Renling Jin and Saharon Shelah- Possible Size of an Ultrapower of omega

8/3/2019 Renling Jin and Saharon Shelah- Possible Size of an Ultrapower of omega

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Possible Size of an Ultrapower of 1

Renling Jin2 & Saharon Shelah3

AbstractLet be the first infinite ordinal (or the set of all natural numbers) with theusual order


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[15] is recommanded for the general theory of the Boolean ultrapower of arbitrary

models. Here we give the definitions and facts needed in this paper to keep it self-

contained.

Throughout this paper we use ,,, etc. for infinite cardinals, , , , etc. for

ordinals, and k,l,m,n, etc. for natural numbers. Let A and B be two sets. We

denote by AB for the set of all functions from B to A (except in the case when B is

a Boolean algebra). We also write for exponents in cardinal arithmetic, and this

should be clear from the context. Let P() be the set of all subsets of of size < .

Let D, E, F, etc. denote filters or ultrafilters, and let B,C, etc. denote Ba or cBa,

i.e. Boolean algebras or complete Boolean algebras.

We shall not distinguish a Ba (B; , , , 0, 1) from its base set B. For any S B

let

S (

S) be the least upper bound (greatest lower bound) of S in B, provided

it exists. By an anti-chain in B we mean a subset A B such that for any a, b A,

a = b implies a b = 0. A Ba B has -c.c. iff every anti-chain A in B has size < .

1-c.c. is called also c.c.c.

We write CB to denote that C is a subalgebra of B and for any S

C, if

S = c in C and

S = b in B, then b = c. Hence we shall not distinguish

and

in C or in B.

A B is called a -complete Ba or a -cBa iff for any S B, |S| < implies

S B.

B is complete iff it is -complete for every . 1-complete is also called countably

complete. Given a B, let a0 denote a and a1 denote a. Let C B. Then a

sequence {a : < } in B is called -independent over C iff for any P(), for

any h 2 and for any c C {0} one has

c (

{ah() : }) = 0.


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A sequence in B is -independent iff it is -independent over {0, 1}. A sequence

{C : < } of subalgebras ofB is called -independent iff for any P() and

for any a C {0}

a = 0.

We shall omit when = . Let B be -complete. Then for any S B we denote

by S B the -complete subalgebra ofB generated by S. For any B let B denote

the completion ofB. A Ba B is called -free iff there exists a -independent sequence

{a : < } in B such that

B = {a : < }.

The cardinal above is called the dimension ofB. Note that if


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Suppose D is a filter on and let I be the dual ideal ofD. We often write P()/D

instead of P()/I, the quotient Boolean algebra of P() modulo I. For any A

let [A]D denote the equivalence class of A in P()/D.

Suppose D is a countably complete filter on . Then the Boolean algebra B =

P()/D is countably complete, and for any An one has

[

n

An]D =

n

[An]D.

Let E be an ultrafilter on B and let

F = {A : [A]D E}.

Then it is easy to see that F is an ultrafilter on extending D. We want to showthat B/E and /F are isomorphic.

For any f let f be a function from to B such that f(n) = [f1(n)]D. Then

it is easy to check that f B. Let i be the map from /F to B/E such that

i(fF) = fE.

Lemma 0.1. (folklore)

The map i is an isomorphism from /F to B/E.

Proof: We show first that i is a well-defined one-one function, which preserves the

order. This is because that for any two functions f, g , one has

fF < gF { < : f() < g()} F

m


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Then we show that i is onto. Given any t B, by countable completeness we

could choose An inductively such that {An : n } is a partition of and

[An]D = t(n). Define an f such that f() = n iff An. Clearly, i(fF) = tE.

2

Remark 0.2. By this lemma to find an ultrafilter F on such that /F has some

desired properties, it suffices to find a countably complete filter D on and an ultra-

filter E onB = P()/D such that B/E has such properties.

1. Strong Limit Cardinal as the Size of an Ultrapower of

An ultrafilter F on is called uniform iff |A| = for every A F. An ultrafilterF on is called regular iff there exists a family C F such that |C| = and for any

infinite subfamily C0 C,

C0 = .

The study of the cardinality of ultrapowers started in the 1960s. The subject is very

closely related to the regularity of ultrafilters. Regular ultrafilters were introduced in

[4] and [7]. It was proved there that if F is a regular ultrafilter on , then /F has

size 2. So it is natural to ask whether it is possible to have an ultrafilter F on

such that the size of /F does not have the form 2 for any . In fact, Chang and

Keisler asked this in the following forms (see page 252 of [1]).

Question 1.1. Is it possible that the cardinality of an ultrapower of is a singular

(strong limit) cardinal?

Question 1.2. Is it possible that the cardinality of an ultrapower of is a strongly

inaccessible cardinal?

The original form of Question 1.1 in [1] has no requirement for the cardinality to

be a strong limit cardinal. Since the cardinal 2 could be singular, we would like to

make it more specific by requiring the singular cardinality be also a strong limit.


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Obviously, a positive answer to either Question 1.1 or Question 1.2 would imply

the existence of a uniform non-regular ultrafilter. Since the existence of a uniform

non-regular ultrafilter was unclear in the early 1970s, when the first edition of [1] was

published, people were more interested in a general question of Chang and Keisler

concerning the existence of uniform non-regular ultrafilters. A lot of work has been

done since then for solving the general question. On one hand, Prikry [17], Ketonen

[9], Donder [2], etc. showed that one may need to assume the consistency of some

large cardinals to construct a uniform non-regular ultrafilter. For example, Donder

proved that if there is no inner model containing a measurable cardinal, then (1) every

uniform ultrafilter on a singular cardinal is regular, (2) every uniform ultrafilter on

a regular cardinal with (+)K = + is regular, where K is the Dodd-Jensen core

model. On the other hand, Magidor [14], Laver [13], Foreman, Magidor and Shelah [3],

etc. showed that it is possible to construct a uniform non-regular ultrafilter with the

help of some large cardinal axioms. For example, in [3] it is proved that, assuming

is a huge cardinal and < is a regular cardinal, there is a forcing extension

preserving every cardinal , in which there exists a uniform (fully) non-regular

ultrafilter on +. However, as far as we know, neither Question 1.1 nor Question 1.2

has been answered yet. In this section we are going to give positive answers to both

questions by assuming the consistency of a supercompact cardinal. First, we need to

introduce the Laver-indestructibility of a supercompact cardinal. See [5] or [6] for the

basic facts of supercompact cardinals

A supercompact cardinal is called Laver-indestructible iff remains supercompact

in any -directed closed forcing extension. The reader should consult [12] to see how

one makes a supercompact cardinal Laver-indestructible by a -c.c. forcing of size .


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Theorem 1.3. Suppose is a Laver-indestructible supercompact cardinal. Let

be such that


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where we denote A0 = A and A1 = A for A . The existence of such a sequence

is guaranteed by


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We now want to show that

Ah() A I, where = dom(h). Suppose not.

Then

(

Ah() A) Hg.

Hence there exists h P such that

h

Ah() A H g.

This implies

h

Ah() H g and h A Hg.

Let = dom(h). For any one has

h Ah() H g and h Ah

() Hg.

Hence h() = h(). So h h P. But this contradicts

h A Hg. 2(Claim 1.6.2)

Let a = [A]D. By Claim 1.6.1 the sequence {a : < } is -independent. Let

C = {a : < }. Then by Claim 1.6.2 C is dense in B = P()/D. 2

Now we are ready to prove the theorems.

Proof of Theorem 1.3: Let 2 be such that = . Let D be a -complete

filter on obtained in Lemma 1.6 such that the Ba B = P()/D has a -free dense

subalgebra C B of dimension . It is clear that |C| = and every element a in B

is the least upper bound of an anti-chain in C. Since C has +-c.c., every anti-chain

in C has size . Hence |B| |C| = = . This shows that for any ultrafilter E

on B one has |B/E| = .

We now want to show that there exists an ultrafilter E on B such that |B/E| .

Let {a,n : < , n } be a -independent sequence in C such that

C = {a,n : < , n }.


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For each < let t B be such that

t(0) = (

n

a,n) and t(n + 1) = a,n (

m


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Note that for any successor C is atomic, for any limit C is not -complete, and

for any < |C| < . It is easy to see that C =


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Clearly, {(t)E : < } is a strictly increasing sequence of length in B/E. Hence

|B/E| . We need to show that |B/E| .

Claim 1.5.2. For any maximal anti-chain {b : < } in C there exists a <

such that

{b : < } E.

Proof of Claim 1.5.2: Note that |C| < for any < . Using the inaccessibility

of one could show that there exists a < such that {b : < } is a maximal

anti-chain in C. Hence

{b : < } E. 2(Claim 1.5.2)

Claim 1.5.3. For any t B there exists an s C such that tE = sE.

Proof of Claim 1.5.3: Since C is dense in B and C has +-c.c., then there exists

a maximal anti-chain {b : < } in C, which refines {t(n) : n }, i.e. for any

< there exists an n such that b t(n). By Claim 1.5.2 there is a <

such that

{b : < } E. Define s C such that

s(0) = (

{b : < , b t(0)}) (

{b : < })

and

s(n + 1) =

{b : < , b t(n + 1)}

for every n . Here we use the fact that C is -complete and < . It is now easy

to check that

[[t = s]] =

n

(t(n) s(n))

{b : < } E.

Hence |B/E| |C| = . Now the theorem follows from Lemma 0.1. 2

2. -Archimedean Ultrapowers

Let L be any first-order language including symbols for number theory, say, N, +, ,

such that


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Proof: By Lemma 1.6 there exists a -complete filter D on such that B = P()/D

contains a -free dense subalgebra C of dimension . Without loss of generality let

C = {a,n : < , n },

where {a,n : < , n } is a -independent sequence in C. For each let

C = {a,n : < , n }

and let B = C. Note that C0 = B0 = {0, 1}. Since C has +-c.c., so does B. This

implies that

B = B =

. Also for each < one has

|C| ||


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Hence

a (

n


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where B = B and E = E, because B =

. Hence |B/E| .

On the other hand, if x B/E, then there is a < such that x B/E. Hence

|{y B/E : y x}| |B/E| |B| < .

This shows that B/E is -Archimedean. Now the theorem follows from Lemma 0.1

2

Next we show a different way to construct -Archimedean ultrapowers. The con-

struction of a -Archimedean ultrapower in Theorem 2.8 needs only to assume a

measurable cardinal.

Lemma 2.7. Suppose and is a -supercompact cardinal in V. Let U be a

-complete normal ultrafilter on P() in V and let j be the elementary embedding

from V to M = VP()/U induced by U. Suppose B V is a cBa and j(B) = B C,

i.e. B = {j(p) : p B} is completely embedded into j(B). Then in VB there exists a

-complete filter D on (P())V such that P((P())V)/D = C.

Proof: Let G B be a V-generic filter and let H C be an M[G]-generic filter.

Let j be the embedding from V[G] to M[G][H] defined by letting j(x) = (j(x))GH.

It is well-known that j is an elementary embedding. Note that M[G] V[G] but

M[G][H] is generally not a subclass of V[G]. For any A (P())V in V[G] let

i(A) = [[j (j(A))G]]C,

where

A is aB

-name for A and j

= {j() : < } M

M.

Claim 2.7.1. i is a Boolean homomorphism from P((P())V) to C in V[G].

Proof of Claim 2.7.1: It suffices to show that i is well-defined function. The rest

follows from the fact that j is an elementary embedding. Suppose A1 and A2 are two


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B-names for same set A in V[G]. Then there exists a p G such that

p B A1 = A2.

By the fact that j(p) = p one has

p BC j(A1) = j(A2)

or

p B C j(A1) = j(A2).

Since p G, then

V[G] |= C (j(A1))G = (j(A2))G.

Hence in V[G]

[[j (j(A1))G]]C = [[j (j(A2))G]]C. 2((Claim 2.7.1)

Lets define a filter D in V[G] by letting

D = {A (P())V : i(A) = 1C}.

It is easy to see that D V[G] is a -complete filter on (P())V.

Claim 2.7.2. P((P())V)/D = C in V[G].

Proof of Claim 2.7.2: It suffices to show that i is an onto map. Given any a C,

let a be expressed as [F]U for some F : P() B in V such that F is not equal

to any single p B modulo U. Let A be a B-name for a subset of P() in V such

that for each P() one has [[ A]]B = F(). Since j = [d]U, where d is the

identity function from P() to P(),

[[j (j(A))G]]C = [[[d]U (j(A))G]]C = [[[ A]]B : P()]U =

[F() : P()]U = [F]U = a.

Hence i is an onto map from P((P())V) to C. 2


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Theorem 2.8. Suppose is a measurable cardinal such that < 2 for any < 2.

LetB be the Boolean algebra for adding Cohen reals or adding random reals.

Then in VB there exists an ultrafilter F on such that /F is 2-Archimedean.

Proof: Obviously, is -supercompact. Let j be the elementary embedding induced

by a -complete normal ultrafilter U on . For any let B denote the cBa for adding

Cohen reals (or random reals). Then j(B) = Bj() = B C, where C is isomorphic

to the cBa for adding 2 Cohen (or random) reals in VB because |j()| = 2. By

Lemma 2.7 there exists a -complete filter D on such that

P()/D = C

in VB . For any let B denote the cBa for adding Cohen reals (or random reals)

in VB . Note that C = B2 . Since B has c.c.c. for any , then B

2 =

with


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References

[1] Chang, C.C. and Keisler, H.J., (1990) Model Theory 3rd ed., North-Holland, Amsterdam, (2nded., 1977).

[2] Donder, H.-D., (1988) Regularity of ultrafilters and the core model, Israel Journal of Mathe-matics, 63, pp. 289322.

[3] Foreman, M., Magidor, M. and Shelah, S., (1988) Martins Maximum, saturated ideals, andnon-regular ultrafilters. Part I I, Annals of Mathematics, 127, pp. 521545.

[4] Frayne, T. E., Morel, A. C. and Scott, D. S., (1962) Reduced directed products, FundamentaMathematicae, 51, pp. 195228.

[5] Jech, T., (1978) Set Theory, Academic Press, New York.[6] Kanamori, A., Reinhardt, W. and Solovay, R., (1978) Strong axioms of infinity and elementary

embeddings, Annals of Mathematical Logic, 13, pp. 73116.[7] Keisler, H. J., (1964) On cardinalities of ultraproducts, Bulletin of American Mathematical

Society, 70, pp. 644647.[8] Keisler, H. J. and Schmerl, J. H., (1991) Making the hyperreal line both saturated and complete,

The Journal of Symbolic Logic, 56, pp. 10161025.[9] Ketonen, J., (1976) Non-regular ultrafilters and the cardinality of ultrapowers, Transactions of

American Mathematical Society, 224, pp. 6173.[10] Koppelberg, S., (1980) Cardinalities of ultraproducts of finite sets, The Journal of Symbolic

Logic, 45, pp. 574584.[11] Kunen, K., (1980) Set Theoryan introduction to independence proofs, NorthHolland, Ams-

terdam.[12] Laver, R., (1978) Making the supercompactness of indestructible under -directed closed

forcing, Israel Journal of Mathematics, 29, pp. 385388.[13] Laver, R., (1982) Saturated ideals and non-regular ultrafilters, Patras Logic symposion

(Metakides, ed.), North-Holland, Amsterdam, pp. 297305.[14] Magidor, M., (1979) On the existence of non-regular ultrafilters and the cardinality of ultra-

powers, Transactions of American Mathematical Society, 249, pp. 97111.[15] Mansfield, R., (1971) The theory of Boolean ultrapowers, Annals of Mathematical Logic, 2,

pp. 297323.

[16] Monk, J. D. and Bonnet, R. (editors), (1989) Handbook of Boolean Algebras, Volume I, NorthHolland.

[17] Prikry, K., (1970) On a problem of Gillman and Keisler, Annals of Mathematical Logic, 2,pp. 179187.

[18] Shelah, S., (1970) On the cardinality of ultraproducts of finite sets, The Journal of SymbolicLogic, 35, pp. 8384.

[19] Shelah, S., (1978) Classification Theory and the Number of Nonisomorphic Models, NorthHolland, Amsterdam.

Department of Mathematics, College of Charleston, Charleston, SC 29424

e-mail: [email protected]

Institute of Mathematics, The Hebrew University, Jerusalem, Israel

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903

Department of Mathematics, University of Wisconsin, Madison, WI 53706


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Sorting: The first address is the first authors and the last three are the second

authors.

Renling Jin and Saharon Shelah- Possible Size of an Ultrapower of omega

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