correctly computed (Theorem below) seems to have been missed, as the argument is not dissimilar to other arguments in this area. If is singular and Jo

Transcription

1 On Successors of Jonsson Cardinals J. Vickers Dept. of Mathematics, University of Bristol. Bristol BS8 TW, England. P. D. Welch Graduate School of Science & Technology, Kobe University, Rokko-dai, Nada-ku Kobe 657, Japan. Abstract We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jonsson cardinals, assuming O Sword does not exist. Namely, if is a Jonsson cardinal then + = +K, provided that there is no non-trivial elementary embedding j : K?! K. There are a number of related results in ZF C concerning P() in V and inner models, for a Jonsson or singular cardinal. x Introduction An inner core model, built under some assumption on the limitation of size of the universe V of sets, reects many of the cardinality and conality properties of V. Such \Covering Properties" are well known, and such results of Jensen's, which assert that if O # does not exist, then singular implies +V = +L, and from which also follows the fact that weakly compact implies +V = +L, are now seen as precursors of a family of theorems related to generalisations of L - the so-called core models, also built under some assumption that will imply the model's rigidity. (By a model M's rigidity we mean that there is no non-trivial elementary embedding j : M?! M.) With the Dodd-Jensen core model (which can be viewed as being built under the assumption there is no inner model of a measurable cardinal) the possibility was open for further properties to reect from V to the \generalised constructible hierarchy" of K DJ : namely Erd}os, Jonsson, and Ramsey properties. Recall that a cardinal is Jonsson if every algebra A = ha; (f n ) n<! i; A (where f n is a sequence of nitary functions) has a proper elementary subalgebra A of cardinality. (See [3]). Ramsey, and Jonsson cardinals are Ramsey (and so Jonsson) in K DJ by results of Jensen [] and Mitchell [5]. Ramsey cardinals are already weakly compact (see Jech [2]) and it was known that weakly compact implied +V = +K DJ. We denote here by K the core model for measures of order zero, (see [4]), where measures are allowed, as long as they have no measure set concentrating on smaller measurables. It is known (under the working assumption that K is rigid, which we abbreviate as \ O sword does not exist" or \:O s "), that the successors of singulars are computed correctly by the Covering Lemma for that model and that the same holds for successors of weakly compacts (by essentially the same argument as for K DJ ). That successors of Jonsson cardinals are AMS Class: 3E55, 3E, 3E45

2 correctly computed (Theorem below) seems to have been missed, as the argument is not dissimilar to other arguments in this area. If is singular and Jonsson, for example by a result of Prikry [6], if is a limit of measurable cardinals, then +V = +K by the Covering Lemma for K. So we restrict ourselves in the theorem to regular. In the subsequent theorems we prove some related ZF C results: that for a regular Jonsson if V = K then V + = K + (Theorem 2); that V 6= L[A] for any A (Lemma 3); and for singular cardinals of uncountable conality: if V = K then either is Jonsson or V + = K +. Some further related ZF C results could be mentioned: a Jonsson cardinal cannot be a successor cardinal by work of Tryba, [], and Woodin. Shelah, has shown, also in ZFC, that a regular Jonsson is weakly -Mahlo [8] and even weakly!-mahlo [7]. The set theoretical notation is standard. We assume familiarity with the construction of such models, and use simplied covering lemma style arguments, involving \lifting-up" embeddings. x2 Theorem (ZFC+ \O Sword does not exist"). Suppose is a Jonsson cardinal. Then + = +K. Proof As intimated in the introduction we may assume that is a regular limit cardinal. Suppose the theorem false let = +K < + and let D :?! be a monotone conal map. (D exists since by the Covering Lemma for K cf( +K ) =.) Appealing to the Jonsson property of in V, let X hk ; E K ; Di where card(x) = ; X \ 6=. [Note a substructure Y with Y \ = would not be proper as K j= \ is the largest cardinal"]. Let j : hh; E; Di?!! hk ; E K ; Di come from the inverse of the transitive collapse on X. Then, by virtue of the above, and D: () j 6= id; j() = ; j\on \ H is conal in. Let = On \ H. Now form the coiteration of (Hj; Kj) to nal models (H ; M ) setting H = Hj; M = Kj, with iteration maps H i;j ; M i;j (i j ), and indices h iji < i. (2) crit( H ). Proof As Kj = Hj, this could only fail if = +H and = with E H 6= ;: If = +K, then by elementarity of j we should have E K 6= and EH E K. By the initial segment condition on premice this would imply E H = E K contradicting the denition of coiteration. (3) H is a proper initial segment of M. In fact: (i) =, (ii) H \, (iii) There is a cub C so that i < j 2 C =) M ij ( i) = j = j where i = crit(e i ). Remark: In the terms of the mouse ordering, (3) says that Hj < K j. We know Hj Kj by virtue of the embedding j. (The Dodd-Jensen Lemma [4], x2-, Lemma 3 rules out the possibility of the reverse strict inequality. Otherwise suppose Hj = Kj. 2

3 Then a standard argument shows that both H \ ; M \ and, and this would contradict (iii).) Proof We may \shift" or \copy" maps of the iteration from H to H over on the K-side in a familiar way: set j = j; K = Kj. Now assume j i : H i?!! K i is dened. If ii+ H : H i?!! Ult(H i ; Ei H ), then dene ii+ : K i?! K i+ = Ult(K i ; j i (Ei H )) and j i+ : H i+?! K i+ by j i+ ( H ii+ (f)(h i )) = ii+(j i (f))(crit(j i (Ei H ))). We take direct limits at limit stages. It is straightforward to verify that we have a! -commuting system. Note that there has been no truncation on the H-side of the coiteration, so all ultrapowers are formed using functions from the models concerned and are! -preserving. Set ~ = j : H?! K and set K ~ = K. Let ~ = ; : Kj?! K. ~ Then ~ j = ~ H, by commutativity. So suppose H = M for some. Let H = H ; M = M and dene the copy maps j i ; ij ~; ~ ; K ~ as above. So ~ : H?! K ~ and ~ : M?! K. ~ Note that we can construe the iteration of Kj = M?! M as also an iteration of the full K to some K by using the same indices and measures. There is thus M ; : K?!! K; K M. We now extend the map ~ to a map : K?! W, with ~ ; W K ~ by using a \pseudo ultrapower"or \long extender" construction. H H S S SSw j H M???? M - M -?? K - W [Namely, we form a domain. D = fha; fijf 2 ^ domf 2 K (= H K ) ^ a 2 ~ (dom(f))g. We set ha; fi e I hb; gi i ha; bi 2 ~ (fha; vijf(u) 2 g(v)g) and _E(ha; fi) i a 2 ~ (fujf(u) 2 Eg) ; - = nally set D = hd; I; e; _Ei. By induction on -structure we may prove a Los Theorem for -formulae : if ha ; f i; : : : ; ha n ; f n i 2 D and ' 2 then D j= '[h??! a i ; f i i] ()?! a i 2 ~ (f~ujk j= '[ f??! i (~u)]g). Then I satises the axioms for equality, and D satises extensionality. If e is wellfounded then there is an isomorphism [ ] : D g!w where W is a transitive class, and [x] = [y] i x I y. 2 e We dene : K?! W by (x) = [h; fh; xigi]. is then a conal map of the universal weasel K into the L[E] model W. Note that W is also iterable: this is clear for internally dened iterations of W. Note that we get iterability of the model W virtually for free: any set-sized initial segment of W is iterable by any iteration denable within W (by elementarity of ~ ); but in the region of linear iterations by measures, any iteration whatsoever can be \embedded" back into a \universal" iteration of W that can be dened internally to W. (This \universal iteration" procedure, and the construction of the \embedding back" of a general iteration is described in full detail for non-overlapping extenders in [9] Def. 2.8 and Lemma 2..) If all initial segments of W are iterable, then clearly W is iterable. Clearly W is also universal. ~K 3

4 Claim e is wellfounded. Proof This follows from the fact that cf() = >!. If e were illfounded, let ha i ; f i i, for i <!, be a sequence witnessing illfoundedness. Let be a suciently large cardinal with ff i g Kj. As is regular, there is a transitive set t 2 K with t [ i a i. Let X Kj; card(x) < ; with t [ ft; f i g i2! X. Let : M g!x be the inverse of the transitive collapse, with (f i ) = f i. Then check, if we coiterate (M; K ) to (P; Q) with maps = MP ; = K Q, then K absorbs M (as K = Kj which is \universal" for mice of smaller size than ). Hence P Q, and (t) = (t) = t. Then for all i <! ha i+ ; a i i 2 ~ (fhu; vij (f i+ )(u) 2 (f i )(v)g). Set (f i ) = g i. If the iteration K ;Q took < many steps, we may embed the mouse Q back into a -universal iteration of K. That is, there is a mouse Q 2 K and a map : Q?! Q. Set g i = (g i ). Then still (t) = t and ha i+ ; a i i 2 ~ (fhu; vijg i+ (u) 2 g i (v)g). But now for all i <! ~ (g i+ (a i+ )) 2 ~ (g i (a i )) - a contradiction. QED(CLaim). Also crit( ) = crit( ) = crit(~ ) = crit(j) =, whilst W extends K. ~ But W is a simple iterate of K (as we are below O pistol, all universal weasels are simple iterates of K), and crit( KW ) = crit(~) j() > as K ~ extends Kjj(). This is a contradiction to the Dodd-Jensen Lemma. Hence H is a proper initial segment of M. Standard regressive set arguments show that (i)-(iii) hold. QED(3) Now consider the coiteration of (H ; M ) where H = H and M = K. Let the nal models be (H ; M ) with maps H and M. Then for i <, exactly the same indices and measures are used as in the coiteration of (H ; M ). (4) H () = ; On \ H = On \ H =. Proof The rst follows from H \ and the inaccessibility of. But similarly H j= ZF? and cf(on \ H) = cf() =, and by induction on i one may show On \H i = On \ H =. QED(4) There is a \primed" version of the diagram above obtained by adding a prime to all models and maps (other than j!), which we think of as coming from copying the map H to a ~ via j, where now ~ : M?! K ~, and ~ : H?! K ~. (5) H is a proper initial segment of M ; and < ( + ) M. Proof By coiteration M extends Hj. Let C be as in (3)(iii) & let i = min C. Then for i < j 2 C +M j j has conality +M i i. As M is the direct limit model, we have cf(( + ) M ) <. However On \ H = and this has by design conality equal to. Hence ( + ) M 6=. But ( + ) M since otherwise the next stage would involve a truncation of the model H. But in the comparison of H with K there can be no truncation on the H side. QED(5) Let M ~ be the shortest initial segment of M so that On \ M ~ whilst! n+ ~M <! n~ for some n <!. Clearly for such an n m n =) M!m~ =, and ~ M M is a sound mouse. And M ~ extends H. 4

5 (6) ~ () = and ~ \. Proof By the same reasoning as in (4) above: ~ \ (since H \ ) and ~ Kj?! ~K j ~ ( ). Again each stage of this iteration leaves both and xed by the inaccesibility of. So ~ \. Let Q = ~ M jj and Q = ~ K jj. (7) ~ Q : Q?! Q conally. QED(6) Proof Because j\ is conal in, by (4),(6), the commutativity of the primed diagram, and the denition of ~ M. (8) There is k ~ Q, and a mouse N ~ K jj with (i) k : M ~?! (n) N (ii) N is sound above and! n+ N <!n N. QED(7) Proof As Q & Q both think is the largest cardinal and cf() >!, we have the right conditions to apply the Extension of Embeddings Lemma (cf. x3.2 of []). This gives the existence of such a k and N. QED(8) But as N is sound above and! n+ N, there is A 2 (n) (N) coding N; A. But ~K is a universal weasel extending Nj = K ~ j. A simple comparison argument of N with K ~ shows that A 2 ~ K and so codes a wellorder of length On \ N. But = ( + ) ~ K! A contradiction. QED Theorem 2 Let be a regular Jonsson cardinal, and suppose V = V K. Then V + = V K +. Proof Suppose X. Let K V be an elementary substructure of some suciently large, so that (i) K K ; X (ii) card (K) < + ; (iii) if =df +K then cf ( ) = : Let D :?! be a conal function monotonic into. We use most of the notation of the previous theorem. Let j : hh; E; D; Xi?!! hk; E K ; D; Xi arise from the Jonsson property, with = H ; ran j \ 6= : Then D :?! conally where = +H. Again consider the rst many steps of the coiteration of H = H with K = M yielding models hh i ; M i ji i, and maps i;j H ; M i;j. Note that although H; K are not mice, both model \V = K ", so this makes sense. Then set i H(X) = X i. Then for a cub set C as in (3)(iii) above, we have crit(i; H ) i (if the former map is non-trivial) and so X \ i = X i \ i. As H i j= \ V = K " we have X i \ i 2 P( i ) H i = P( i ) M i, and a pressing down argument yields that ij M(X i \ i ) = X j \ j on a stationary set of i < j 2 C. Consequently X 2 M. As in (3), dene the \copy" systems ij ; ~; j i ; ~ = j : H?!! K, ~ as before, where now ~ : K?!! K. ~ Again this makes sense as we are simply applying full measures below 5

6 to form ultrapowers. Indeed there is ~ : V?!! V ~ obtained by applying the same sequence of measures to form an iteration of all of V. Notice that ~(Y ) = ~ (Y ) for any Y 2 P() \ K: Set X ~ = ~ (X ). As H j= \ = +K ", and cf(), the same is true in H. However < ( + ) M (for the same reasons as before) so there is a least initial segment M ~ of M with On \ M ~ whilst! n+ ~M <!n~ for some n. Noting that X M 2 M ~ we nish as before : Let Q = M ~ k ; Q = (K k ) ~K we may \lift-up" ~ Q :?! Q conally, to a k : M ~?! (n) N as in (6) & (7) above. Then X ~ 2 N, and there is a code for N in K ~ = df K ~V. But X ~ = ~ ( (X) H = ~(j(x)) by the commutativity of the copy system. Hence X ~ = ~(X) = ~ (X). As V ~ j= \~(X) 2 K" we have X 2 K, by elementarity of ~. QED If is a regular Jonsson cardinal, then it is not hard to see that V + 6= V L[K] +. This is immediate from the following observation, (noticed independently by Welch and Mitchell) which is short (and relevant) enough to repeat here: Lemma 3 ([2]) If is a regular Jonsson cardinal, then for all A, V + 6= V L[A] +. Proof Suppose otherwise and choose A contradicting the theorem, and work in L[A]. Then if j : L [A]?! L [A] is non-trivial, then using the regularity of, we can perform a coarse \Lift-up" (as at (3) of Theorem above) and extend j to ~ : L[A]?! L[A]. This is absurd as we have dened in hl[a]; 2i using a class term in the language of set theory the embedding j from the parameter (indeed we can easily eliminate the parameter as it is \the largest Jonsson cardinal"). But it is impossible in ZF C to dene in hv; 2i an elementary embedding from a proper inner model into the universe. (In fact even allowing the use of parameters which we do not need here). See []. QED One can compare this with the fact that for no A is Ramsey in L[A]: Ramseyness implies that A # exists. A result similar to Theorem 2 for singular cardinals follows. Its proof is a much simplied Covering Lemma argument. Theorem 4 Let be a singular cardinal of uncountable conality. Suppose V = V K. Then either is Jonsson or V + = V K +. Proof We suppose is not Jonsson in V. Then by a result of Prikry [6], the measurable cardinals of V (and hence K) below are bounded below. Let be the supremum of these measurables, if there are any; otherwise let =. Let = cf(). Let X. Without loss of generality we may suppose = X= cf (). Let Y H + be such that (i) [ X Y ; (ii) Y \ is unbounded in (iii)! Y Y ; (iv) = Y! <. These are possible by our assumptions on. Let : hh; K; Xig!hY; K \ Y; Xi. We compare (K; K) to models (M ; K ) with maps ij ; ij. Let () =. Note by the Covering Lemma for K, K j= \ is singular". As K j = K j, and there can be no truncations on the K side of the coiteration (by universality of K), then K = K, i.e. K does not move. Let be the least so that 6

7 j = K j. Let h i j i < i be the critical points of measures used on the K = M side of the coiteration. M () = k + and M = Ult(M k ; E k ) where crit(e k ) = k <. (Proof K j= is singular, so cannot be amongst the f i j i < g as the latter are all inaccessible in M. As no ordinal 2 (; ) is measurable in, we can only have that f i j i < g\ is bounded below, and that maxf i j i < g = k for some k < and the result follows.) As some truncation occurs in the comparison of K with K we have that! n+ M k <! n M for some n <!, and M = df M is the closure of k [ fp g under good (n) M (M) functions. For P a premouse, let H (n)p (Z) be the (n)p hull of Z. Then in particular: (2) fx \ j < g H (n)m ( k [ fp g). M (3) There is ~ K j ; ~ : M?! (n) M where M is a mouse, M j = K j, and with the properties : (i)! n+ M and M is sound above. (ii) fx \ () j < g H (n)m (( k ) [ f(p )g). M To prove (3) we cannot quite quote the Lift-Up Lemma (Lemma 3 of []), as that supposed (there, which we want to set here to ) was a successor cardinal of M. However cf() >! and is a limit cardinal of M, and an inspection of the proof of the Lemma shows this to be sucient for our (weakened) conclusion (i). By (i) there is A (n) (M) coding M, and by a comparison argument we see A 2 K and hence M 2 K. Let f :! conally, with f 2 K. If g() = df hf ; x i where F is a good (n)m -function, and x 2 [( k )] <! is S chosen, using (ii), so that X \ f() = F M (x ; (p )), M we see that g 2 V = K. Hence X = < F M (g() ((g() ) is in K. QED 7

8 References [] H-D. Donder, R.B. Jensen & B.J. Koppelberg, Some Applications of the Core Model, in \Set Theory and Model Theory"; Ed. R.B. Jensen & A. Prestel. Springer Lecture Notes in Mathematics, vol 872, 98, [2] T. Jech, Set Theory, Academic Press, New York, 978. [3] A. Kanamori, The Higher Innite, Perspectives in Math. Logic, Springer Verlag, Berlin, 994. [4] R.B. Jensen, The Core Model for Measures of Order Zero, Circulated manuscript, 989. [5] W. Mitchell, Ramsey Cardinals and the Core Model, in J. of Symbolic Logic. 44, [6] K. Prikry, Changing Measurable into Accessible Cardinals, Dissertationes Mathematicae, 68, 97, [7] S. Shelah, More on Jonsson Algebras & Colouring, to appear in The Archive for Mathematical Logic. [8] S. Shelah, Cardinal Arithmetic : Chapter IV, Oxford Logic Guides. [9] J.R. Steel & P.D. Welch, 3-Absoluteness and the Second Uniform Indiscernible, in Israeli J. of Math. 4 (998), [] J.Tryba, On Jonsson cardinals with uncountable conality, Israeli Journal of Mathematics, 49, 984, [] J. Vickers & P.D. Welch, On Elementary Embeddings of an Inner Model to The Universe, to appear in the J. of Symbolic Logic. [2] P.D.Welch, Some remarks on the Maximality of Inner Models, to appear in Proc. Logic Colloquium 998, Praha, Ed. S.Buss, P. Hajek & P.Pudlak. 8