An Introduction to Extenders and Core Models for Extender Sequences

Logic Colloquium '87 H a . Ebbinghaus et el. (Editors) 0 Elsevier ScienLe PublishersB.V.(North-Holland), 1989

I37

AN INTRODUCTION TO EXTENDERS AND CORE MODELS FOR

EXTENDER SEQUENCES Peter Koepke Universitit Freiburg, West Germany Abstract. This article surveys the theory of core models for non-overlapping coherent sequences of extenders. These models of set theory arise canonically when one wants to construct inner models for strong cardinals. Strong cardinals are defined in terms of elementary embeddings of V. and we define extenders a s a way of coding elementary embeddings. The natural inner model for a strong cardinal is of the form L[E] , where, in L[E], E is a coherent sequence of extenders. We obtain such sequences E = recursively if each EKV is a suitable extender on the core model K[Ep]. We give a definition of the core model K[F] from iterable premice, which are small "L[ E] -like" structures. Without proofs, we cite the fundamental properties of this family of core models. If a model L[E] for a strong cardinal does not exist (''7L[E]'') we can define a canonical core model K[ Fcan]. K[Fcan] is the largest core model. Assuming ?L[ El, # K[Fcan] satisfies analogues of the properties of L when 0 does not exist: K[Fcan] is rigid, i.e.. there is no non-trivial elementary embedding II :K[ Fcan] -t K[ Fcan] , and "[ 'can1 satisfies a weak covering theorem. The coarse ( = non-finestructural) characterisation of core models together with some fundamental facts on core models suffice for certain applications. We show: If the existence of a successor cardinal which is Jonsson is consistent. then the existence of a strong cardinal is consistent (relative to ZFC).

138

P. Koepke

Introduction. The study of consistency strengths in axiomatic set theory involves the construction of certain models of set theory out of given models. The method of forcing allows to extend ground models by generic sets. Smaller models are obtained by forming inner models of set theory within given universes. The well-established program of reducing consistency strengths to large cardinal axioms (see [6]) typically proceeds as follows: To show that a principle A is equiconsistent with a large cardinal axiom B (relative to ZFC) one (a) gives a forcing construction of a model of ZFC+A from a ground model of ZFC+B, and (b) shows that within any model of ZFC+A we can define an inner model of ZFC+B. Part (b) corresponds to the intuition that by suitably restricting the universe large cardinal properties should grow stronger. a d e l ' s model L of constructible sets is the paradigm of an inner model. It is distinguished as being the smallest inner model of ZF. L is definable as the union of a hierarchy whose structure is exceedingly uniform. L is a model a of the generalized continuum hypothesis (GCH). Jensen's finestructure analysis of the L=-hierarchy produced many important combinatorial principles in L. Unfortunately, L ' s role as an inner model for large cardinals is restricted since there are no measurable cardinals in L [lo]. The decisive step towards larger inner models with a constructible structure was taken by Dodd and Jensen in defining the core model K [4]. Large cardinal properties below measurability are compatible with "V = K" . The definition of K readily generalizes to include measurable cardinals. and also measurable cardinals of higher order (see [91). In this paper we consider the notion of a strong cardinal which was introduced by R.Jensen. Like measurables. these cardinals are defined using elementary embeddings of the universe. We define an inner model K[Fcan] which captures the

Extenders atid Core Models for Extender Sequences

139

large cardinal strength of the universe below a strong cardinal. K[ Fcan] relates to a strong cardinal much like K relates to a measurable cardinal. K[Fcan] is the canonical member of a whole family of inner models. the family of c ~ r e models for non-overlapping extender sequences. Extenders are a generalization of the concept of a normal measure on a measurable cardinal. They were invented by A.Dodd, R.Jensen, and W.Mitchel1. This family of core models satisfies natural embedding properties, and a weak covering theorem holds for K[Fcan]. We apply these in establishing a lower bound for a consistency strength: Theorem. Assume that the existence of a successor cardinal which is Jonsson is consistent with ZFC. Then the existence of a strong cardinal is consistent with ZFC. Although the core models for extender sequences are defined here without using any finestructure. the deeper results depend heavily on finestructural techniques. So far, these techniques are of much greater complexity than the finestructure for L. However, ongoing research indicates that by convenient re-structuring of K[Fcan] as the union of some hierarchy one can get along with an "L-like" finestructure theory. This paper is structured as follows: First we introduce strong cardinals and show, that they are of maximal order of measurability. Extenders are introduced in 5 2 . In 53 we consider extendability. i.e. whether an extender possesses an extension map with a transitive target model. Next we show how extenders can be used to approximate arbitrary elementary embeddings. This is used to re-formulate the notion of strongness within ZFC. $ 5 introduces the model L[E] for a strong cardinal. To test the "L-likeness" of L[E] we begin to prove the continuum hypothesis (CH) in L[E] and are led to the consideration of iterable premice. S6 studies iterable structures, and 57 develops (part of) the theory of iterable premice. With this we are able to conclude the proof of CH in

P. Koepke

140

L[E]. $ 8 contains an informal argument why one is lead to consider core models if one wants to recursively build a model L[E] for a strong cardinal. The next paragraph quotes the fundamental theorems on these core models, and finally in 5 1 0 we prove the result on the consistency strength of successor Jonsson cardinals, strengthening a result of [7]. The theory of extenders presented here is basically that of R.Jensen in [5]. Throughout this paper we use standard set theoretical notation. We work in ZFC, i.e., Zermelo-Fraenkel set theory with the axiom of choice. By ZFC- we denote the system ZFC but without the powerset axiom. Relative constructibility is done with the J-hierarchy. If d is a finite sequence of predicates let rudx(X) denote the closure of X under the rudimentary functions and the functions x xnAi (see [ 21 ) . The J[d]-hierarchy is defined as:

JO[x] = @

;

Ja+l[x] = rUdd(Ja[x]u{Ja[x]});

JA[x] =

U

a
Ja[x].

U Ja[d]. We identify Ja[d] and L[d] aeon with the structures and respectively. A

for limit

A.

Then L[d] =

structure is amenable if vxcM xnBieM. So Ja[d] and L[d] are amenable. A map n:M-tN where M,N are €-structures is called cofinal, if vyrN 3xrM yrNn(x). If n:M+N is zo-elementary (we write n:M+ N for this) and cofinal, then n M-N is =0

zl-elementary. z0-elementary cofinal maps a low to map amenable predicates: Let be amenable, and let n:M

N

-t

=0

be cofinal. N transitive. Then there is a unique sequence

B

such that rt: -. is zo-elementary; B is defined by Bi= U{n(xnAi)lxsM}. Bi .

is amenable, and we define: n(Ai):=

Extenders and Core Models for Extender Sequences

141

1. Strong Cardinals. A most-important unifying principle in the theory of large

cardinals is their characterisation in terms of elementary embeddings. Scott [ 101 proved: K is a measurable cardinal iff there exists an elementary embedding n: V-rM where M is transitive. nPK = id, and ~ ( K ) > K .

Measurability can be strengthened by stipulating that M has certain largeness properties, e.g. by requiring that M satisfies some closure conditions or that M contains certain sets. This way, one obtains most of the large cardinal notions above measurability. Let us fix some notation: n:M-rN at K means that n is an elementary embedding between transitive e-structures M.N. KCM. n r K = id, and ~ ( K ) > K . K is called the critical point of n. A particularly natural strengthening of measurability is given by: 1.1 Definition. A cardinal K is strong if for all xeV there exists n:V-M at K such that xrM and n(K)>rank(X). One could show that the clause n ( ~ ) >rank(x) can be omitted from the definition. If K is strong, it is measurable. Indeed, its Mitchell-order O ( K ) of measurability takes the largest possible value (see [ 81 ) : 1.2. Theorem. Let

K

be strong. Then

O(K)

= (2K)+.

is Proposition 1.2 of [ 8 ] . For the converse, Proof. 0(K)<(zK)+ take V,, U > K , which reflects sufficiently many properties of 2 ~ be V. In particular. the functions V W O ( V ) and v ~ should absolute between V, and V. Take n:V-rN at K such that VacN. It then, since V,cN, O(K) = suffices to show that oN (~)>((2~)+)~; ((2K)+)N = (zK)+. Let U:={X~K(KEU(X)}.Let u:V-"M at K be the ultrapower of V by the measure U. Define T:M.+N by o(f)(K) -r n(f)(K). T is elementary, and the following diagram commutes: ON(K)

142

P. KOep ke

(1) P(K)nM = P(K) = P(K)nN.

(2)

Tl'K+1

= id, by the definition of

T .

( 3 ) t#idpM. since UeM.

Let 6 be the critical point of

T.

(4) 6>(2K)'.

Proof. Assume 6 < ( 2 K ) Minstead. Let f:(2K)M-

P(K) be

bijective, feM. T(f): (2K)N- P(K) is bijective. Since Gerange(r), we have for c<(zKIM: T(f) (6)#T(f) ( T ( Q ))=T(f ( t ) ) = =f(t). Hence T(f)(6)tP(K), contradiction! qed (4) k + M (5) 6 > ( ( 2 ) 1 Proof: Assume not. Then (2K)M 5 6 < ( ( 2 K ) + ) M . There is fcM, such that f:G-P(K) is bijective. This gives a contradiction as in (4).

qed (5).

= id.

(6) T / ' o ( K ) M

Proof: Let

t
some measure D on

T(D)=D, by (1) and ( 2 ) . O(D)'=O(D)N, ( 1) )

.

(7)

T(o(KIM)

Hence

T

x,

D~M.

since P(K)nM = P(K)nN (by

( f )=o ( T (D))N=o(D)M=b.

qed ( 6 )

> o(KIM.

Proof. o ( U ) = O ( K ) ~by , the definition of o(U). Since UaN, o(K) > ~ ( u ) ~ = ~ ( u ) = o ( K ) M. qed ( 7 ) By ( 6 ) , ( 7 ) , o ( K ) ~is the critical point of T : ~ = o ( K M) .

>

o ( K ) ~2 ( ( 2K ) + )M ,by (5). Hence o ( K ) ~

( ( 2 K ) + ) N , as required.

QED An upper bound for the strength of strong cardinals will be proved after we have looked at extenders.

Extenders and Core Models for Extender Sequences

143

2. Extenders.

Definition 1.1 is not formalized within ZFC, since it involves quantification over proper classes. To reduce this to set-quantification we need a method to code elementary embeddings by sets. These codes are provided by extenders. Even more important than this reduction will be the possibility to manipulate elementary embeddings through set theoretic operations on their codes. The prototype of an extender is a measure U on a measurable cardinal K. The ultrapower embedding n:V-rUM by U, M transitive, satisfies: ( 1 ) M = {n(f)(K)lf:K+V}; ( 2 ) U = {XCK~KE~(X)}. By (1). the target model M is "z0-generated" from range(n) and the additional ordinal K. By ( 2 ) , U contains information on the behaviour of K in relation to the n-images of elements of

..v .

Now extenders code information about target models which are zo-generated from the image of V together with (possibly long) initial segments of the ordinals. We will be able to code more complicated elementary embeddings. Extenders will be used to "approximate" embeddings, for example those witnessing that some K is strong. In 4.l(iii) we see that if n:V+M at K and xeV nM. f < n ( ~ ) then , there is n':V+M' at K with the same f and n ' is coded by an properties, i.e., xeV nM'. T<&(K), 1 extender. First we give a definition of extenders using embeddings. This will be equivalent to measure-like properties of the constituents of an extender. We consider extenders only on structures which are suitable for ultrapower-like constructions: 2.1. Definition. Let M= be a transitive structure. M is suitable, if M is closed under rudimentary functions, and if M satisfies the axiom of choice ( A C ) . Models of the forms Jb[d] and L [ A ] , p a limit ordinal, are P suitable. as are transitive models of ZFC-.

P. KOep ke

I44

2.2. Definition. Let M be suitable. Let KEOnnM and U E O ~ . E = is an extender at K , V EM with extension n : b N iff: n:M-tN is x0-elementary and cofinal; (i) (ii) N is an extensional structure whose wellfounded part is transitive; (iii) ~ + and l v are initial segments of On N; (iv) nrK = id, ~(K)"K; (v) n(K)>Nu; N= {V(f)(K,a)lae[Y]<@,

(Vi)

(vii) E ~ = {XCP(KX[K]

la[

f:KXIK]lal + M, fcM};

)nMIcNn(x)).

for ac[v]<~.

Notation: la[:= card(a), [v]":={acvl lal=n}, [ u ] < ~ : = eN are the interpretations of >.> , E in ={acv I 1 a[< @ } . >N, N. Observe that N might be non-standard. Functional evaluation as in n(f)(K.a) has an obvious non-standard interpretation. If E is an extender as above, then K is the critical point of E, and v is the length of E. E is an extender at K,V on M, if there is some n:M+N as above. E is an extender, if it is an extender on V. M is extendable 2 E, or is extendable, if there is n:M+N as above with N well-founded, hence transitive. To work with extenders, some machinery for relating the components Ea of E is useful: 2.3. Definition. Let as[OnIm, b~[On]", acb. Let b =

. . ,pn},

{pl.. j(l)<

pl<.

. . .< j ( m ) .

. .
a={p j (1)'.

Then, for u={cl,

*

. s P j (m)}

3

. . . , gn}e[0nln, S1< . . .
Uab:'{tj(l),'..,tj(m)}' For f:KX[Klm+ v define fab:Kx[K]n-t For XCKX[ K]" Set Xab:={EKXIK]nl Then we can prove a Los theorem:

v

set:

by fab(T,u)=f(T,Uab). EX}.

2.4. Theorem. Let E = be an extender at K , U on M with extension r:M+N. Let + ( vl, . . . ,vl) be a Zo-formula, and let

17(

fl)(K,al), . . . ,n ( fl) (K,al)eN with aie[ u ]

fi :Kx[ K]

lail

+ M

, fieM. Take

~ E [ u ] < ~such

that

.

Extenders and Core Models for Extender Sequences

145

Extensions are uniquely determined up to isomorphism. 2 . 5 . Lemma. Let n:M+N and no:M+N'be extensions determined by the extender E at K,V on M. Then there exists a canonical isomorphism a:N+N' such that n ' = OOTT :

Proof: Let 9(vl,. . .vl) be a zo-formula,
one

extension So if M is extendable by E there is exactly n:M+N by E which we denote by n:M+EN. We now formulate the extender axioms El-E5, for M a suitable structure. KEOnnM, VaOn, and E = :

EI: E~

is an ultrafilter on P(KX[K] laJ)nM, for a e [ v ~ < ~ .

E2: (Coherence property):

Let a c b e [ ~ ] < ~XEP(KX[K] ,

Then X E E ~ M X abeEb.

E3: (Uniformity): (x{pl}#Epl, for all

(
1

)nM.

P. Koepke

I46

E4: (First normality property):

Let f:KX[K] IaI+K, feM such that {~~(T,u) I f( T ,u)=t} aEa.

E5: (Second normality property) : -

Let f:Kx[ K ] I a I + x , f6M such

that {~f(T.u)~max(u)}~Ea. Then there is b ~ [ v ] < ~ . b>a, such that {lfab (T,u)EU}EE~. These axioms characterise extenders:

2.6 Theorem. Let M,K,u, E = be as above. Then E is an extender on M at K , U iff M,E,K,u satisfy El-E5. Proof. ( + ) Let n : M+N be an extension of M by E. It is straightforward to verify El-E5, using the property xeEa t--( <~,a>en(x). (t) is proved by an ultrapower-like construction. We only sketch the lengthy argument. Assume El-E5. Let “ M = { < f , a > l a s [ ~ ] < ” , f : ~ x [lal+M, ~] faM}; the idea is that will become n(f)(K,a) in an extension n by E. on %: a,aub b,aub {<~,u>lf (T,U) = 9 ( 7 U)}eEaUb b,aub a,aub {lf ( T U)}EEaUb(T.u) e g

Define relations “

C )

+-

and

et--t

9

3

.. - The structure M = with +

#

interpreting “=” satisfies a

Los theorem:

(1) Let s(vl,. . . , vl) be zo, , . . . , eM. and b = h

..

alu . . .ual. Then k9( ,..., ) iff

..

This is proved by induction on the complexity of 9. For zrM define T(z):=, where constz: KX{@}-~M is defined by constz(t,@)=z. Fact (1) implies: (2) Let s(v l,...,vl) be z0 and zl,. . . ,zleM. Then Mcq(z l,...,zl) iff

...... Mks(n(zl) ,..., n(zl)).

Extenders and Core Models for Extender Sequences ( 3 ) Let 9 be 2,.

zl,,..,zleM. Then

MCV~S (3, Q zl,. . . ,zl) iff Proof:

(t)

b:= Uai.

.-.

-.

147

-.

M c v "(3. ~ ~u ( zl),. . . ,n ( zl)) .

follows from (2). ( + ) Let

.-.

EM,

yi = . Set

{lMk?(fl alb (T,u),. . . ,fk "kb(T,U),Z~,. . . , zl)} =

i

Hence

..,

M/"

% satisfies the identity axioms and extensionality. Let

be the quotient of

by

";

if M is a proper class, we have

to use "Scott's trick" for representing equivalence classes by sets. Let a:M/'- w N be an isomorphism, such that the well-founded part of N is transitive. Define n:M+N by n ( z ) = a(z(z)/").We leave it to the reader to check, using axioms El-E5, that R:M+N is an extension of M determined by E.

QED

Axioms El-E5 just talk about P(lc)nM instead of the whole of M. This implies: 2 . 7 . Lemma. Let M,N be suitable structures, such that P(K)nM = P(K)nN. If E is an extender on M at K , V then E is an extender on N at K , V . Consider an extension n:M-tEN, and let < M , h be amenable. Since u is cofinal, we can use the remarks at the end of the introduction to define unique predicates

B

= n(A) on N, such

that U: -+ is z 0-elementary and
P. KOep ke

148

2.8. Lemma. Let n:M-rN be an extension by E = at K,V. Assume that is amenable for all a c [ ~ ] < ~ Then . P(K)nM = P(K)nN. Proof. (c). Let X€P(K)nM. Then x=n(X)nKeN. ( 3 ) . Let xeP(K)nN. x = n(f)(K.a)

for some ae[v]
{~
x = {a
= ( < T , u ~ Carf(T,u)}eEa}, ~M by 2.4. Then x M , since is rudimentarily closed and amenable. QED

3. Extendability.

Since we want extenders to code elementary embeddings with transitive target models we develop criteria for the extendability of extenders. 3.1. Lemma. Let n:M+N be an extension of M by an extender E at u
K,U,

Let

be an extension of fd by an extender E at K,;,

;:&fi

;5 on"fi . Let a:fi

-r

M be Z0-elementary such that

J a l ) n f i ( x e ~ ~ o(x)e~ a(a))* Vas[;~<~Vxr~(ix[~] Then there exists a':fi+N 2,-elementary such that noa =

and

a'(K)

= K,

air;

=

UP;.

Hence if M is extendable by E. then

Proof: Let S(V l,....vl) be xO, n(fl)(K,al)

alon.

, . . . , a(f,)(K,al)eR.

fi is extendable by E.

and let Let b = alu . . .ual

Extenders and Core Models for Extender Sequences

149

3.2. Definition. Let E be an extender on M at K , V . E is countably complete iff for all sequences XieE ,(ieXi, for i
complete extender E at

fi

by an extender

K.V

at K,;,

on M . Let where

be a map such that A c dom(o), zo-elementary, and

I v a e [ ~ ]
n

fi.+fi be an extension of

R and G are countable. Let a c dom u ) , upfi:fi-.M is

)"A ( x e ~ ~ + + ~ ( x ) a ~ ~ ( ~ ) ) .

Then there exists a':fi+M

zo-elementary such that UrR = o D o n .

Hence A is extendable by E. Proof : Let <libe an enumeration of all such

-

for iea(xi), for i < w . Let that xafi. xeEa. a(xi)EE

1

q(vl,

. . . ,vl)

be 2, and n(f,)(K,b,)

, . . . ,n(fl)(i,bl EN. Let c =

blu . . . ubl. Then: fiCS(n(fl)(i.bl) , . . . . a(fl)(K,.bl)) blc * x:= {lRWf, (f,U),. . . ,f;lC(t.u)))cBc j <~,6oa(c)>ea(x), since is one of the ;

P. Koepke

150

3.4. Theorem. Let E be a countably complete extender on M. Then M is extendable by E. Proof: Assume M,E were a counterexample. We can assume that M is a set: The proof of 2 . 6 shows that extendability is equivalent to the well-foundedness of a certain relation + e

z.

So

. %

formed for M,E is ill-founded; e formed for some initial part of M which is a set is ill-founded, hence there is a set counterexample to the theorem. Let M,EeHo. e a sufficiently big regular cardinal. Let o : f i

+ZwHg

such that

M,E,K,v.

E

- - fi is transitive, fi countable, and a(fi,E,~,w)=

is an extender on

fi.

In

fi, fi

is not extendable by

E . Since extensions are uniquely determined up to isomorphism (2.5). M is not extendable by

in V.

But the assumptions of Lemma 3 . 3 are satisfied, hence extendable. Contradiction!

i , E is QED

The following provides us with countably complete extenders: 3.5 Theorem. Let V be extendable by the extender E at K , V . Then E is countably complete. Proof. Let R : V + ~ M and E'=n(E). Let xieEai, for i
-

and < ~ , a ~ > e n ( x ~Set ) . q : = n -1rU{n(ai)li~n(x~). for i> where WEM. Let W = {fl3n~n(x~))}. W is partially ordered by reverse inclusion 2 . The existence of q implies that > is ill-founded in V. So n (xi) E E '

-

> is illfounded in M, and there is ;EM which satisfies (1). So, with 6 = t): M c ~ I + ( K ) 36:U{n(ai)lien(x,). Hence: ~ T < K 3b:U(aili E Xfor ~ i
Extenders and Core Models for Extender Sequences

151

We conclude this chapter with two lemmas showing that extendability may be preserved when we change the structure on which a certain extender lives: 3.6 Lemma. Let V be extendable by an extender E at K , V ; let n:V-rEN. Let M be suitable such that "McM. Then M is extendable by E. and nrM:M-rEMi.where M'=U{n(x)lxaM}.

Proof: KMcM implies: if R(f)(K,a) n(g)(K,b), geM, then there is such that n(f)(K,a) = n(f')(x.a). This shows E

f'EM

nrM :M-rEM'

QED .

.

3.7 Lemma. Let E be an extender at by E iff H is extendable by E.

K . V .

Then V is extendable

+

K

Proof: Assume n:V+M is an extension of V by E. where M is not well-founded. So there are fi,ai,ai"~]
transitive card(fi) =

k H K

+. Let

K,

uPK+l = id, and u(Zi) = fi, for i
n':H ++ H' be an extension of H K

( fi+l) Ksai+l)

+

by E.

I(

'M n(fi)(Kiai)

QED .

4.

Approximating Elementary Embeddings by Extensions

We derive extenders from elementary embeddings and show that the corresponding extensions retain some prcqerties of the original embeddings. With this we can define strong cardinals in terms of the existence of extenders. We then show that strong cardinals are consistencywise very small in relation to supercompact cardinals.

P. KOep ke

152

4.1 Theorem. Let n:M-+N be an elementary embedding of transitive models M,N of ZFC-. Let IT,

and

U < I T ( K ) .

K

be the critical point of

Define E = by:

la[

XPEa t-t XEP(KX[K] ) n M and €n(X). Then: (i) E is an extender at K , V on M , and M is extendable by E. Let n t :M-+EMi . (ii) There is an elementary embedding u : M ' - + N defined by k(f)(K,a) -+ n(f)(K,a). So u r K + 1 = id, o r v = id, and n = uOn' :

(iii) Let gsv be strongly inaccessible in N . Then V n M ' = Y V n N . and U r V n M ' = id. Y Y

Remark. In (iii), the potential of the extender notion surfaces, whereas before we had just verified that extenders satisfy the properties of normal measures.If n:M-rN is "g-strong". i.e.. V c N , $ < v < I T ( K ) . then the extender E derived 'Q from IT preserves this degree of strength, i.e., if n 4 : M - + E N ' then V c N ' and g < n ' ( ~ ) .So extenders can be used to express Y large cardinal properties much stronger than measurability. Proof: Let x:= {n(f)(K,a)laa[vliW, f:KX[K] J a J + ~ faM}. , (1) X 4 N . Proof: Assume N c 3v09(vo,n(fl)(~,a),...,IT(f1)(K,a)), where n(fi)(~,a)eX. There exists f o a M , M

c

fo:,x[~] lal-+M such that:

, . . . , fl(T.u))

vT
-+

s(fo(7.U),f1(T,U) ....,fl(T,u))). s(n(f,)(~,a),n(f,)(~,a) . . . . .n(fl)(K.a)), with +

Then N

!=

n(fo)(K,a)EX. Let a : M ' a X , ( 2 ) IT =

M'

qed(1) * transitive. Define n ' : M - r M i

by n'(x) = u - ' o n ( x ) .

UOIT'.

o r K + 1 = id, = id. (4)K is the critical point of n ' , ~ ' ( K ) ) _ v .

(3)

Extenders and Core Models for Extender Sequences ( 5 ) M * = {na(f)(K,a)lae[v~<~. f:KX[K]

(6) X@Ea

lal-+~,

feM}.

@n'(X), for ac[v]
t-+

Proof: xcEa By ( 3 ) - ( 6 ) , proved.

C)

153

la[

)nM.

<~,a>en(x)H <~,a>eu-'on(x) = ~'(x). qed(6) is the extension of M by E. So (i) and (ii) are

M such that, in M, frq:q-+Vvis onto for (iii) Take frM, f:K-+Vk, every strongly inaccessible Q < K . f is strongly inaccessible in

N. and so n(f)rt:t+VN is onto.

f 8 is strongly inaccessible in M ' since either a(a) = I , or 8

is the critical point of upf'

is given by n'(f)(6)

So n'(f)rf:pVM' is onto. Then

0 .

t

-+

n(f)(s),

6 < 8 . This is an

c-isomorphism between transitive structures, and urVM' =

= idpVM': t f'+

Vy

.

t

QED

We now reformulate the notion of strong cardinal within the system ZFC, using extensions: 4.2. Definition. A cardinal K is strong if for all zeV. zcOn there exists an extender E on V at K and some v>sup(z) such that V is extendable by E and if n:V-rEM then zeM. This definition is informally equivalent to Definition 1.1; the following argument could be formalized in a set theory which allows to quantify over classes, e.g. Gdel-Bernays set theory. Suppose K is strong in the sense of 1.1. For zeV, zcOn there is n:V+N at K , such that z c N , sup(z)
P. Koepke

I54

"zeM, where

IT:

V-rEM"is in this situation equivalent to

3aE[V]
w

where

la'+

P(K) v t < V

{wwp)(T,u)

{IC

C:KX[K]~

E

fa'au{c)(~,u)}

is defined by

-r

E

E

Q.

Next we show that strong cardinals are much weaker than supercompact cardinals: 4.3. Theorem. Let K be 2K-supercompact. Then there are cofinally many T < K which are strong within VK.

Proof: By 2K-supercompactness take n:V-rM at

such that

K

(2K)M~M.By standard arguments it suffices to show that, in M, K

is strong in V

ff(K)'

Let xcV

n(K)'

k (2 )McM. n:H +-rZw(H K

xcOn.

n:=

nrH +cM, since K

+)M. Let v be strongly n(K) such that X E ~ In . M, define

+)M, and xe(H

R ( K )

inaccessible in M, V by:

XEEa w XCP(KXIK]lal) and E;(X). By 4.1, E is an extender on H at K , V , ~

and H

K

by E; moreover, if a:H ++E N, then xaN. K

+

is extendable

K

-

E E ~ ( ~ )since , rank(E)a and N>N. Hence xeN. QED

5. An Inner Model for a Strong Cardinal. We shall consider a model L[F] whose predicate F consists of extenders whose corresponding extensions witness the strongness of a cardinal K. F will contain extenders at K and at other ordinals. We first define the notion of a sequence of extenders: 5.1. Definition. F is called a natural sequence if every z a F is of the form z = where h.ueOn, a r [ ~ ] < ~ ,

Extenders and Core Models for Extender Sequences

x

P(AX[A]

lal

155

) . Define an order function o = oF:On-rOnu{-}

O F ( h ) : ' S U P { v ~ < X , a , h , V > E F ) . Let dom(F):={lv
by For

Edom(F) set FAv:=, where FAva:= {xl€F}. For ~ , p € Odefine n FrK:={EF~A:= {EFlh<~or ( A = K and v < ~ ) } .

5.2. Definition. Let M be a suitable structure and F a natural sequence. F is called a (non-overlapping) sequence of extenders on M iff: (i) FAv is an extender at h , v on M for all edom(F); (ii) If A < h ' , o ( h ) > l , o ( A i ) > l then o ( A ) < h ' . M is extendable by F, if M is extendable by every extender edom(F). FAv ' F is countably complete if every F A v , cdom(F), is countably complete. Condition (ii) ensures that the extenders at A don't "overlap" the critical point A ' . So extensions at h i don't affect extenders at A . The following coherency condition is fundamental for working with extender sequences: 5.3. Definition. Let F be a sequence of extenders on M such that M is extendable by F. Let be amenable. We say that , <~,fl>~dom(F), if the following is coherent at ~ , p where holds : Let n: where , F t =IT(F). Then F'rK+1 = Fr, KP i.e. oFrK = oF,PK, oF,( K ) = p , and Fhv = F,!! for all < h . v > E dom(F'), A S K . is coherent if is coherent at K , U for all <~.p>~dom(F). We will see in 6.1-6.3 that being a coherent sequence of extenders is first-order definable, modulo the extendability of . Coherency allows to cut off the extenders with . If critical point K at some X , P through an extension by F K 9M , can try to push some undesired property of F holds at ~ , p we this problem up by extending at K . P . By transfinite iteration of this method we are sometimes able to smooth out some problem until it vanishes.

P. Koepke

I56

Coherency allows to define a model 5.4. Theorem. Let L[ F]= such that o~(K)=x-.Then L[F] C " K is Proof. We can assume that F= FnL[ F] . eL[F]. In L[F], form some aeon. FK, w a

for a strong cardinal: be a coherent structure strong". Let xeL[ F] ; xeJa[ F] for a:L[F] L[ F '1.

Ja[ F]=J,[ F'] by the coherency of L[ F] at XEJa [F] = Ja [F'] c L[F'].

-)FK, oa K , o ~ .Hence

QED

The above model L[F] can be shown to be very "L-like". it satisfies the generalized continuum hypotheses (GCH) and various other combinatorial properties of L. A s a test for L-likeness we intend to give a proof of the ordinary continuum hypothesis 2 O = w 1 (CH). We begin the proof until we are led to the notion of iterable premouse. We develop the theory of iterable premice in the subsequent two chapters, and we are then able to conclude the proof of CH in L[F].

5.5 Theorem. Let L[F] be a coherent structure. Then L[F]cCH. We give the initial steps of the proof: Work in L[F] and assume V = L[ F] . For acw take an Na = J Xa

< Na

a

d

F] with asNa. Let

such that aeXa and Xa is countable. Let oa:MazXa where

Ma is transitive. Then Ma is of the form Ma = J

[Fa], with

P (a)

p(a)
and aeMa.

In general, Fa will be incompatible with Fb , so that we cannot conclude the proof as in the L-case. We have to develop methods to compare different Ma and Mb: Ma will be an iterable premouse. We study these structures in the subsequent chapters and postpone the conclusion of the proof of 5 . 5 until chapter 7.

Extenders and Core Models for Extender Sequences

157

6. Iterable Structures.

P ( d Fa] formed in the CH-argument are

The structures Ma = J extendable. since u a :Ma+

J

E w a(a)

[ F] . Moreover. the target

models of the resulting extensions are extendable, and so on. This process can be carried on into the transfinite: Ma is iterable. We will study iterability for structures where F is an extendersequence on M and is amenable. We first convince ourselves that being a (coherent) extendersequence on M is uniformly nl(M), so that along an iteration the predicates of the target models are again (coherent) extendersequences. 6.1 Lemma. There is a nl-formula B,(v.w) such that for all suitable structures M, all K,u€OnnM. and E=: E is an extender on M at ,. where E = {JxeEa}.

+

K , U

iff I=@~(K,u),

Proof. (Sketch). We can view the extender axioms El-E5 as axioms for the structure , and we have to show that El-E5 on

are uniformly equivalent to n l < ~ , statements ~> in CI

K.U.

Now E 1 - E 5 have an obvious nl-structure, except for the use of notions like [ u ] < ~ KX[K] , l a l , xab, fab. Some detailed study of these operations shows that they are closely related to rudimentary operations, and that in El-E5 they are used in a Al-manner. The complete argument is somewhat involved and is omitted for the sake of brevity.

QED

A s an immediate corollary we get: 6.2. Lemma. There is a nl-sentence 0, such that for all suitable structures M and all natural predicates F: F is a sequence of extenders on M iff kB2. The coherency of at K.U. where is amenable. can be expressed by two axioms C1 and C2 as follows:

P. Koepke

158

C2 expresses that o ~ , ( K ) < uWe . transform this, using Los Theorem : tt

vt(t
3t
(n(f)(Kla)<°F, (K) -S n(f)(K.a) = f ) H

Va€[U ] <@ vf :KX[ K ] I al+On

3c
+ {Ifa*au{t)(,,u)

I

( { f (T .U)
= pr{t}'

( ~ ) } E F ~ , { ~)}*

where pr:KX[K]l+K is defined by < T , { L ) > .+ L . We define to be the last of these equivalences. With the techniques of 6.1, C1 and C2 can be seen to be uniformly equivalent to nl-formulae. We get the following analogon of 6.2: 6.3. Lemma. There is a nl-sentence O 3 such that for all suitable structures M and all FcM such that F is a sequence of extenders on M, M is extendable by F. is amenable: is coherent iff c g3. In view of lemmas 6.1-6.3 we agree that in future whenever we say E is an extender at K , U on M (K.U E M), F is an

,.,

extendersequence on M, or is coherent we mean that or satisfy the nl-statements @l.@2.@3 respectively. 6.4. Definition. is an extender structure, if M is suitable. is amenable. FcM, and F is a sequence of extenders on M. is a coherent extender structure, if also C @,. We shall now study iterations of extender structures.

Extenders and Core Models for Extender Sequences

159

6.5 Definition. Let be an extender structure. A

system <<li,>is called an iteration of with indices

(i) (ii) (iii) (iv) (v)

< < K ~ , v i+l, ~ > ~

provided:

each is an extender structure; nij:Mi+Mj is 2,-elementary and cofinal; the nij commute, i.e. cik = "jko"ij for iis the transitive direct limit of ,.

The structures are called iterates of . The iteration is called normal if for all iSj,> be an iteration as above. Then: (i) The iteration is uniquely determined by the sequence <l i+l of indices.

(ii) Each n :+is zl-elementary. ij (iii) If K < K ~for all i+l
-

limits.

QED

6.7. Definition. An extender structure is called

iterable if for every iteration <<l i,> of : is extendable, (i) if e = e+l then e (ii) if e is a limit ordinal then the direct limit of <,> is well-founded. This means that every iteration of can be freely continued. We consider some criteria for iterability: 6.8. Lemma.Let a: + be a xo-elementary embedding of extenderstructures, and let I = <</i, >be an iteration of M with indices < < ~ ~ , v ~ > l i + l < e =Let - . be iterable.

P. KOep ke

I60

(ii) u 0= u . (iii) < K ~ , U ~ > = C Y ~ ( < K ~ . U ~for > ) , i+l
the obvious limits. At successor stages use 3.1.

QED

An immediate corollary is: 6.9. Lemma. Let a: -P be a Zo-elementary embedding of extenderstructures and assume that is iterable. Then is iterable.

Countable completeness implies iterability: 6.10. Theorem. Let a:+be a Z0-elementary embedding of extenderstructures, and assume that G is countably complete. Then is iterable. ijj> be a Proof: Assume not; let I = <<li, +ij counterexample to iterability, i.e. 6.7(i) or ii) fails for I. Let IeH where p is a regular cardinal. Let X .( H X cc cc’ countable, such that I e X . Let p : & X , fi transitive. Let T = <<
-

-

i ’ ~i,li,>= p-’(~).

to the iterability of

in V.

T

is a counterexample

8<01.

By recursion, define a sequene of maps u

such that u i = ujonij for iSj-+2 0

ao:=uo(p/’fio).

If ui is defined and i+l
161

Extenders and Core Models for Extender Sequences 0

-

i+ 1 : +

such that ai = ai+loii,i+l.If ai is 2O

defined for all i < A , A a limit ordinal < 6 , define a h : fiA-tN by aA(iiA(x)) = Oi(X). Distinguish two cases: Case 1. 6 = c+l. o t : < f i ,Ft>-t

r

. so by 3 . 3 ,

=0

is e

extendable. Then 7 is no counterexample to the iterability of

, contradiction.

Case 2: Lim(B). -

Let Re,be a direct limit of i. Define a-:fi-+

e

e 2, N by

So fig is well-founded , and again 1 is no

a-(iii(x))=ai(x). e

counterexample to the iterability of
to>. Contradiction. QED .

6.11. Corollary. If is a countably complete extender structure then is iterable. Iterability is absolute between uncountable ZFC--models: 6.12. Theorem. Let W be a transitive model of ZFC-, wlcW. Let eW be an extender structure. Then is iterable iff is iterable in W. Proof. (a) is obvious. (e).Assume that is iterable in W but not in V. As in the proof of 6.10 there exists an iteration

T

=

-

<<li<6>,>of Ro, Fo with indices -

<li+l<6> and a map

(1)

u0

such that

ao:~2. for some

ao;

0

(2)

7 is a counterexample to the iterability of ;

( 3 ) &a1.

We shall argue, that in W there exists an iteration I = <<li,> of with indices < < ~ ~ , v ~ > l i + lsuch < e > that

can be "embedded" into I in the

P. Koepke

162

following sense: there exists a sequence


I i.

ui : + Zo, with d(i)
-

(5) i. By the

-

embeddings ui, I cannot be a counterexample to the iterability of

,

contradiction.

> above: We are done if we show the existence of I , < ~ ~ l i < gas In W, we can construct an iteration

I = <<l i,>of with index <li+1and a strictly increasing sequence of ordinals <8 such that: (6) d(0) = 0;

(7) if edom(Fd(i)) there exists jc[d(i).d(i+l)) that

< K ~ , v ~=> n d(i),

such

j()*

Now define by recursion on i, such that

u

0

is already defined. Assume oi is defined and i+l
-

-

u i ( < K ,ui>)cdom( ~ Fd(i)).

that

-


embedding

u

By (7), there is je[ d( i) .d( i+l) ) such

-

-

T ~ ( ~ ) , ~ O U ~ ( < K ~ , V ~ By >). 3.1,

there exists an

:di+l,pi+l>+ ,such that ZO

on^,^+^

-

"j,j+l0"d(i -

"i+loni,i+l

To define

- * j+1,d( i+l ) OU ' O"i, i+l - "j+1,d( i+l )On j ,j+lO"d( i ) ,jOCi - f?d(i),d(i+l)ouiS as rewired. -

for limit A < e proceed similarly.

QED .

The next theorem shows that by iterations we can compare coherent extender structures. This is the fundamental method in the theory of inner models for extendersequences, core models, etc. :

Extenders and Core Models for Extender Sequences

163

6.13. Definition. Let . be extender structures. . are comparable if (i) o ~ ( K )= o ~ ( K ) for all KeOnnMnN; (ii) FKvanMnN = GKvanMnN for all KeOnnMnN, v
6.14. Theorem. Let ,be iterable coherent extender

* *

structures which are sets. Then there are iterates ,

* *

of ,, respectively. which are comparable. Proof. It will simplify notation if in iterations

-

<<I i, > with indices < < K ~v, > 1 i+l we 1 > be an arbitrary pair of ordinals; in case allow for < K ~ . v ~ to < ~ ~ , ~ ~ > e dio) mwe ( Fthen require that Mi+l = Mi and ni,i+l -

idPMi. It is clear that this notion of iteration is equivalent to the ordinary one. Now assume that the theorem is false with counterexamples ,.Let e be a regular cardinal > card(M),card(N). We define iterations

IM = <<lije>,>and IN = <<lije>,> of ,with common indices < < ~ ~ . v ~ > l i < Let e > . i ~ > ~ be fixed. Then , are defined and not comparable. 1 Let K~ be the smallest KEOnnMnN such that o ( K ) # o ( K ) . or Fi Gi i such that there are V < O . ( K ) , a c [ ~ ] li,and defines IM and IN. for i
-

P. Koepke

164

Let X (H

e:=

such that card(X)
ind

+

= f-’(ind),TM

= f-’(IM),TN = F-’(IN).

( 2 ) frB = id, f(6) = e . ( 3 ) f()=, f()=.

-

(4) ind = < < ~ ~ , v ~ > l i (2). ,

,T

is the iteration of determined by

ind. So:

(5) 7, = < < < M i , F 1 > J i < B > , < n i j 1 i ~ j ~ g > > .

(6) nge = fPM;.

Proof. Let xsMg. -

Let x =

-(%),%sMi,i<6. ie ITee (x) = nie(%)) = rrie(f(%)) = f(ri;(G)) = f(x). Similarly: (7) age = frNg. (8) K ;

=

IT

qed(6).

-

8.

Proof. Assume K ; < Z . Then e e e = f(K-) = e = e and f(Kg) = ~ 6 But . is moved by at least one of ee +I*aSS+l IT-

K-

( K - )

O ; ~ ( K ; )

IT--

K-

e Contradiction. Now assume

~ ~ for > all e i>e, so g = f(6) =

8.

*

Since ind is a normal index,

K->;.

e

is not moved by

ee and

IT-

Contradiction.

ace.

But n e e ( 6 )

qed(8).

e = ;is moved by both e defined according to Case 1 above: there i s a e [ ~ - ]
Since

and “ i s ,V - has been

K-

xeM-nNe e such that x

-

-

-

E

X E F ~ ++ ~ -en-~ ee+

e

Fe_ evga and

But:

l(x)

~n,,(x),

e ev-a e Contradiction.

e

-

e G%a-

since

K

i

>v- for

e

i>6

XEG-

QED .

Extenders and Core Models for Extender Sequences

165

7. Iterable Premice. 7.1. Definition. Let D be a natural sequence and 6 = rank(D) = sup{q I cD or eD}. M = J,[ F,D] is called a D, if premouse (i) M c "F is a coherent sequence of extenders", and (ii) if <~.v>edom(F) then ~ > 6 .

over

The low part of M is lp(M):= , : H where T = min({KIcdom(F)}u{m}). M is iterable, if is iterable. J,[F] is a premouse if J,[F,@] is a premouse over @ . Iterations of a premouse M don't move lp(M) and preserve the predicate D: i>be an iteration of the 7.2. Lemma. Let <, premouse J,[F,D] over D. Let T = min{KIedom(F)}. Consider Mi = Ja,[ F a,Di], i<8. Then: Mi = id, and lp(M) = HT . (i) nOiPlp(M) __ (ii) Mi = J,,[F',D] is a premouse over D. Proof. (i) Let xrlp(M). There are feM. ~lf(t)ef(C)) "codes" z. M i= (f:v+z onto. z transitive, ve , c q ( < Q ,C>ea-f ( g )ef (c ) ) ) . Similarly, nOi (f) is uniquely = v , troi(a) = a determined by roi(v) and noi(a). But troi($) (6.6(iii)), so IT 01.(f) = f. nOi(x) = noi(f(0)) = f(0) = x. This

-

Mi also implies lp(M)cHT . There are feMi, ? < T , zsMi such that z is transitive, f:v+z is onto, and f(0) = x. Again let a:= {lf(t)cf(~)}. Then: Mi c sf.z(f:g+z onto, z transitive, ~t . c q ( ea-f ( t )ef ( c ) ) ) . Since noi:M+Mi is Z1-elementary. and since noi(~) = f,noi(a) = a, M satisfies the same

Now let x~H:~.

statement. So there are 2,zeM such that (P:g+z onto, ( < gt,c )ea-f( transitive, v f ,~ <

s ) s f ( c) ) ) .

z

By Mostowski

a

isomorphism theorem, P = f. and so x = f ( 0 ) = E(O)slp(M). follows easily from (i). QED

.

(ii)

P. Koepke

166

Comparable premice belong to the same constructible hierarchy, and we get a very nice comparison theorem. 7.3. Lemma. Let M = J,[F.D], N = J [G.D] be comparable premice P over D, a s p . Then F = GnM and J,[F,D] = J,[G,D].

Proof. Easy.

QED

7.4. Theorem. Let M = J,[F,D],

N = J [G,D] be iterable premice

* P ,

over D. Then there are iterates M ,N of M.N respectively which are of the form M * = J,*[F*,D],

* F .

N* = Jp*[F*,D] for a

single predicate Proof. 6.14, 7.2(ii), and 7.3.

QED .

We are now able to complete the CH-proof from chapter 5 . 5.5 Theorem. Let L[F] be a coherent structure which is extendable by F. Then L[F]kCH. Proof. Assume V = L[F] Let Xa

+

.

For aco take Na =

[F] with acNa. Ja(a) Na such that aeXa and card(Xa) = a. Let oa: Ma 1 Xa,

M~ transitive. M~ = Jp(a)[Fa],

where p(a)

Ma= Jp (a)[ Fa] is an iterable premouse. Proof. M a is a premouse, because Ma and Na satisfy the same extendersequence- and coherency-axioms. By 3.5, F is countably ( 1)

complete. By 6.10, is iterable. Define a transitive, reflexive relation 5

*

qed( 1) on P(u) by

a<*b iff P(u)nMa c P(u)nMb. (2)

<*

is linear on P(w), i.e. \da,bcu(a<*b or b<*a).

Proof. Let a,bcu, and let Ma,&.

"M, "Mb

be comparable iterates of

By 7.4, Gac;lMb or P c i a . Say Lac$.

Then P(w)nMa =

P(u)nia C P(u)nib = P(o)nMb, hence aS*b. qed(2 1 . ( 3 ) Every aco has at most countably many predecessors under < * .

Proof. Since {blbS*a}

c Ma. qed(3). By a standard argument on the cardinality of linear. transitive, reflexive relations, ( 3 ) implies that QED . card(P(o) )so1.

Extenders and Core Models for Extender Sequences

167

To prove further results on the coherent structure L[F] one has to study iterable premice in greater detail. A very important fact is that an iteration map is the "least" zo-elementary map from a premouse into an iterate: 7.5. Lemma. Let M be an iterabel premouse over D. Let <.>be an iteration of M. Let a:M+ MA be

2,-elementary.

Then a(~)>n,,(~)

for all CEOnn.

=0

Proof. For n,< Y T ? I~isj> of M" and maps un:M"-rZ Mn+l. nn:M"-rZ Mn+l by recursion. 0

0

,0 :=nij for isj
-20

Mn+l

ao:=a,

no:=noh.

are defined, apply 6 . 8 to an,In:

I ijh>,
There is an iteration In+l=<
M~:=M;,

u n+l:Mn+Mn+l

such that

A h nn+' :=nn+l.

an+'onn

isj> of Mn+' = nn+'oan.

Set

Oh

Mn+2.,Mn+l and * A Oh Define for m
is part of an iteration of M, and <,> since M is iterable, it has a transitive direct limit. Let MU,be this limit. Assume that the lemma fails, i.e., there is CeOnnM such that So ~ ~ ( e ) < n ~ ( eBut ) . then there is an infinite

~(c)
descending e-chain in MU: OW lu 0

(C)

n

>

= n

n

(c)

>

nlUaO(C) = n2%?Y0(c) = n2%??O(Q) > 2U 1 0 3W2lO 30211

> n

a a ( t ; ) = n

n u a ( ( ) = r

a a n ( Q ) >

3W2lO > n a a a ( c ) , etc. Contradiction.

QED

As an immediate corollary one gets: 7.6. Theorem. If n:M-+N and u:M+N are both iteration maps, then R = a . So we can define n m to be the unique iteration map n m :M-tN. Let us state, without proofs, two more fundamental facts on premice :

168

P. Koepke

7 . 7 . Lemma. Let P be a common iterate of the iterable premice M and N. Let nMp:M+P, nNp:N+P, and assume that range(nMp) c range(nNp). Then N is an iterate of M by a normal iteration.

By the lemma, every iterate of M can be reached by a normal iteration. The proof of the lemma relies on comparison arguments and the minimality of iteration maps. It is easy to see that the hypothesis range(nMp) c range(nNp) is necessary for the lemma. We conclude this chapter with a condensation result which can be used to prove the full GCH in L[F]: 7.8. Lemma. Let M = J , [ F ] , N = J [GI be iterable premice. Let P u:M+N be Z1-elementary such that ~ ( K ) = K and P(K)nN c range(n). Then o~(K)
8 . Core Models Approximating Models with a Strong Cardinal.

If we want to derive the existence of an inner model for a strong cardinal the obvious models to look for are of the form L[F] where L[F] is a coherent extendable structure with OF["] = m for some KeOn. In general, even under very powerful combinatorial assumptions, there seems to be no way of finding such a predicate F directly. Therefore one will try to define F by determining its components F recursively. KlJ Assuming we have already defined Fr we have to have a criterion for choosing FKv correctly. Eventually, FKw is to be an extendable extender in L[F]. So in choosing FKV we have to anticipate subsets of K which will only be introduced into L[F] at later stages of the construction. Now this is no problem, if sufficiently often we can find an FKw which is an extendable extender on the whole of V. This is the case if there is a strong cardinal K in the universe: 8 . 1 . Theorem. Let K be strong. Then there is a coherent inner model L[F], with oF (K) = m .

Extenders and Core Models f o r Extender Sequences

169

For a proof see [3,S23]. First a coherent sequence E is defined recursively consisting of extendable extenders E on TV V . For a measurable cardinal I under consideration, we put extenders ETv onto the sequence for as long as possible. If this goes up to m -( 0E(T)-), this concludes the recursion. Otherwise proceed to the next measurable cardinal > oE (7). Case 1: oE (T) 2 K for some T < K . Let U be a measure on the measurable cardinal K . By an indefinite iteration of V by U we obtain a coherent structure L[Ei] from L[E] with oE'(7) = m ("iterating U out of the universe"). Let u:L[E'] + L[F] be an iteration of L[E'] by E i o such that U(T) = K . Then L[F] is a s required. Case 2 : oE (T)
K

to show that for

all peon there are endextensions Fp of E r < K , O > with oFP(,)>p (although the details of the argument are a bit subtle the main intuition is that a strong cardinal is able to supply us with arbitrarily strong extenders). The condensation lemma 7.8 is then used to show that for a proper class &On, if p,p'eB then the predicates

FPnL[FP]nL[FP'] and

FP'nL[FP]nL[FP']

agree on their common domain. Then the predicate is as required for Theorem 8.1.

F =

U Fp P EB

Now if we have not got this profusion of real extenders to construct F from, we will have to require that each F K V is an extender on some sufficiently,large auxiliary model, which can already be defined at stage < K , v > of the recursion. These auxiliary models, which capture P ( K ) of the final L[F], will be provided by core models. The following line of thought is intended to motivate the definition of these models: Assume we have, by some method, constructed our natural sequence F. For simplicity assume that all components FKV are countably complete (in the sense of 3.2). Now assume the

I70

P. Koepke

construction failed, i.e. L[F]!= "F is not a coherent sequence of extenders". Since the right hand side is z1 (6.2, 6.3), there is aGOn such that: (1) Ja[F]k "F is a coherent sequence of extenders", and (2) Ja+l[F]t "F is not a coherent sequence of extenders". The counterexample in (2) can be taken to be a subset CCK for ~[ F] , and c codes a some <~,v>cdom(F): C E J ~ +F]\Ja[ counterexample to the extender properties of FKV or to its coherency. Now the elements of Ja+l[F] are in a certain sense definable over J [F]. and for simplicity of this argument let a us make the assumption that c is xl-definable over Ja[F] from some parameters, and K % a . Let n:Ja[F] Ja,[F']. Setting D = FP, G = F'\D. the *FKV

structure M = Ja ' [G,D] is a premouse over D. M is iterable by the countable completeness of F ( 6 . 8 ) . c is z over M. since n -1

is zl-elementary and n r K = id. By an argument of Dodd and Jensen there exists an iterable

+

premouse M+ = JP+l[G+,D] over D such that J [G ,D] is an P iterate of M (this difficult result is called the continuation lemma since it allows to continue M to a longer iterable premouse - modulo an iteration of M).

-

crz (J G+,D]) c M+, and we have located c in the low part of -1 P [ an iterable premouse over D. So in taking all low parts of iterable premice over D together we get a domain which contains counterexamples like c in case such exist. This motivates the following: 8.2. Definition. Let D be a natural sequence. If D is a set. define K[D]:= U{lp(M)IM is an iterable premouse over D). If D is a proper class, define K[D] := U K[DnVa]. aeon 8.3. Theorem. K[D] is an inner model of ZFC + "V=K[D]", and

is amenable, The proof of 8.3. involves the above-mentioned "continuation lemma". Let us just give an indication of techniques used by showing that the powerset of w exists in K[D]. First use

Extenders and Core Models for Extender Sequences

171

"comparison iterations" as in 7.4 to put all subsets of w in K[D] into the low part of a single iterable premouse M over D. By the continuation lemma there is an iterable premouse M+=

J p + d F+,D]. such that JP [F+,D] is an iterate of M.

Then

P(o)nK[D] = P ( w ) n J [F+,D] E lp(M+) c K[D], as required. P With these new models we are now able to formulate the "local" criterion for the successive choices of FKV as described in the beginning of this chapter: 8.4. Theorem. Let F be a natural sequence such that each F KV is countably complete. Then is an extendable coherent extenderstructure provided the following holds: Whenever <~,v>~dom(F), then FKV satisfies the extender axioms (El-E5) and the coherency axioms (Cl,C2) with respect to K[ F ~ < K . v > and ] Fr. (H + ) K

The proof of this theorem follows the line of argument that led us to the definiton of the structures K[D]. but a complete proof is far beyond our means here. In particular the case when the counterexample c is not zl(J [F]) but more ...a

complicated necessitates finestructure theory and a different, finestructure preserving notion of iteration. This method of proof also shows that the "intermediate" models K[Fr] satisfy that Fr is an extendable coherent extendersequence. We have arrived at the notion of core model: 8.5. Definition. A model K[F] is called a core model if K[F]c "F is an extendable coherent extendersequence". If K[F] is a core model. the predicate F is called strong.

9. Fundamental Properties of Core Models. Core Models are "L-like" models of set theory. The family of core models possesses natural embedding and extension properties, and there is a unique canonical core model which is the largest of these models. The canonical core model can

P. Koepke

172

be considered to be a direct analogue of Gijdel's model L: If there is no inner model with a strong cardinal ("lLIE]"), the canonical core model has many properties of L under the assumption .O # . Though natural, these results are very difficult to prove. Usually, finestructure theory and iteration theory have to be combined, and we are unable to include any of this in the present paper. If these fundamental properties are taken for granted it is however sometimes possible to apply higher core models to certain problems, just like the Covering Theorem for L can often be used without mentioning finestructure theory. Such an application will be given in the next chapter. 9.1. Definition. ''7L[E]'' stands for: there is no inner model with a strong cardinal. Most proofs about core models presuppose .L[E] ; .L[E] allows to define maximal core models which are technically important. Let us assume .L[E] for the rest of this chapter. This is no restriction for most applications: if we want to show that there is an inner model of a strong cardinal, assume .L[E] and work for a contradiction. Let us remark that the analogy to L and becomes even closer if we would admit models L[E] for a strong cardinal into the family of core models and define a real 041 ("zero-pistol") which transcends models L[ E] as'0 transcends L. We would then get the subsequent theorems under the weaker assumption .O'.

But working with lL[E] saves us the exact

definition of the set 0 ' . 9.2. Theorem (,L[E]). Let K[F] be a core model. Then K[F] c GCH. Also other L-like principles like 0 , o , . . . in K[ F] .

hold

The proof of the continuum hypotheses (CH) could be done like the proof of Theorem 5 . 5 .

Extenders and Core Models for Extender Sequences

173

9.3. Theorem (,L[E]). Let F be strong and <~,v>edom(F). Then: (i) F ~ < K , u is > strong.

K)nK[ F] . 9 . 4 . Theorem (,L[E])

(The Limit Theorem). Let F be a natural sequence such that Fr is strong for all <~,u>edom(F). Then F is strong. 9.5. Theorem (.L[ E] ) . Let F be strong, and let n :K[F]+W be an elementary embedding into a transitive class W. Then W = K[F'], where F a =n(F).

dom(F)=dom(G).

.

Let K[ F] , K[ G] be core models with Then IK[ F] I = I K[G] I and FnK[ F] = GnK[ F] .

9.6. Theorem (,L[ E] )

By 9.4 and 9.6 one can define a unique canonical core model by choosing extenders recursively at the smallest possible critical point: 9.7. Definition. Assume .L[E]. Then there is a unique strong sequence Fcan satisfying:

1'(

'can c ' [ ' c a n ] ; (ii) for KeOn, u = o (K) there is no strong sequence G Fcan with G l \ < l ( , u > = Fcanr and <~,u>edom(G).

So extenders with critical point K are included into Fcan for as long as possible; only then the next critical point is considered. K[Fcan] is called the canonical core model. 9.8. Theorem (,L[E]). If n:K[Fcan]+W is elementary, W transitive, then n is the iteration map of an iterated ultrapower of K[ Fcan] .

an immediate corollary one has the following rigidity property of K[ Fcan]: 9.9. Theorem (,L[E]) There is no non-trivial elementary embedding n :K[ Fcan]+K[ Fcan] As

.

P. Koepke

I14

9.10. Theorem (,L[E]). If G is a strong predicate which is a set and G c K[G] then there exists an iterated ultrapower n:K[Fcan] + K[F] such that G is an initial segment of F, i.e. G = Fpv for some ordinal Y . Finally, we have a weak covering theorem: 9.11. Theorem (.L[ E] ) . Let K be a singular cardinal in V. Then

10. An Application to Jonsson Cardinals. 10.1. Definition. Define J ( A ) by: every first-order structure P of cardinality A whose type ( = number of relations, functions, and constants) is < p possesses a proper elementary substructure of cardinality A . A is a Jonsson cardinal if Jw(A). Details on Jonsson cardinals can be found in [1,7.3.2]. Every Jonsson cardinal is > o w . The successor of a regular cardinal is not Jonsson [ll]. Here we will obtain some information on successors of singular cardinals being Jonsson. 10.2. Lemma. Let A be a Jonsson cardinal and let p be a regular cardinal < A . Then J ( A ) . I.( Proof. Consider a structure S = >with functions fy. We may assume that the fv contain Skolem functions for S . Define (partial) functions F,G:

( ~ ) < ~ - rby h

+

+

F(u,jf)

= f,(2),

if

~ ( 2=) sup({fY(~)lY
v
< p + , >be

Jonsson take X

<

+

a counterexample to Jw(p ) . Since A is >such that card()[)= A , X#h.

Case 1: p c X. Then ( . X S since G witnesses that X is closed under the Skolem functions f for S . Case 2 :

p

c X. Hence

+ + , and since we threw in the + +

Xnp # p

counterexample to Jo(u+) we must have sup(Xnp ) < p U{fy(2)lV
Then Y 4

S,

. Let Y:=

card(Y)=A, and Y#A since

Extenders and Core Models for Extender Sequences

+

sup(Ynp+)= sup(Xnp ) < p

175

+ . So X or Y is the desired Jonsson

substructure.

QED

10.3. Theorem. Assume the existence of a successor cardinal which is Jonsson is consistent with ZFC. Then the existence of a strong cardinal is consistent with ZFC. Proof: Let A be a cardinal such that A + is a Jonsson cardinal. By our initial remarks, A is singular. It suffices to show that the theory "ZFC + there exists a strong cardinal" possesses an inner model or a set model within our universe V. So let us assume .L[E]. We distinguish two cases: Case 1: There is K < A such that o ( K ) 2 A . Fc an In this case we want to find a set model for a strong cardinal. Let F:= Fcan. We shall consider a suitable Jonsson substructure of the iterable premouse M = J +[F]. Since A

K[F]CGCH there is an enumeration of P(x)nM, with p < A . Choose a "cardinality function" D:AxA

+ +

+ A
4

such that D"(Ax{a})

= a for all a. Define a first-order structure S = >.where the au are to be constants. A

By 10.2, we can apply the principle J (A')

cc

X 4

S

fi

= J

A

S:

there is

A + and XpJ +[F]. Let a : f i fX 4 M,

with card()()=

transitive;

to

+[a]

A

for some predicate

critical point of a. (1) a < A . Proof. Consider the function D. (2) a > ~ ,P(K)nM c range (a). Proof. Since X contains all the a".

P . Let

fi

a be the

qed (1) qed (2)

(3) a(A) = A .

Proof: Since card(XnA) = A . (4) ~ ( a ) < A s by (3). (5)

JA[q=

qed(3)

JA[F].

Proof: Since arJA[P] :JA[ of Lemma 7.8.

e] +

JA[F] satisfies the assumptions

qed (5)

176

P. KO ep ke

( 6 ) JJFI 4 Ja(,)[F1. Proof: By (5). at'JU[F]:JU[F]+

since

apa

J

[F]. aPJU[F] = idpJa[F].

Zw o(a)

= idpa.

qed ( 6 )

(7) JU[F] P ZFC-. Proof. Let us for example present the proof of the replacement scheme in J,[F]. Assume: ( * I J,[F] != Vxaa 3!y ?(x.Y). z:={ycJU[F] IJa[F]k3xsa y(x,y)}~JO(,,[F]. By (6) and ( * ) : z ={Y~Jo ( a )[ F] I Jo(a)[ F]C3xca~(x.y)). So J [F]!=3zvy(ycz ct 3xeay(x,y)). o(a) By ( 6 ) : Ja[F]k3zVy(ycz tt 3xra?(x,y)). qed ( 7 ( 8 ) JU[F] satisfies the powerset axiom. Proof. Let xaJU[F], and assume P(x)nJU[F]4Ja[F]. Take feJA[F , f:Tt+P(x)nJA[F]. T ~ U ,since otherwise we could find a cofinal map from T into a in Jn[F] , but the critical point u must be regular in JA[F] . o(f):a(r)trP(x)nJAIF]. Then o(f)(a) # a(f)(o($)) = f($) for all $ < T , so f:THP(x)nJn[F] was not chosen onto. Contradiction. qed(8) (9) J,[F] is a coherent structure. Proof: Since L[F] is coherent, and coherency is a nl-notion

-

qed( 9 1

(6.3).

(10) JU[ F] is extendable by F. Proof: L[F] is extendable by F. Then use 3.3.

qed(10)

Ja[F] is the required set By theorem 5.4 applied within J,[F], model for a strong cardinal. It remains to consider: ( K ) < A for all K < A . Case 2 : oF can Here we want to derive a contradiction to our initial assumption .L[ El. Set F = Fcanrh. By the weak Covering Theorem 9.11 and theorem 9.3: (11) A + = A+K[F].

+ +

Consider the structure S:= , where D:AxA +A A

cardinality function from case 1. (12) S c "V=K[F]".

is the

Extenders and Core Models for Extender Sequences

177

Proof: L e t xaHKLF]. By 8 . 3 . t h e r e i s an i t e r a b l e premouse M h

over F i n K[ F] such t h a t xclp(M)

.

A downward Uwenheim-Skolem

argument shows t h a t w e can assume M

By J u:HzX,

take X


c

HKiF].

qed( 12)

A

HKLF]. L e t A H t r a n s i t i v e . L e t a be t h e c r i t i c a l p o i n t of O . and l e t (A+)

such t h a t card(X)=A+ and X

;r!

-1( F ) . A s i n c a s e 1 w e g e t : (13) a < h . n ( A ) = A , n ( a ) < h .

i:=a

K[ F1

(14) H = H Proof:

+

A

( c ) I f xeH, t h e n , by ( 1 2 ) , xelp(M) f o r some i t e r a b l e

premouse M over

t,

c o n s i d e r e d i n s i d e H . By t h e a b s o l u t e n e s s

theorem 6 . 1 2 , M i s an i t e r a b l e premouse over u n i v e r s e , so xeK[ ii'] (2)

i n the

.

L e t x e H K I F ] . There i s an i t e r a b l e premouse M over A

ii' w i t h

xelp(M). By a downward Gwenheim-Skolem argument, assume t h a t card(M)
r

Let

GcM.

I f G=@, t h e n MeH and w e a r e

be minimal such t h a t edom(G).

K

+

I t e r a t i n g M i f n e c e s s a r y w e can assume t h a t K E ( A . A ) . Now

H c Card(K)
i,G b e

with f e l p ( N ) . L e t

comparable i t e r a t e s of M , N

r e s p e c t i v e l y . f e G ; b u t f @ %because , o t h e r w i s e feM and n o t be measurable i n M . c

x e H:=Htc (15)

P

So

c

%,

H:=Hflc

and w e g e t

H.

qed( 1 4 )

is strong. A

e "t is

a c o h e r e n t sequence of

e x t e n d e r s , and V i s extendable by Lemma 3 . 7 i m p l i e s : K [ F ]

el'.

could

h

Proof. By ( 1 4 ) . H K i p ]

by

K

F".

c "F i s c o h e r e n t , and V i s e x t e n d a b l e qed( 1 5 )

P. Koepke

178

By (15) and theorem 9.10 there exists an iterated ultrapower n:K[Fcan]+K[Fi] such that P = F'rh'. We can also assume, possibly by further iterating K[Fi], that all extenders in F t with critical points

-

>A+

are countably complete.

We shall now construct an embedding G:K[ F']+K[%] such that

,..

a P H = o r HK[F'I = +

0.

h

Let IQI :=(lfcK[F'], f is a function, dom(f)eH, xeu (dom(f)) } .

,.

Define relations % and E onlQl by: a iff ea ( { I f ( u) = g( v)} ) , :

iff co({lf(u)eg(u)}).

.-

The "structure" Q:= satisfies a L O theorem: ~ (16) Let 'f be a formula of set theory and ..., eIQI.Then Qk='f(, . . . , < fl.xl>) iff , Eo({IK[F']CS(fl(ul) ,...,fl(u~))}). Proof. By induction on the complexity of 9. The induction is clear for atomic formulae and for the case of propositional connectives. Let us consider the induction step for 9 I 3vot. Then : Qk~vo~(vo, , . . . , < fl,xl>) Q k i ( ~ f o , x o ~ , ~ f l , ..., x l ~ ), . for some . EU({lK[F']kt(f~(u,) ,....fl(ul))}) E U ( { < U ~ , . . . , U ~ > ~ K [ F ' ] ~ 3v0t(vo,fl(~l)~.~ * ,fl(Ul))}) E Q ( { < u ~ . . . . , u ~ > ~ K [ F ' ] ~ 3 v ~ i ~ v 0 , f l ~ ~ l ~ ~ . . . ~ f l ~ ~ 1 ~ ~

Extenders and Core Models for Extender Sequences

I79

QC3(>,, ...'< fl,xl>), and Q ~ 3 v o t ( ~ o . < f l , ~ 1 ~ ,). ..., (17) The relation is well-founded on Q. Proof. Assume instead that there exist sQ for neu ~ , ( { < u ~ +un> ~ ,I fn+l( u ~ +E ~ fn( ) un)} ) . There exists an iterable premouse M = J8[G,Firg] over F'Pg for some Q > A such that fnslp(M) for n < w . We can assume that G is countably complete, and sinie F' is countably complete above A we can view M as an iterable premouse over P = F'PA. A downward Uwenheim-Skolem argument shows that there is an iterable premouse N over

with card(N)
n 1 9

zn+1 (un+l)"(un)

1

and

I

= { fn+l(Un+l)Efn(un) 1* 9

Assume N = J [ E , P ] . r)

Case A : dom(E) = fl Then N c H. For ne~(

= { a descending €-chain.

So we have:

Case B: dom(E)pfl. Let K > A be minimal such that <~,O>~dom(c). A < K < A + , and by (11). K is singular in H. Let PeH be an iterable premouse over

i? such that lp(P) c

"K

is singular". .-"z

Iterate N,P up to comparable structures N.P. with iteration z

map p:N+N. By an argument similar to the proof of (14) we see

c $. We can assume that card($)
P. KOep ke

I80

z

Now the proof of 6.12 shows that there exists an iterate P

=

of P such that P EH and

?

s

can be embedded into P by a map

;.

Note that all the embeddings fix the domains dom(fn), n
.,. .,. = { < u n + l . u n > ~ a o ~ o ~ ( f n + l ) ( ~ ~E+ loooop(f,)(Un)} ) * .,.

*

Hence aopopofn+l(xn+l) E uopopofn (xn ) forms a descending s-chain. Contradiction. ¶ed( 17) By (16). Q satisfies the equalitiy axioms, so = is a congruence relation on Q. Let [f,x] denote the %-equivalence class of elQl. Let Q/=:={[f,x]lElQI) with the induced relation ?/%. More precisely one would employ Scott's trick to represent equivalence classes by sets. Q/% is a well-founded, extensional structure. Let T:(Q/=) 2 W be the Mostowski isomorphism onto W transitive. .,.

Define U:K[F'] + W by ~(z)=T([{},~]).

It remains to see that

a

extends

u:

Extenders and Core Models for Extender Sequences

181

(19) If eQ and frH then T([f,x]) = o(f)(x).

Proof. By induction on -

..,

E .

Suppose (19) holds for all

z

€-predecessors of . (c). Let z€~([f,X]),with Z = T([g,y]). e~({lg(u)~f(v)}). Define gl:dom(g)+H, gtrH,by g'(c)=g(c). if g(c)eH, and g'(c) = @ , else. Claim. < g ,y>%. } 2 {ulg(u)*H} 3 {ul3vg(u)ef(v)}. v)}) because co((~g(u)~f(v)})~ 1). qed(C1aim) cH, and we can use the inductive

z = T([9,Y]) = 7([9',Y]) = 0(9')(Y) e o(f)(x). Conversely, let zau(f)(x). z = o(idrd)(z), for some transitive dsH with zsu(d). o(idPd)(z)Ea(f)(x) a e ~({~id~d(u)ef(v)}) .., * r([idPd,z]) E ~([f~x]). By inductive assumption, ~ ( [ i d r d , ~ ]=) o(idrd)(z) Hence Z E T ( [ f,x] ) .

= z.

qed( 19)

z

(20) u 2 u .

Proof. For ZEH,

z

o ( z ) = T ( [ { < z . @ > } , @ ] ) = u ( { < z , @ > } ) ( @ ) o(z). =

qed(20 1

.., (21) FrA = F = FcanrA. z

-

Proof: FrA = u(F'rA) = o(F) = Fcanr~. Consider

-

K[Fcan]

3 K[F'] 2 K[?]

.

By theorem 9 . 8 . O O B is an iteration map of K[Fcan]. Its critical point is < a < A. call it K . Then o ( K ) > o - ( K ) , because Fcan F but o (K) = o _ ( K ) , by (21). Fcan F Contradiction!

z

OOB

is an iteration at

K,

QED.

182

P. Koepke

References [l] C.C. Chang and H.J. Keisler, Model Theory (North-Holland. Amsterdam, 1973). [2] K.J. Devlin, Constructibility (Springer, Berlin, 1984). [3] A.J. Dodd, The Core Model, Lecture Note Series 61 (London Math. SOC., Cambridge, 1982). [4] A.J. Dodd and R.B. Jensen. The Core Model / The Covering Lemma for K / The Covering Lemma for L[U], Ann. Math. Logic 20(1981) 43-75 and 22(1982) 127-135. [5] A. J. Dodd and R.B. Jensen, The Core Model for Extenders, Handwritten notes, Oxford. (1978-1980). [6] A. Kanamori and M. Magidor. The Evolution of Large Cardinal Axioms in Set Theory, in: G.H. MUller and D.S. Scott, eds., Higher Set Theory, Lecture Notes in Math. 669 (Springer, Berlin, 1978) 99-275. [7] P. Koepke. Some Applications of Short Core Models, Ann. Pure and Appl. Logic 37(1988) 179-204. [ 81 W.J. Mitchell, Sets Constructible from Sequences of Ultrafilters, J. Symbolic Logic 39(1974) 57-66. [9] W.J. Mitchell, The Core Model for Sequences of Measures I, Math. Proc. Camb. Phil. SOC. 95(1984) 229-260. [lo] D.S. Scott, Measurable Cardinals and Constructible Sets, Bull. Acad. Pol. Sci., Ser. Math. Astron. Phys. 9 (1961) 521-524. [ll] J. Tryba, On Jonsson Cardinals with Uncountable Cofinality. Israel J. of Math. 49 (1984). 315-324.