collapse. Let R be V[G H]-generic subset of Col(1, ).

Denote V[G H R] by V1. We shall define a precipitous ideal satisfying () in V1.

Let j0 = j, N0 = N, 0 = , U0 = U. Set N1 = Nj0()0 /j0(U), 1 = j0(), U1 = j0(U)

and let j1 : N0 N1 be the canonical embedding. Continue the definition for all n < .

Set j, N to be the direct limit of < jn, Nn|n < >. Then j() =

n


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Define a filter U in V[G H R] as follows:

A U iff for some r G H R for some n < in N

r pn j(A

)

where pn is the name of condition in the forcing C(j(+)) defined as follows:

let h

be the name of the generic function from onto (+)V in Col(, j())/G H R.

Set pn = {< j(h

(0)), , 1 >, < j(h(1

)), , 2 >, . . . , < j(h

(n)), , n+1 >}. The

meaning of the above is that the value on of j(h(m))-th function from j() to j()

is forced to be m+1.

It is not difficult to see now that U is a normal precipitous ideal on 1 and a generic

ultrapower with it is isomorphic to a V[GH R]-generic extension of N[G, H, R].

Also for a generic embedding j, j|V = j. Hence j() = which is of cofinality in

V.

As in [J-M-Mi-P], it is possible to extend U to the closed unbounded filter with the

same property. Using the Namba forcing, it is possible to construct a precipitous ideal

satisfying () on 2. Starting from a measurable which is a limit of measurable, it is

possible to build such an ideal over an inaccessible or even measurable. Since then it is

possible to change the cofinality of the ordinal of cofinality + to in N and that is what

was needed to catch all ns in the above construction. We do not know if one measurable

is sufficient for a precipitous ideal satisfying () over > 2.

Note also that by Proposition 1.5 ifI is -preserving, then cfVj() . In particular,

if I is presaturated then cfVj() = +.

Let us now return to the construction ofN S+ 1-preserving. Let < + be

an ordinal. Define M as in the construction of N S 1-preserving for an inaccessible .

Now cfVi() will be . We shall define, in V[G, H], U(, ) extending U(, ) satisfying

conditions (i) and (ii) from page 23. To do so simply combine the definition there with the

definition of the example. We leave the details to the reader. The rest of the construction

does not differ from an inaccessible cardinal case.

The above results give equiconsistency for NS| (singular) or something close to

equiconsistency for N S, but for > 2. For = 2, we do not know if the assump-

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tion of the existence of an (, 1 +1)-repeat point (or of an (, 1)-repeat point for N S02

)

can be weakened. By [G3], a measurable is sufficient for the precipitousness of N S02 and

a measurable of order 2 for N S2 . Let us show that 1-preservingness requires stronger

assumptions.

Lemma 2.5. Suppose that > 1 is a regular cardinal, 2 = +, 20 = 1. I is a normal

1-preserving ideal over so that { < |cf = } I. Then 0() 2 in the core

model.

Proof: Without loss of generality, assume that 0() = ++. Denote by K( F) the

core model with the maximal sequence F. We refer to Mitchell papers [Mi 1,2] for the

definitions and properties of K( F) that we are going to use.

The set A = { < | is regular in K( F) and of cofinality in V} is I-positive.

If the set A = { A| is not measurable in K( F)} is bounded in , then 0() 2.

Suppose otherwise. Let j : V M be a generic elementary embedding so that the set

{ < |cf = } belongs to a generic ultrafilter GI. Then j(A) is unbounded in j().

Pick the mimimal j(A) .

Claim. cfK(F)() = (+)K(

F).

Proof: Suppose otherwise. Then, since is regular in K( F) and j|K( F) is an iterated

ultrapower of K( F) by F, is a limit indiscernible of this iteration. By Proposition 1.5.

M V[GI] M. Since is not a measurable in K(j( F)), it implies that cfK(F) > .

But then, using M V[GI] M and the arguments of [Mi 2] and [G4], we obtain some

with 0F() (1)K

( F). of the claim.

By [Mi 2], (+)K(F) = (+)V. But in V[GI], cf = and hence cf(

+)V = which

contradicts Proposition 1.5.

Lemma 2.6. Suppose that > 1 is a regular cardinal, 2 = +, 20 = 1 and I is a

normal 1-preserving ideal. If there exists an -club C so that every C is regular in

K( F), then 0F() 1 in K( F).

The proof is similar to Lemma 2.5; just consider the 1-th member of j(C) .

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Theorem 2.7. Assume GCH, if N S02 is 1-preserving then 0F() 1 in K( F).

The proof follows from Lemma 2.6.

References

[A] U. Avraham, Isomorphism of Aronszajn trees, Ph.D. Thesis, Jerusalem, 1979.

[B] J. Baumgartner, Independence results in set theory, Notices Am. Math. Soc.

25 (1978), A 248-249.

[B-T] J. Baumgartner and A. Taylor, Ideals in generic extensions. II, Trans. of Am.

Math. Soc. 271 ( 1982), 587-609.

[G-J-M] F. Galvin, T. Jech and M. Magidor, An ideal game, J. of Symbolic Logic 43

(1978), 284-292.

[G1] M. Gitik, Changing cofinalities and the nonstationary ideal, Israel J. of Math.

56 (1986), 280-314.

[G2] M. Gitik, SC H from 0() = ++, Ann. of Pure 43 (1989), 209-234.

[G3] M. Gitik, The nonstationary ideal on 2, Israel J. of Math. 48 (1984), 257-288.

[G4] M. Gitik, Some results on the nonstationary ideal, Israel Journal of Math.

[G5] M. Gitik, On generic elementary embeddings, J. Sym. Logic 54(3) (1989),

700-707.

[H-S] L. Harrington and S. Shelah, Equiconsistency results in set theory, Notre

Dame J. of Formal Logic 26(2) (1985), 178-188.

[J-M-Mi-P] T. Jech, M. Magidor, W. Mitchell and K. Prikry, Precipitous ideals, J. Sym.

Logic 45 (1980), 1-8.

[Mi 1] W. Mitchell, The core model for sequences of measures I, Math. Proc. Camb.

Phil Soc. 95 (1984), 229-260.

[Mi 2] W. Mitchell, The core model for sequence of measures II, to appear.

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[Mi 3] W. Mitchell, Indiscernibles, skies and ideals, Contemp. Math. 31 (1984),

161-182.

[S1] S. Shelah, Proper forcing, Springer-Verlag Lecture Notes in Math, 940 (1982).

[S2] S. Shelah, Some notes on iterated forcing with 20 > 2, Notre Dame J. of

Formal Logic.

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moti gitik and saharon shelah- cardinal preserving ideals

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