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On the GCH At Measurable Cardinals
ON THE GCH AT MEASURABLE CARDINALS Kenneth KUNEN* University of Wisconsin, Madison, Wisconsin, USA
Theorem. Suppose that there is a measurable cardinal, K , such that at least one of the following holds: (i) 2 K > K + . (ii) Every K-complete filter over K can be extended to a K-complete ultrafilter. (iii) There is a uniform K-complete ulirafilter over K + . Then for all ordinals 0, there is a transitive proper class, N , such that N l= [ZFC t there are 0 measurable cardinals]
.
We consider the proof from (i) to be the main result of this paper. That the conclusion follows from (ii) and (iii) will be proved by the same method and with little additional effort. This theorem is an improvement over some results in 589-10 of [2], which showed that the existence of Solovay’s 0t followed from (i) and from (ii), and that the full conclusion of the theorem followed from the assumption that K is strongly compact. Familiarity with [ 2 ] is assumed here throughout. Both the statement of the theorem and the proof are understood to be formalized within Morse-Kelley set theory with the axiom of choice (see the appendix to Kelley [ 11 ). We leave to the reader the simple modifications necessary to carry out everything within ZFC. Note that we may forget completely about assumption (iii), since it implies that either (i) or (ii) hold. Thus, from now on we shall assume that K is a measurable cardinal satisfying (i) or (ii). We shall, in fact, derive the conclusion of the theorem directly from Lemma 4. The proof of Lemma 4 from (i) uses Lemma 1, and from (ii) uses Lemma 3. Lemma 1 combines some modifications, due to C.C. Chang and to
* The author is a fellow of the Alfred P. Sloan Foundation. 107
108
K. KUNEN
A. Hajnal, of Godel's proof of GCH in L.
; Lemma 1. I f a 5 K + , 3 g T(K),b C: ORD,and < L [a,b, $1 ) < K + .
K,
then card(T(K)
fl
Proof. Let h = i.By a Lowenheim-Skolem and collapsing argument,
P(K)nL [a,b, 3 ] which has power
cu {LI1[a n v, x, 3 ]
:p
< K + A v < K + A X c K + A T = A},
K+.
To prove Lemma 3 , we need an unpublished combinatorial result, due to J. Ketonen, which we include with his permission. =
Lemma 2 (Ketonen). There is a family Q C K~ such that 9 = 2 K ,and such that whenever h < K , gt (t < A) are distinct members of 9, and p t (t < A) are any ordinals < K , then 377 <: KVt < h [gt(v) = p t ] . Proof. Let A , (a < 2 K ) be almost disjoint subsets of K . Let f , : A , + K be such that V p < K [card({{ € A a : fa({) = p } ) = K ] . Let s,, (77 < K ) enumerate { S C K: S < ~ } . L e t g= {g, : ~ ~ < 2 ~ } , w h e r e g , ( t ) = f , ( { ) w h e n s ~n A , = {{}, and g&) = 0 if card(st f A,) l # 1. Lemma 3. I f (ii) of the theorem holds and 6 < ( 2 K ) + ,then there is a K-complete ultrafilter, U,over K such that i$(K) > 6 (where iz : V + Ult,(V,?L )). Proof. We may assume 6 2 2 K .Let 9 be as in Lemma 2 ; let Q = {g, : a < 6 }, in 1-1 enumeration. Let U be such that whenever a K + , let U be any K-complete non-principal ultrafilter over K . iG(K)> 2K 2 K * , so there are g, E K~ for a < 6 such that { [ : g,@)
ON THE GCH AT MEASURABLE CARDINALS
109
Definition 1. For any ordinal 8, a &set is a set b C ORD such that b has order type 0.8, and such that for each 5 E b, 5 > sup(b 5). Definition 2. I f b is a &set, f < 0, and n < o, y(b, f , n ) is the o * f + n t hordinal in b and (a) h(b, 0 = sup({y(b, f , m ) : m < a}). 2(b, f ) = {x E A@, #9 : 3m V k > m [y(b,f , k ) E x (b) g(b)=(a(b,q):q
>K A
V r < u [rK < u] }.
n
a
We shall eventually show that if 8 < K , b is a &set, and b C Kw.8+1, then
Y f < 8@(b,f ) . This will prove the theorem for 8 < K . The full theorem will then follow immediately, since for any 8 one may carry out the entire proof within Ult,+,(V, U),where ?L is any K-complete non-principal ultrafilter over K .
Lemma 5. If M is a transitive model for ZFC U a K-complete ultrafilter over M E M, u E KO,and p <(I, then i Z h M ( u ) = u . K, Lemma 6. If b is a &set, 8 < K , and g < 8 is such that VQ < f [h(b,Q) < K + + ] and V n Vq [f < q < 8 + y(b, 77, n ) E KO], then @(b,4). Proof. Let 6 = sup({h(b,Q) : 77 < g}). S < K++. In the special case that 6 < K , the lemma is standard and just uses the fact that K is measurable. In the general case, let M and U be as in Lemma 4, and apply the proof for 6 < K within Ultl(M, CU nM). Definition 4. If a C ORD and X C_ L [a], then H(a, X) = { y E L [a] : y is definable in L [a] from elements of a U XU {a}}. Lemma 7. Let a C K, be of order type a, where a < K . Say a = { u1 : f < a}, in increasing enumeration. Then for each f < a , card(uE n H(a, K1+l ul))
< K+.
Proof. We use induction on f
-
< a.Assume the lemma holds for 77 < [. Let
110
K. KUNEN
<
<
p = sup {uo : q t } .If p = ul, the induction step is trivial, so assume p at. Let A = H(u, KC p ) . card@ nA) < K + . Let T be the transitive collapse of A, j the isomorphism from A onto T , 6 = j ( p ) , and b = j(u). 6 K + + , and
-
<
j(T)=Tfor T E K [ + ~- @ + 1). Let M a n d U be as in Lemma4. Lemma 7 for t will follow if we can show that for all n E ut f H(u, l Kt+l at), j(n) < n M ( ~ ) Say . n is definable in L [a] from elements of s U r U {a},. where s is a finite subset of u n uC and r is a finite subset of KC+l u C .Then j(n) is definable in L [ b ] from elements of j ( s ) U t U { b ] . Now if n < u t, then j(n) is fixed by: i n M , since all elements of j ( s ) U r U { b }are. Thus, j ( n )< " M ( K ) . We now prove the theorem, using the remark following Definition 3. Suppose 8 < K , b is a 8-set, and b C Kw,+l. Let t < 8. Let A = H(b, KW+ sup({h(b, q) : q < t } ) ) and , j be the isomorphism fromA onto the transitive collapse of A . For t < q < 8 and n < w , j ( y ( b , q, n)) = y ( b , q. n ) E K O ,and for q < 8 and n < w, j(b, q, n) < K++ by Lemma 7. Hence, by Lemma 6, @(I@), t).It follows that @(b,t),since A iL [ b ] and @(b,t ) is a property of b and t expressible within L [ b ]. Thus, V 4 < 8@(b, t).
it
-
-
ig
-
References [ I ] J.L. Kelley, General topology (Van Nostrand, Princeton, N.J., 1955). [ 21 K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic l(1970) 179-227.
Theorem. Suppose that there is a measurable cardinal, K , such that at least one of the following holds: (i) 2 K > K + . (ii) Every K-complete filter over K can be extended to a K-complete ultrafilter. (iii) There is a uniform K-complete ulirafilter over K + . Then for all ordinals 0, there is a transitive proper class, N , such that N l= [ZFC t there are 0 measurable cardinals]
.
We consider the proof from (i) to be the main result of this paper. That the conclusion follows from (ii) and (iii) will be proved by the same method and with little additional effort. This theorem is an improvement over some results in 589-10 of [2], which showed that the existence of Solovay’s 0t followed from (i) and from (ii), and that the full conclusion of the theorem followed from the assumption that K is strongly compact. Familiarity with [ 2 ] is assumed here throughout. Both the statement of the theorem and the proof are understood to be formalized within Morse-Kelley set theory with the axiom of choice (see the appendix to Kelley [ 11 ). We leave to the reader the simple modifications necessary to carry out everything within ZFC. Note that we may forget completely about assumption (iii), since it implies that either (i) or (ii) hold. Thus, from now on we shall assume that K is a measurable cardinal satisfying (i) or (ii). We shall, in fact, derive the conclusion of the theorem directly from Lemma 4. The proof of Lemma 4 from (i) uses Lemma 1, and from (ii) uses Lemma 3. Lemma 1 combines some modifications, due to C.C. Chang and to
* The author is a fellow of the Alfred P. Sloan Foundation. 107
108
K. KUNEN
A. Hajnal, of Godel's proof of GCH in L.
; Lemma 1. I f a 5 K + , 3 g T(K),b C: ORD,and < L [a,b, $1 ) < K + .
K,
then card(T(K)
fl
Proof. Let h = i.By a Lowenheim-Skolem and collapsing argument,
P(K)nL [a,b, 3 ] which has power
cu {LI1[a n v, x, 3 ]
:p
< K + A v < K + A X c K + A T = A},
K+.
To prove Lemma 3 , we need an unpublished combinatorial result, due to J. Ketonen, which we include with his permission. =
Lemma 2 (Ketonen). There is a family Q C K~ such that 9 = 2 K ,and such that whenever h < K , gt (t < A) are distinct members of 9, and p t (t < A) are any ordinals < K , then 377 <: KVt < h [gt(v) = p t ] . Proof. Let A , (a < 2 K ) be almost disjoint subsets of K . Let f , : A , + K be such that V p < K [card({{ € A a : fa({) = p } ) = K ] . Let s,, (77 < K ) enumerate { S C K: S < ~ } . L e t g= {g, : ~ ~ < 2 ~ } , w h e r e g , ( t ) = f , ( { ) w h e n s ~n A , = {{}, and g&) = 0 if card(st f A,) l # 1. Lemma 3. I f (ii) of the theorem holds and 6 < ( 2 K ) + ,then there is a K-complete ultrafilter, U,over K such that i$(K) > 6 (where iz : V + Ult,(V,?L )). Proof. We may assume 6 2 2 K .Let 9 be as in Lemma 2 ; let Q = {g, : a < 6 }, in 1-1 enumeration. Let U be such that whenever a K + , let U be any K-complete non-principal ultrafilter over K . iG(K)> 2K 2 K * , so there are g, E K~ for a < 6 such that { [ : g,@)
ON THE GCH AT MEASURABLE CARDINALS
109
Definition 1. For any ordinal 8, a &set is a set b C ORD such that b has order type 0.8, and such that for each 5 E b, 5 > sup(b 5). Definition 2. I f b is a &set, f < 0, and n < o, y(b, f , n ) is the o * f + n t hordinal in b and (a) h(b, 0 = sup({y(b, f , m ) : m < a}). 2(b, f ) = {x E A@, #9 : 3m V k > m [y(b,f , k ) E x (b) g(b)=(a(b,q):q
>K A
V r < u [rK < u] }.
n
a
We shall eventually show that if 8 < K , b is a &set, and b C Kw.8+1, then
Y f < 8@(b,f ) . This will prove the theorem for 8 < K . The full theorem will then follow immediately, since for any 8 one may carry out the entire proof within Ult,+,(V, U),where ?L is any K-complete non-principal ultrafilter over K .
Lemma 5. If M is a transitive model for ZFC U a K-complete ultrafilter over M E M, u E KO,and p <(I, then i Z h M ( u ) = u . K, Lemma 6. If b is a &set, 8 < K , and g < 8 is such that VQ < f [h(b,Q) < K + + ] and V n Vq [f < q < 8 + y(b, 77, n ) E KO], then @(b,4). Proof. Let 6 = sup({h(b,Q) : 77 < g}). S < K++. In the special case that 6 < K , the lemma is standard and just uses the fact that K is measurable. In the general case, let M and U be as in Lemma 4, and apply the proof for 6 < K within Ultl(M, CU nM). Definition 4. If a C ORD and X C_ L [a], then H(a, X) = { y E L [a] : y is definable in L [a] from elements of a U XU {a}}. Lemma 7. Let a C K, be of order type a, where a < K . Say a = { u1 : f < a}, in increasing enumeration. Then for each f < a , card(uE n H(a, K1+l ul))
< K+.
Proof. We use induction on f
-
< a.Assume the lemma holds for 77 < [. Let
110
K. KUNEN
<
<
p = sup {uo : q t } .If p = ul, the induction step is trivial, so assume p at. Let A = H(u, KC p ) . card@ nA) < K + . Let T be the transitive collapse of A, j the isomorphism from A onto T , 6 = j ( p ) , and b = j(u). 6 K + + , and
-
<
j(T)=Tfor T E K [ + ~- @ + 1). Let M a n d U be as in Lemma4. Lemma 7 for t will follow if we can show that for all n E ut f H(u, l Kt+l at), j(n) < n M ( ~ ) Say . n is definable in L [a] from elements of s U r U {a},. where s is a finite subset of u n uC and r is a finite subset of KC+l u C .Then j(n) is definable in L [ b ] from elements of j ( s ) U t U { b ] . Now if n < u t, then j(n) is fixed by: i n M , since all elements of j ( s ) U r U { b }are. Thus, j ( n )< " M ( K ) . We now prove the theorem, using the remark following Definition 3. Suppose 8 < K , b is a 8-set, and b C Kw,+l. Let t < 8. Let A = H(b, KW+ sup({h(b, q) : q < t } ) ) and , j be the isomorphism fromA onto the transitive collapse of A . For t < q < 8 and n < w , j ( y ( b , q, n)) = y ( b , q. n ) E K O ,and for q < 8 and n < w, j(b, q, n) < K++ by Lemma 7. Hence, by Lemma 6, @(I@), t).It follows that @(b,t),since A iL [ b ] and @(b,t ) is a property of b and t expressible within L [ b ]. Thus, V 4 < 8@(b, t).
it
-
-
ig
-
References [ I ] J.L. Kelley, General topology (Van Nostrand, Princeton, N.J., 1955). [ 21 K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic l(1970) 179-227.