Inner Models and Large Cardinals [Reprint 2011 ed.] 9783110857818, 9783110163681

Table of contents :
Preface
1 Fine Structure
1.1 Acceptable J-Structures
1.2 The Σ1-Projectum
1.3 Downward Extension of Embeddings Lemmata
1.4 Upward Extension of Embeddings Lemma
1.5 Iterated Projecta
1.6 Σ*-Relations
1.7 Σ0(n)-Embeddings
1.8 Substitution and Good Functions
1.9 Standard Parameters
1.10 Two Applications to L
1.11 More on Downward Extensions of Embeddings
1.12 Witnesses and Solidity
Notes
2 Extenders and Coherent Structures
2.1 Extenders
2.2 The Hypermeasure Representation of Extenders
2.3 Amenability
2.4 Coherent Structures
2.5 Extendibility
2.6 Strong Cardinals
Notes
3 Fine Ultrapowers
3.1 The *-Ultrapower Construction
3.2 Some Special Preservation Properties
3.3 When F Is Close to M
3.4 Extendibility
3.5 k-Ultrapowers
3.6 Pseudoultrapowers
Notes
4 Mice and Iterability
4.1 Premice
4.2 Iterations
4.3 Copying and the Dodd-Jensen Lemma
4.4 Comparison Process
4.5 Some Iterability Criteria
4.6 Bicephali
Notes
5 Solidity and Condensation
5.1 Cores and Coiterations
5.2 The Solidity Theorem
5.3 Consequences of Solidity
5.4 The Canonical Ordering of Mice
5.5 Condensation Lemma
5.6 Upwards Extensions to Premice
Notes
6 Extender Models
6.1 Extender Models and Iterations
6.2 The Canonical Ordering of Weasels
6.3 Universality
6.4 The Model Kc
6.5 0**
6.6 Weak Covering
Notes
7 The Core Model
7.1 Inductive Definition of K
7.2 Steel’s Definition of K
7.3 The Existence of K
7.4 Embeddings of K and Generic Absoluteness
7.5 Weak Covering for K
Notes
8 One Strong Cardinal
8.1 Premice
8.2 Properties of Mice
8.3 Extender Models up to One Strong Cardinal
Notes
9 Overlapping Extenders
9.1 Premice and Iteration Trees
9.2 Copying and the Dodd-Jensen Property
9.3 Solidity and Condensation
9.4 Uniqueness of Weil-Founded Branches
9.5 Towards the Ultimate Model Kc
Notes
Bibliography
Index

Citation preview

de Gruyter Series in Logic and Its Applications 5 Editors: W. A. Hodges (London) · R. Jensen (Berlin) S. Lempp (Madison) · M. Magidor (Jerusalem)

Martin Zeman

Inner Models and Large Cardinals

W G DE

Walter de Gruyter Berlin · New York 2002

Author Martin Zeman Department of Mathematics University of California at Irvine 261 Multipurpose Science & Technology Bldg. Irvine, CA 92697 USA Series Editors Wilfrid A. Hodges School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London El 4NS, United Kingdom

Ronald Jensen Institut für Mathematik Humboldt-Universität Unter den Linden 6 10099 Berlin, Germany

Steffen Lempp Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388, USA

Menachem Magidor Institute of Mathematics The Hebrew University Givat Ram 91904 Jerusalem, Israel

Mathematics Subject Classification 2000: 03-02; 03E35; 03E45; 03E55 Keywords: Inner model, Extender, Fine structure. ©

Printed on acid-free paper which falls within the guidelines of the A N S I to ensure permanence and durability

Library of Congress — Cataloging-in-Publication Data Zeman, Martin, 1964— Inner models and large cardinals / Martin Zeman. p. cm. - (De Gruyter series in logic and its applications ; 5) Includes bibliographical references and index. ISBN 3-11-016368-3 1. Constructibility (Set theory) 2. Large cardinals (Mathematics) I. Title. II. Series. QA248 .Z46 2001 5U.3'22—dc21 2001047562

Die Deutsche Bibliothek — Cataloging-in-Publication Data Zeman, Martin: Inner models and large cardinals / Martin Zeman. - Berlin ; New York : de Gruyter, 2001 (De Gruyter series in logic and its applications ; 5) ISBN 3-11-016368-3

ISSN 1438-1893 © Copyright 2001 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting using the Authors' T£X files: I. Zimmermann, Freiburg — Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen - Cover design: Rainer Engel, Berlin.

Preface

The purpose of this book is to give an introduction to the theory of fine structural inner models. These models are constructible relative to coherent extender sequences and admit a fine structure theory analogous to that of Gödel's constructible universe. The theory of such models originates in Jensen's analysis of the constructible universe and in the work of Dodd and Jensen on the core model, and was later pursued by Jensen, Mitchell, Steel and others. It is not possible to explain the point of inner model theory on just a few lines, but I would like to draw attention to at least one of its aspects. In many cases, this theory can be viewed as a counterpart to the theory of forcing in the following sense. While forcing allows us to construct various kinds of set-theoretical objects from large cardinals, inner model theory enables us to analyze the complexity of such objects in terms of large cardinals. Such analysis can be then used to determine lower bounds for the consistency strengths of various set-theoretic principles. Let me illustrate this with the following example. It is well-known that Prikry forcing makes a measurable cardinal ω-cofinal. On the other hand, inner model theory can be used to show that measurability is necessary for this task: If a regular cardinal can be turned into an ωcofinal one, then this cardinal must be measurable in an inner model. Other examples involve, for instance, the consistency strength of the failure of the singular cardinal hypothesis, that of the failure of the combinatorial principle square or the consistency strength of projective uniformization. There is also a body of applications to descriptive set theory, that are of a slightly different nature. Here, various definable subsets of reals can be characterized in terms of inner models. For instance, the set C2„+2 is precisely the set of reals of the minimal inner model for η Woodin cardinals, and scales for various pointclasses can be directly constructed using inner models, avoiding any reference to determinancy. My intention is to give an account of inner model theory that would serve as an introductory text for those not familiar with the field and that could also serve as a text book for graduate students. At the same time, I attempt to provide the reader with a glimpse of some current developments. The book can informally be divided into three parts. The first part comprises Chapters 1 through 3 and gives a detailed introduction to the general fine structure theory of acceptable structures. This theory can be developed abstractly, without any reference to inner models or large cardinals. The second part, consisting of Chapters 4 through 8, contains both the full core model theory for measures of order 0 as well as instructions for generalizing this theory for models that can contain up to one strong cardinal (Chapter 8). The third part of the book is just Chapter 9 and provides an introduction to the theory of Jensen extender models that are beyond one strong cardinal. The main difference between the theories below and beyond one strong cardinal rests in the complexity of the comparison process. In the former case, comparison arguments can be carried out using mere linear iterability of the models in

vi

Preface

question, which considerably simplifies the whole theory. One can say that inner model theory at this level has reached its definitive version. In the latter case, however, the comparison process is highly non-linear and the essence of the iterability problem is the choice of cofinal well-founded branches through iteration trees. This problem belongs to one of the central open questions in inner model theory, as all iterability proofs known today can be carried out only under smallness conditions ranging at the level of (several) Woodin cardinals. In this book, however, we will not touch this problem at all, and merely derive various consequences of iterability that determine the inner structure of extender models. This exposition is based on Jensen's handwritten notes [Je88], [Jel] and [Je97], The presentation of the material, however, often diverges from that in Jensen's notes, as it takes into account later developments. Most of the results presented here (and all relevant ones) are due to Dodd, Jensen, Mitchell, Steel and others, and only a few techniques from Chapter 6 and 9 are due to me; a more detailed account of credits can be found in notes at the end of each chapter. Also, many proofs that I present can be replaced by much simpler proofs if we work with measures of order 0, so the reader might get an impression that I made the things more complicated than they really are. My intention, however, is to present methods that easily generalize to higher models. I should also include some words on how to use this book. In first three chapters I give a very general account of fine structure theory. Knowledge of this theory helps to understand fine structural aspects of inner models in general; however, this generality is superfluous if we work at the level of measures of order 0. This might confuse those readers who are beginners in fine structure theory. For this reason, I recommend first reading the beginning of Chapter 1 that precedes the exposition of Σ *-relations, then go directly to Chapter 2 and after that learn only the basic facts about fine ultrapowers from the beginning of Chapter 3. To make the first reading of Chapter 3 easier, it is helpful to assume that η — 0, i.e. that mQ]m < κ where κ is the critical point of the ultrapower in question. The above assumption enables the reader to avoid the use of Σ *-theory (or any substitute for it), as fine ultrapowers are nothing more than coarse ultrapowers in this case. Having accumulated the knowledge of the above-mentioned material, one can proceed directly to Chapter 4 where the theory of extender models for measures of order 0 begins. Later in the course of the reading, one can return back to the first three chapters to learn new facts about general fine structure theory and fine ultrapowers; this should be done precisely at those points where deeper knowledge of this theory is necessary for understanding the arguments. Chapters 4 through 7 constitute a compact exposition of the core model theory for measures of order 0. Although it is possible to skip over various parts of them, the full understanding of the theory requires the knowledge of the methods used in the proofs of the basic lemmata on the comparison process, that of the solidity theorem, condensation lemma and the weak covering theorem, as well as the construction of Κ and the verification of its basic properties. These topics, however, constitute the bulk of the four chapters mentioned above. Chapter 8 is intended for those readers who would like to work with inner models up to one strong cardinal and avoid non-linear iterations. As we have already mentioned, it contains a recipe for generalizing the theory for measures of order 0 to the core model

Preface

vii

theory up to one strong cardinal. This generalization is merely an elaboration on the previous theory and Chapter 8 provides the reader with only those lemmata that do not follow from the methods developed previously in a straightforward way. Thus, the reading of this chapter requires a reasonable knowledge of the previous theory. Chapter 9 is an introduction to Jensen fine structure theory for higher extender models; this theory is based on the so-called λ-indexing. The chapter builds on the fine structure theory developed in the first three chapters of the book in its full generality, but is relatively independent of the remaining chapters. This should enable the reader familiar with the basics of inner model theory who is interested in Jensen extender models to proceed to Chapter 9 directly. I choose to present Jensen extender models based on λ-indexing and *-iterability that is also based on λ-indexing, as this considerably simplifies the general iterability theory including the Neeman-Steel lemma, thus somewhat facilitating the reader's first contact with such models. Acknowledgments. I would like to thank all those people who encouraged, helped and supported me during the preparation of this book. Since the list of these people would be too long, I mention just three of them, to whom I am particularly indebted: Ronald Jensen, from whom I learned set theory, Ralf Schindler, who shared the office with me during the last two years and who was always free to discuss the topic with me and Michael Bruening, who proofread the whole text. I am also grateful to the following institutions, where various parts of this book were prepared and/or which provided me with a financial support: Humboldt University Berlin (Mathematical Institute), Friedrich-Wilhelm University Bonn (Mathematical Institute), Vienna University (Institute of Formal Logic), Slovak Academy of Sciences (Mathematical Institute) and Deutsche Forschungsgemeinschaft (DFG). Bratislava and Vienna, September 2001

Martin

Zeman

Contents

Preface

ν

1

Fine Structure 1.1 Acceptable J-Structures 1.2 The Σι-Projectum 1.3 Downward Extension of Embeddings Lemmata 1.4 Upward Extension of Embeddings Lemma 1.5 Iterated Projecta 1.6 Σ *-Relations 1.7 Σ ωξ for any r ' < ωξ whose power set in is larger than that in S ^ (since τ ' > r ) and it follows immediately that f T i e ^>ωξ+η+ν s o

we

have a uniform bound for all such functions.

If the height of Μ is ωα for some limit a, then acceptability is equivalent to the statement (QSHS&

1= Ψ).

Otherwise we have to state (1.1.1) explicitly for the last level. Hence, the desired condition is then {U, A, B) is an amenable ./-structure & ( ß ? ) ( l i m ( ? ) — • J^ l—iΙΛ \ = f ) X, & \-\(r [ ( ß ? ) ( f ic is limit"! limit) \/ ν ml φ] where φ is the sentence ( ß f ) ( 3 β < C)[lim(/3) & Of η < ζ)(η > β

succfo)) & φ'(β, ζ)]

(1.1.2)

1.1 Acceptable J-Structures

5

and φ'(β, ζ) the formula (Vr < ß)[(3u e S*)(u i

& u c r ) —• (3 / e S*)(f

:τ ^

ωβ)].

φ is clearly a Q-sentence, hence (1.1.2) is a Μ and Μ is acceptable, then so is M. Σι

b) If π : Μ —• Μ and Μ is acceptable, then so is Μ. This holds in particular if π is α Σο cofinal map. Lemma 1.1.5. Let Μ = be acceptable and let ωρ e Μ be a cardinal in Μ. Given u e Jg, any a € Μ which is a subset of u is in fact an element of Jg . Proof Since u e there is a surjective map g : r —> u in for some r < ρ. Set ä = g~l"a. Then a d z and a e ä e . But if ä $ J^, then there is an / :τ ωξ, where ξ is such that ä e — \ notice that ξ > ρ. This contradicts the fact that ωρ is a cardinal in . Consequently, a e . •(Lemma 1.1.5) Lemma 1.1.6. Let Μ be as above and ωρ be a successor cardinal in M. Let a c Jg be such that card (α) < ωρ in Μ. Then a e Proof. Let γ < ωρ and g e Μ be such that g : γ ζ = /(£)

• «(?) e Sf

a. Define / : γ —>• ωρ by A +1

- Sf.

Then / e M. Claim, f is bounded in ωρ. Let • Μ such that π D π andn{p) Σο

Then there is a unique π :

° = p. Moreover, π : Μ —> Μ. Σ]

Proof. Uniqueness. Assume that π has the above properties. Let χ e M. Then x = hjj(i, {ξ, p)) for some i e ω and ξ < ωρΜ• Let Η be Σο (Μ) such that

9

1.3 Downward Extension of Embeddings Lemmata

(3ζ)Η(ζ,χ,ί,ξ, ρ) defines the Skolem function h ^ (this involves a slight abuse of notation). Η has a uniform definition, so let Η have the same definition over M. Pick ζ such that H(z, x, i, ξ, ρ). Since ft is Σο-preserving, we have H(ft(z), fr(x), i, ft (ξ), ρ), i.e. π(χ) — Ημ(ϊ, (ft(ξ), ρ)) — Ιιμ(ϊ, ), ρ))· Hence, there is at most one such ft. Existence. The above proof of uniqueness suggests how to define the extension ft. Here we show that such a definition is correct. We first observe: Claim. Suppose that (3i, z)(i (Ξ (ο & χ — (i, z) & (3y)(y = A Ä (i, (ξ, ρ)))).

14

1 Fine Structure

So there is an ί'ο £ ω such that for every χ e Mp we have λ: e d i f f A^(io,x). Note that p the latter is a rudimentary relation over M . Similarly, the identity and membership relations as well as the membership of Β and D can be expressed in a Σι-fashion over Μ in p, and therefore in a rudimentary fashion over M p . More precisely, we introduce relations / , Ε, B* and D* over d as follows xly xEy B*(x)

k{x)

=

k(y)

k(x)

e

k(y)

^ ^

D*(x)

B{k(x)) •

D(k(x))

and set D=

(d,I,E,B*,D*).

Thus D encodes the structure M: / is a congruence relation on D, Ε represents the membership relation and k is the Mostowski collapsing isomorphism between D / / and Μ. We denote the Σ ι -satisfaction relation for D by f . (The symbol for = is interpreted in Ö as / , the symbol for e as E, and the symbols for B, D as B* and D*, respectively.) More precisely, for x\,..., χι e d and i e ω we have f ( i , (ati, ...,xe))

η+ι,χη+1,χ) where each m" +1 is either some or some and ψ is, a. propositional composition of Σ| n ) -formulae. Let Ql(z"+\x), i = 1 , . . . , m, be the relations defined over Μ by these formulae and Q(zn+l,x) be the propositional combination of the Q"s corresponding to the structure of t/r. Then can be expressed as (3wi)(VU2)...(3U>I 6MI)(VU) 2 eu2)...Qi(v,w,y)

(1.6.2)

(where each w, is either some u/, or some yi,), which is the desired form. If is Σ ι Q \ , . . . , QV)), we can write its definition as in (1.6.2), hence Qx is a propositional combination of Ql~, i = I,... ,m. If we now replace Ql~ by Q', (ν, w) by ( ΰ " + 1 , wn+l) and y by xn+l, we get the form (1.6.1), which says that R(xn+\ ..., χ °) is Σ Λί'· ' for i < η Σι and π \ Hn- : Μη·Ρ —» Μη·Ρ, where ρ e RnQ, ρ = π(ρ) and I e ω. Then

L e m m a 1.7.2. Let π \ Μ

π : Μ —>

Μ.

^e

Proof We shall apply Lemma 1.7.1 here. Let q e Γ ^ and q = n{q). Then clearly q e VnM. Since ρ e there are Ζ e H^ and j e OJ such that q(0) = h^ij, (z, p(0)». Let ζ = π (ζ). Then q (0) = hM(j, (z, p( 0))). Now Αψ\ϊ,χ)

has a uniform definition independent of x, the right side of (1.8.1) gives us the required functionally absolute

definition.

D(Lemma 1.8.10)

This together with Lemma 1.8.2 has a significant consequence. _ _ /\ Lemma 1.8.11. Let Μ, Μ be acceptable and π : Μ —> Μ α Σ[ -preserving map. Let R(vn, ...,v°)beaJ:[n)

(M) relation and R(vn,... 1

the same definition. Finally let F" (z'j ,..., n

functions and F (z'^,...,

z[),...,

z'[), • • •, F°(z\',...,

,v°) the Σ

(M) relation by

F°(z z'^) the good Σ{η)(Μ) functions by the

same functionally absolute definitions. Let äj e h'^ and aj = π (äj) Then R(h(äu

..., äe),...,

F°(ä ι,...,

ät)) ijf R{Fn{a\,

...,ae),...,

(i = 1

F°(au..

,...,£).

-, ae)).

Proof By induction on the defining schema of F. The first step is to prove this for F such that each Fl is a έ\'\μ) function to H'M. By Lemma 1.8.1, there is an effective method which, given a definition of R, say ψκ, and functionally absolute Σ definitions of Fj, say φ^, produces a Σ formula φ satisfying φ(ζ\\...,φ

Μ v(n)

and

I \π"Η"-•= w Μ

n HΜ M.

Remark. This clearly implies π \ Hn- : Mn'p —> Μη·ρ cofinally for all p, hence Σ π : Μ —>- Μ. ° Σ For Σ „ ( Μ ) relations which are subsets of H ^ we can in fact do much better: Lemma 1.8.15. Let Μ be acceptable and ρ e RnM. Let A c H^ be α Σ η + ι ( Μ ) relation. Then A is Σ j"'(M). Proof. We proceed by induction. The case η — 0 is trivial. Let us suppose now that the statement holds for η. Suppose also w.l.o.g. that« is odd. Let ρ e Rn+l and A c be a Σ„ + 2 (Μ) relation in q, i.e. A(x) (/ + l ) f o n € n). Hence (1.8.3) holds and therefore A is and hence Σ ( " + 1 ) ( Μ ) .

1.9

Σ\η>(Μρ)

D(Lemma 1.8.15)

Standard Parameters

It is easy to see that h\j([On Π M] 0 is the least such that the statement (1.9.1) fails. Hence P^ - RnM φ 0. Let q be the Μ, where m > η. Then (m-1) σ \

Μ—>M. (m)

Proof. By induction on Σ -formulae. For Σ j" 1-1 ^-formulae this holds by our assumption. The induction step for boolean combinations of formulae is trivial. Now suppose Ν and κ = cr(7r). Then F is an extender of length λ on F

T(/c) Π Μ and Ν, π are uniquely determined. Proof. Note first that F(k) = λ. Hence, once we have proved that F is an extender, we will know that its length is λ. To see that F is an extender, suppose Bai Um is uniformly primitive recursive in a\,..., am e Π Μ. Then π(κ

η Bau...,am)

= π{κ)

η βπ(α,),...,τT(am)

since π is Σο-preserving: by the fact that κ is primitive recursively closed, Ba has the same canonical Σι-definition over y^1 and V. This definition is preserved by π, since the suitability of Μ guarantees that the map π : j"u -am ),...,π(am) is elementary. Hence, F(K Π Βαχ

am) = λ η βπ·(αι),...,π(α„) = λ Π Βχηπ(αι)

ληπ(αΜ)·

50

2 Extenders and Coherent Structures

The right equality holds since λ is primitive recursively closed: Β π ( α ,) l)

3r(am)

the same canonical Σ , definition over y ^ ) - ' the first part of the lemma. Now suppose that π' : Μ —» Ν'. Note first λ Π π(χ) = F(x) = λ Π 7γ'(λ:)

(a])

, J*

n(am)

π(α,„) then has

and V. This proves

for all x e T W O M .

(2.1.1)

Define a map σ : Ν —> Ν' by σ(π (/)( Ν by π(χ) = [0, cx]

where εχ(η) = χ for all η < κ.

We shall now verify that in fact π : Μ —> Ν. First of all, Los' Theorem gives us F

immediately π \ Μ —> Ν Σο

cofinally.

(2.1.6)

In order to get λ C

(2.1.7)

w f c ( N )

we first prove if a < λ and € (α, id), then (ß, f ) = ( / , id) for some γ < a.

(2.1.8)

Clearly, [a, id] represents an ordinal in Ν. If [β, / ] represents an ordinal in Ν smaller than [ o r , id], then, by (2.1.8), [ß, f ] =N [γ, id] for some γ < a. Hence [ß, f ] = [ / , id], which means that all ordinals below a in Ν can be represented in the canonical form [ξ, id]. Moreover, id] eN [ζ, id] —> ξ e ζ. Thus, the structure N N ({[£, id]; [f, id] e [a, id]}, e ) is isomorphic to (a, e) and, hence, well-founded. Thus a c w f c ( N ) for all a < λ, which immediately gives (2.1.7). Moreover, as wfc(W) is transitive, we also get [a, id] = a. Proof

of

(2.1.8). Let

[ß, f ] e

[a, id]. By Los' Theorem,

(β,a) e F({(

Set A =

{{η\,η2);

f(rh) =

(β,a)

(2.1.9)

m

,

n 2

); / O n ) e

%})·

m}· Then

€

F ( { (

=

{ ( » 7 1 . »72); ( 3 »73 e η2)((ηι,

m

,

V 2

y A 3 m e n 2 ) ( { m , m ) £ A ) } ) m)

e

F(A))},

since the above set is uniformly primitive recursive in A. Hence there is γ e a s.t. (β, γ) e F(A), which by Los' Theorem means that (ß, f ) = (γ, id>. D(2.1.8) To see that Ν is the Zo((V)-closure of λ U rng(7r), we prove [a, f ] = ττ(/)(α)

for every

(a, f ) e D.

(2.1.10)

2.1 Extenders

Proof.

53

Note that A = {{ηι,ηι)·,

Cf(»72)(id(j?i))} =

f(,m) =

χ

κ

κ.

Hence, for every a e λ we have (a, 0} e F(A), which by Los' Theorem tells us that [a,

f ] =

[0,

id]) = π(/)(«).

cj]{[a,

P(2.1.10)

To see that κ — cr(7r)

(2.1.11)

we need to verify two things. Firstly, π f κ = id and secondly, π (κ) > a in Ν for all a < λ. The former follows from the fact that for a < κ, A = {(m>m)\

=

id(772)} =

Κ

χ {α},

hence (0, a) e F(A) = λ χ {α} and by Los' Theorem we get [0, ca] — [a, id], which means π (a) = a. The latter follows from the fact that for a < λ, {(/?i, ηι)\id(»7i)

A =

e

c

K

^

2

χ

)} = κ

κ,

hence {a, 0) e F(A) = λ χ λ and by Los' Theorem we get [a, id] eN means α eN π(κ). It only remains to prove F =

{π(χ)(1λ·,

χ

[0, cK], which •(2.1.11)

(2.1.12)

e 9 ( K ) n M ) .

So pick χ e ?(/

F([(

Ν

m

,n

weakly,

2

); f { m ) =

where

F

F

I f F is whole

is at

F

is whole

iff whenever

m))· κ,

λ.

Then

F

is

whole

iff

• ( L e m m a 2.1.4) then

π(κ)

e wfc(N)·

•(Corollary 2.1.5)

54

2 Extenders and Coherent Structures

If U, U' are suitable, U' C U and F is an extender on U with length λ, then F Γ U' is clearly an extender on U'. With this in mind, we shall briefly say that F is an extender on Μ at /c, λ if Μ is suitable for κ and F is an extender on Τ(κ) Π Μ with length λ. Definition. Let F be an extender on U of length λ. Then, for a primitive recursively closed ν < λ, F|v is an extender on U such that

(F»(jc) = F(i)nv. That F\v satisfies the definition of extender follows from the fact that υ is primitive recursively closed. Definition. Let F be an extender on U at κ, λ and A c λ be closed under Gödel's pairing function. We say that A generates F iff for every / e t / 2 and a < λ there is a g e U2 and aß e A such that e F({ κ is primitive recursively closed. The sequence F = (Fa\ a < λ) is a hypermeasure iff a) each Fa is a κ-complete ultrafilter on T(/c) Π Μ. b) {α} 6 Fafor each a < κ. c) If a < β < λ then [\ η\ € η2} e d) Let ( α ϊ , . . . , α„) e "λ, and 1 < /χ < ...,

< it < η. For χ C κ set

χ* = {', Then χ* e F

,,···,

me>

\ /(ηι)

F. = ηι) 6 F^yis>·

Remark. In a), the κ -completeness w.r.t. Μ is meant, i.e. f ] m g ( / ) e U whenever / 6 Μ and / : γ U for some γ < κ. Proof These conditions are just what we need to carry out the ultrapower construction. Such a construction can be found in [Ka94] and [MS89] and is much like the construction we did above. The conditions a) - e) enable us to prove Los Theorem for Σο-formulae; f) is called normality condition and enables to prove an equivalent of (2.1.8). We define D similarly as above with the only, rather formal, amendment: instead of a e F(x ) we write χ € Fa. The rest of the construction is then almost the same as above. We get a map π : Μ Ν and all of the properties corresponding to (2.1.9) - (2.1.12). In particular, we get: a e π(χ) -ο- χ e Fa for a < λ. Hence, if we set F(x) = π{χ) Π λ for χ e Τ(κ) (Ί Μ, it is easy to see that π : Μ —>- Ν weakly and, consequently, F is F

an extender by Lemma 2.1.1. Clearly then (Fa; a < λ) is a hypermeasure associated with F. •(Lemma 2.2.1) We now adopt the terminology, which we developed for extenders, to hypermeasures. This is useful, since, as already mentioned above, we shall later freely use both the map and the hypermeasure representation, often in the same context, if this simplifies the notation without causing confusion and/or the choice of the representation is inconsequential (e.g. if we form an ultrapower). Hence the following statements have an obvious meaning:

56

2 Extenders and Coherent Structures

a) F is a hypermeasure on Μ at κ, λ; b) π : Μ —> Ν, π : Μ —>· Ν weakly (here we can formulate propositions correF

F

sponding to Lemma 2.1.1,2.1.3 and Corollary 2.1.2); c) F is whole.

2.3

Amenability

In this section we shall prove one useful lemma concerning amenability and introduce two other sorts of amenability, which will play an important role later in connection with the so called E*-ultrapowers. Lemma 2.3.1. is whole. Proof.

Then

Given

Let π : Μ —> Ν, where F (Ν, F) is amenable.

a e

Ν,

κ

= cr(jr)

F Π a e

we have to show

is the largest

Ν.

cardinal

in Μ and

F

We first note the following two

facts: a)

F = it \

(TOO Π

Μ).

b) It is sufficient to prove F Π a e Ν for a e rng(7r), since π is cofinal in N. Suppose (x, y) e FC\a and a = n{a). Then y e | J 2 a, hence Λ; = 7 r - 1 ( j ) e | J 2 ä . Let f \ κ ( J 2 ä be surjective and f e Μ . Such an / exists since κ is the largest cardinal in M. Then / = π ( / ) & Ν. If £ < κ is such that / ( £ ) = x, then y = /(ξ) and χ — /(ξ) Π κ. Hence, F Πα

— {{/(ξ)

Π κ;

/(ξ));

ξ