The Cofinality Spectrum of the Infinite Symmetric Group
Author(s): Saharon Shelah and Simon Thomas
Source: The Journal of Symbolic Logic, Vol. 62, No. 3 (Sep., 1997), pp. 902-916
Published by: Association for Symbolic Logic
Stable URL: https://www.jstor.org/stable/2275578
Accessed: 07-02-2019 12:46 UTC
REFERENCES
Linked references are available on JSTOR for this article:
https://www.jstor.org/stable/2275578?seq=1&cid=pdf-reference#references_tab_contents
You may need to log in to JSTOR to access the linked references.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms
Association for Symbolic Logic
is collaborating with JSTOR to digitize, preserve and extend access toThe Journal of Symbolic Logic
THE COFINALITY SPECTRUM OF THE INFINITE SYMMETRIC GROUP
SAHARON SHELAH AND SIMON THOMAS
Abstract. Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals A such that S can be expressed as the union of a chain of
i proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper.
THEOREM. Suppose that V t GCH. Let C be a set of regular uncountable cardinals which satisfies the jollowing coalitions.
(a) C contains a maximum element.
(b) Iju is an inaccessible cardinal such that ui = sup(C n iu), then ,u E C.
(c) I'li is a singular cardinal such that pi = sup(C n iu), then i + E C. Then there exists a ce..c. notion offorcing P such that VP t CF(S) = C.
We shall also investigate the connections between the cofinality spectrum and pef theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.
?1. Introduction. Suppose that G is a group that is not finitely generated. Then G can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals i such that G can be expressed as the union of a chain of i proper subgroups. The cofinality of G, written c (G), is the least element of CF(G).
Throughout this paper, S will denote the group Sym(co) of all permutations of the set of natural numbers. In [5], Macpherson and Neumann proved that c (S) > NO. In [6] and [7], the possibilities for the value of c(S) were studied. In particular, it was shown that it is consistent that c(S) and 2"O can be any two prescribed regular uncountable cardinals, subject only to the obvious requirement that c (S) < 2No. In this paper, we shall begin the study of the possibilities for the set CF(S).
There is one obvious constraint on the set CF(S), arising from the fact that S
can be expressed as the union of a chain of 2`0 proper subgroups; namely, that
cf (2No) c CF(S). Initially it is difficult to think of any other constraints on CF(S). And we shall show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, the following result is a special case of our main theorem.
THEOREM 1.1. Let T be any subset of o -. {O}. Then it is consistent that Nn C CF(S) if and only if n C T.
Received June 23, 1995; revised November 15, 1995.
Research partially supported by the BSF. Publication 524 of the first author. Research partially supported by NSF Grants.
? 1997, Association for Symbolic Logic 0022-4812/97/6203-0012/$2.50
After seeing this result, the reader might suspect that it is consistent that CF(S) is an arbitrarily prescribed set of regular uncountable cardinals, subject only to the above mentioned constraint. However, this is not the case.
THEOREM 1.2. If ?N c CF(S) for all n c o -,. {0}, then N?1,+I C CF(S).
(Of course, this result is only interesting when 2'0 > N,?,+.) In Section 2, we
shall use pcf theory to prove Theorem 1.2, together with some further results which restrict the possibilities for CF(S). In Section 3, we shall prove the following result.
THEOREM 1.3. Suppose that V t GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.
(1.4)
(a) C contains a maximum element.
(b) If u is an inaccessible cardinal such that u = sup(C n ,u), then ,u c C. (c) If u is a singular cardinal such that u = sup(C n ,u), then Au+ c C. Then there exists a ccc. notion offorcing P such that VP t CF(S) = C.
This is not the best possible result. In particular, clause (1.4) (c) can be improved so that we gain a little more control over what occurs at successors of singular cardinals. This matter will be discussed more fully at the end of Section 2. Also clause (1.4) (a) is not a necessary condition. For example, let V t GCH and let
C = { N,+I I a < co }. At the end of Section 3, we shall show that if X is any singular cardinal such that cf (X) c C, then there exists a c.c.c. notion of forcing P such that
VP t CF(S) = C and 2'0 = -. In particular, 2'0 cannot be bounded in terms of
the set CF(S).
In this paper, we have made no attempt to control what occurs at inaccessible cardinals ,u such that u = sup(C n lu). We intend to deal with this matter in a second paper, which is in preparation. In this second paper, we also hope to give a complete characterization of those sets C for which there exists a c.c.c. notion of forcing P such that VP t CF(S) = C.
Our notation mainly follows that of Kunen [4]. Thus if P is a notion of forcing and p, q c P, then q < p means that q is a strengthening of p. If V is the ground model, then we often denote the generic extension by VP if we do not wish to specify a particular generic filter G C P. If we want to emphasize that the term t is to be interpreted in the model M of ZFC, then we write tM; for example, Sym(co)M.
If A C co, then S(A) denotes the pointwise stabilizer of A. Fin(co) denotes the subgroup of elements ir c S such that the set {n < o Ir (n) 74 n } is finite. If X, c' S, then we define X = * y if and only ifq - I c Fin(co).
?2. Some applications of pcf theory. Let (,i i c I) be an indexed set of regular cardinals. Then I7iEG Ri denotes the set of all functions f such that dom f = I and f (i) c Ri for all i c I. If 5 is a filter on I and J is the dual ideal, then we write either lJiEG tJ/ or lJiI Ei/y for the corresponding reduced product.
We shall usually prefer to work with functions f c fLiE1 Ri rather than with the
corresponding equivalence classes in J7iEJ ~i For f, g, c I7iI ii, we define
f <? g iff {i C I I f (i) > g(i)} CE
We shall sometimes write f <?g g, f <ga g instead of f <?g g, f <j g respectively. If e1 = {q}, then we shall write f < g, f < g. Suppose that there exists a regular
cardinal A and a sequence (fO I a < A) of elements of Hliei Ai such that
(a) if a < /l < A, then f, <_y ffl; and
(b) for all h E flie, Ai, there exists a < A such that h <_ fa.
Then we say that A is the true cofinality of fei ii and write tcf (lJiI i/) =
Furthermore, we say that (f a lo < A) witnesses that tcf (fl / = A. For
example, if q is an ultrafilter on I, then ]7IJ Ai/o is a linearly ordered set and
hence has a true cofinality. A cardinal A is a possible cofinality of Jlie, Ai if there
exists an ultrafilter q on I such that tcf (JiJ Ai/g) = A. The set of all possible cofinalities of Hiei Ai ispcf (Hiei Ai).
In recent years, Shelah has developed a deep and beautiful theory of the structure
of pcf (Hli Ai) when III < min{fi I i E I}. A thorough development of pcf
theory and an account of many of its applications can be found in [13]. [1] is a self-contained survey of the basic elements of pcf theory. In this section of the paper, we shall see that pcf theory imposes a number of constraints on the possible structure of CF(S). (Whenever it is possible, we shall give references to both [13] and [1] for the results in pef theory that we use.)
THEOREM 2. 1. Suppose that (in I n < o) is a strictly increasing sequence of nals such that An E CF(S) for all n < co. Let q be a nonprincipal ultrafilter on co, and let tcf (fln<c, An /_) = A. Then A E CF(S).
PROOF. For each n < co, express S = U1<;. Gi as the union of a chain of An
proper subgroups. Let (f, I a < A) be a sequence in ln<.c An which witnesses that
tcf (fJn<c* An/9) = A. For each a < A, let Ha be the set of all g E S such that {n < co I g E Gn E _q. Then it is easily checked that Ha is a subgroup of S, and that Ha C Hi for all a < fl < A. Suppose that g E S is an arbitrary element. Define f E fn<. An by f (n) = mini I g E Gi}. Then there exists a < A such that f <a faO. Hence g E Ha. Thus S= Ua<, Ha.
So it suffices to prove that Ha is a proper subgroup of S for each a < A. Fix some a < A. Lemma 2.4 [5] implies that for each n < co, i < An and X E [co](, the setwise
stabilizer of X in G. does not induce Sym(X) on X. Express co = Un<(, Xn as the
disjoint union of countably many infinite subsets Xn. For each n < w, choose 7rn E
Sym (Xn) such that g [ Xn 7' rn for all g E G n). Then Uf= < n E S
PROOF OF THEOREM 1.2. By [13, 11 1.5] (or see [1, 2.1]), there exists an ultrafilter
q on co such that tcf (Iin<co Nn/9) = -19+1
If we assume MA,, then we can obtain the analogous result for cardinals X, such that No < , < 2NO. (In Section 3, we shall prove that the following result cannot be proved in ZFC.)
THEOREM 2.2 (MAJ). Suppose that (/{Q I ag < K) is a strictly increasing sequence of cardinals such that Aa E CF(S) for all ag < A. Let q be a nonprincipal ultrafilter
on A, and let tcf (fla<, Ah/g) = A. Then A E CF(S).
PROOF. For each a < x., express S Ui<. Ga as the union of a chain of Aa
tcf (JJa<, /{,/O) ={. For each / < A, let Hfl be the set of all g E S such that f{g < X I g E G (a)} E -. Arguing as in the proof of Theorem 2.1, it is easily
checked that (HA I / < A) is a chain of subgroups such that S = U#<. Hfl. Thus it suffices to prove that Hfl is a proper subgroup of S for each /1 < A. Fix
some /1 < A. Suppose that we can find an element g E S \' U.<, G (a)
Then clearly g V Hfl. But the existence of such an element g is an immediate consequence of the following theorem. -1
THEOREM 2.3 (MAJ). Suppose that for each ag < X, S = Ui<o, H. is the union of
the chain of proper subgroups H... Then for each f E fl<, Oa, S 7' Ua<,. HJ(a)* REMARK 2.4. In [6], it was shown that MA,. implies that c(S) > K,. This result is
an easy consequence of Theorem 2.3.
REMARK 2.5. In [5], Macpherson and Neumann proved that if {H, I n < co} is an arbitrary set of proper subgroups of S, then S 74 U,<. Hn. It is an open question whether MA,< implies the analogous statement for cardinals X. such that
No < X. < 2t?. Regard S as a Polish space in the usual way. Then the proof of
Theorem 2.3 shows that the following result holds.
THEOREM 2.6 (MAJ). Suppose that for each a < X, Ho is a nonmeagre proper
subgroup of S. Then S 74 Ua,<, Ho.
Unfortunately there exist maximal subgroups H of S such that H is meagre. For example, let co = Qf U Q2 be a partition of co into two infinite pieces. Let
H = {g c S lg[Q]Qi] I < No for some i C {1,2}}.
(Here A denotes the symmetric difference.) Then H is a maximal subgroup of S; and it is easily checked that H is meagre.
PROOF OF THEOREM 2.3 (MA,.). We shall make use of the technique of generic sequences of elements of S, as developed in [3]. (The slight differences in notation between this paper and [3] arise from the fact that permutations act on the left in this paper.)
DEFINITION 2.7. A finite sequence (gi... ., g,) E S' is generic if the following two conditions hold.
(1) For all A E [CO]<, there exists A C B E [cow]< such that gi[B] = B for all 1 < i < n.
(2) Suppose that A c [CO]<' and that gi[A] = A for all 1 < i < n. Suppose further that A C B E [co]< and that hi E Sym(B) extends gi j A for all
1 < i < n. Then there exists 2T E S(A) such that gi7r-I extends hi for all
1 < i < n.
CLAIM 2.8. If (g, ... .,g ),(hi, ... , hn) E Sn are generic, then there exists f E S
such that fgif = hi for all 1 < i < n.
PROOF OF CLAIM 2.8. This follows from [3, Proposition 2.3]. From now on, regard S as a Polish space in the usual way.
CLAIM 2.9. The set {K(gi, . , gn) C Sn I (gi, . . ., gn) is generic} is comeagre in Sn in the product topology.
PROOF OF CLAIM 2.9. This follows from [3, Theorem 2.9]. H
CLAIM2.10. If (gi,. Ign+) E S+ 1 is generic, then for eachA A [O]<w,m E
co -- A and 1 < ? < n + 1, the following condition holds.
(2.11)A,m,e LetQ = {iII < i < n + 1,i #? f}. If gi[A] = Aforall i E Q. then there exists B c [co -. A]' such that
(a) m C B;
(b) gi[B] = B for all i E Q; (c) ge[A U B = A U B;
(d) for all ir C Sym (Q), there exists q C Sym (B) such that q$(g1 L B)
g,,(i) [ B for all i E Q.
PROOF OF CLAIM 2.10. For each A E [co]<',m E co - A and 1 < ? < n + 1,
let Cn+1 (A, m, ?) consist of the sequences (gi1. . ., gn+1?) E sn+1 which satisfy (2.11)Ame. Then it is easily checked that Cn+l(A,m,?) is open and dense in
Sn+ I. Hence Cn+1 = nA m e Cn+I (A, m, ?) is comeagre in Sn+h1. Claim 2.9 implies
that there exists a generic sequence (gi,... Ign+1) E Cn+l. So the result follows easily from Claim 2.8. H
DEFINITION 2.12. If a is an infinite ordinal, then the sequence (gi i < a) of
elements of S is generic if for every finite subsequence il < ... < in < a, (gi, , gi ) is generic.
We have now developed enough of the theory of generic sequences to allow us to begin the proof of Theorem 2.3. Consider the chains of proper subgroups,
S = Uj<0c. Hi' for a < a.. We can assume that Fin(co) < Ho' for all a < ti. Let
f E fJ<K, 0Oa. We must find an element 7z E S - Ua<<K H7(a). We shall begin by inductively constructing a generic sequence of elements of S
(gg ? I I gog I .. )ao< K
such that for all a < K, there exist fc(a) < y, < ,O such that go G HY' and g Ha. Then we shall find an element 7t c S such that rgo7-' =* g' for all ce <K. This implies that 7r V Ua,<K Hyf D UK H"()
Suppose that we have constructed go, gay for / < a. For each finite subsequence g
of (goS, g)/ < al), the set {h c SI g-h is generic} is comeagre in S. (See [3, p. 216].)
Since MA, implies that the union of Kc meagre subsets of a Polish space is meagre, the set
{h C S|(gangj J 1 < ac)-h is generic }
is also comeagre in S. So we can choose a suitable go and f (a) < ya < 0 with
go c Ha. The set
C = {h C S | (g gJ: fit/3 < a)-g 0h is generic}
is also comeagre in S. Since H7, is a proper subgroup of S, we have that C KHf' #a 0. (If not, then Hfa is comeagre and hence so are each of its cosets in S. As any two
comeagre subsets of S intersect, this is impossible.) Hence we can choose a suitable gaI E C C- H7. Thus the desired generic sequence can be constructed.
LEMMA 2.13. Let (ga, gI oI < K) be a generic sequence of elements of S. Then there exists a c-centred notion offorcing P such that
1- There exists 2T E Sym(wo) such that ngon- = * g1 for all a < a. PROOF OF LEMMA 2.13. Let P consist of the conditions p = (h, F) such that
(a) there exists A c [co]< such that h c Sym(A); (b) F E [Kl]<w;
(c) for each e F and z E {0, 1}, g [A] = A.
We define (h2, F2) < (Ah, F1) iff the following two conditions hold. (1) hi C h2and F1 C F2.
(2) Let B = dom h2- dom hl and let X h2 L B. Then q(go L B)q-1 gL B for each oa C F1.
Clearly P is ac-centered. Claim 2.10 implies that each of the sets
Dm {(h,F)Im c domh}, m < co
and
Ea={(h,F)c E F}, a < rs,
are dense in P. The result follows. H This completes the proof of Theorem 2.3. H The following theorem goes some way towards explaining why we have assumed that C satisfies condition (1.4) (c) in the statement of Theorem 1.3. (We will discuss this matter fully after we have proved Theorem 2.15.)
DEFINITION 2.14. If 1 is a limit ordinal, then J(>' is the ideal on 6 defined by jbd = {B IThere exists i < 6 such that B C i}.
THEOREM 2.15. Let K be a regular cardinal, and suppose that ( a0 o < K.) is a strictly increasing sequence of cardinals such that 'k c CF(S) for all a < a. Suppose further that tcf (Fa<, //jmb) =A Then either X. c CF(S) or i E CF(S).
PROOF. Suppose that K. CF(S). For each oa < X., express S Ui<, G.' as the
union of a chain of i{ proper subgroups. Let (f i /# < A) be a sequence in IL<K /,>
which witnesses that tcf (fla<, /{a/Jb) A. For each P < A, let G* be the set of
all g E S such that X {o < X I g E Ga)} EjJbd. Arguing as before, it is easily
checked that (Gi t / < A) is a chain of subgroups such that S = Up<; G$.
Thus it suffices to prove that GJ is a proper subgroup of S for each P < A. So suppose thatG = S for some P < A. For each a < a, define Ho = nf{G>y() I a <
y < K}. Then (Ho I a < a.) is a chain of subgroups such that S Ua<, Ho. If
a < K, then Ho, <G(X) and so Ho is a proper subgroup of S. But this contradicts
Suppose that V t GCH, and that yu is a singular cardinal. Let (SO I i < a) be the strictly increasing enumeration of all regular uncountable cardinals 0 such that
0 < ,. Let 3 = fli<, 0i. Then 11 = ,u+. Now let P be any c.c.c. notion of
forcing. From now on, we shall work in VP. Since P is c.c.c., for each g E Hli<, 0i, there exists f E Y such that g < f . Suppose now that (/{Q I a < () is an increasing subsequence of (Oi I i < q) such that 1I I < Ao and supr<s An, = ,u. Let
'F= {f E H A, I Thereexistsh E F suchthat f C h}.
a!<6
Then for all g E Ja<J Ah!, there exists f E i* such that g < f. This
plies that max(pcf (HQa<6 AO)) = y+. By [13, I] (or see [1, 4.3]), we obtain that
tCf (Ha<6 Al/jbd) = t+. In summary, we have shown that the following statement is true in VW.
THE STRONG HYPOTHESIS (2.16). Let ( be a limit ordinal, and let (Aa I a < I) be a strictly increasing sequence of regular cardinals such that 1(5 < Ao. Then tCf (fl<(5 hl/Jbd) = (supa<6 Aa)+
In particular, using Theorem 2.15 and the Strong Hypothesis, we see that the following statement is true in V'e.
(*) If yu is a singular cardinal such that yu = sup(CF(S) n u)), then either cf (u) E CF(S) oru+ E CF(S).
This suggests that we might try to replace condition (1.4) (c) of Theorem 1.3 by the following condition.
(1.4)(c)' If yu is a singular cardinal such that yu = sup(C n u), then either cf(u) E C or + E C.
However, Theorem 2.19 shows that this cannot be done. For example, Theorem 2.19 implies that if
C - { N } U {fN+1 1' < W2, Cf (() = c} U {f02+1 },
then there does not exist a c.c.c. notion of forcing P such that VP l= CF(S) = C. REMARK 2.17. The Strong Hypothesis is usually taken to be the following ently weaker statement.
(2.18) For all singular cardinals yu, pp(ju) = y +.
(For the definition of pp(4u), see [11].) However, Shelah [12, 6.3 (1)] has shown that (2.16) and (2.18) are equivalent.
THEOREM 2.19 (The Strong Hypothesis). Let K, be a regular uncountable cardinal, and suppose that (Aa a < A,) is a strictly increasing sequence of cardinals such that
ha E CF(S) for all a < A,. Supposefurther that (a) K < A0;
(b) E = {(5 < | limo5, (supa<, AOh+ V CF(S)} is a stationary subset of k.
Then r, E CF(S).
PROOF. For each &e < X, express S = U1<> Gil as the union of a chain of Aa
tcf (HQ< a/jbl) = A . Let (f6 I < It+) be a sequence in JJa<6 Aa which
witnesses that tcf (H<a Aa/Jbd) = A. For each 4 < u. let Hx be the set of allg E S such that (s x {a < s I g E Gp.,(a) } E jbd Once again, it is easily checked that
(Hx t < 8a) is a chain of subgroups such that S = U</+ HA . Since A V CF(S), there exists 7rG() < mu. such that H6 = S.
Since X, < A, there exists f E Ha<, At such that f (a) > suptff(6)(a) I a < ( e E} for all oa < a,. Let g E S. Then for each (5 E ,g E Hc(,); and so there exists
y(g,5 ) < > such that g e G., (a) C G7f(a) for all y(g,(5) < a < (. By Fodor's
Theorem, there exists an ordinal y (g) < X and a stationary subset D of E such that y(g,(5) = y(g) for all( EE D. Thismeansthatg E n{G() y(g) a< < .
Foreachy < a,, let Fy = fl{Gf(a)Iy < a < a }. Then (FyIy < a) isachain
of subgroups such that S = Uy< Fy. Finally note that Fy C G.(Y and so Fy is a proper subgroup of S for all y < a,. Thus X, E CF(S).?3. The main theorem. In this section, we shall prove Theorem 1.3. Our notation generally follows that of Kunen [4]. We shall only be using finite support iterations. An iteration of length a will be written as (IP, %(y I i < ae, y < a), where PIR is the result of the first fi stages of the iteration, and for each fi < oa there is some P# -name Qfi such that
I[ Qp is a partial ordering
and P+ I is isomorphic to P# * QDi. If p E Pa , then supt(p) denotes the support of P.
There is one important difference between our notation and that of Kunen. Unlike Kunen, we shall not use VP to denote the class of P-names for a notion of forcing P. Instead we are using VP to denote the generic extension, when we do not wish to specify a particular generic filter G C P. Normally it would be harmless to use VP in both of the above senses, but there is a point in this section where this notational ambiguity could be genuinely confusing. Suppose that Q is an arbitrary suborder of P. Then the class of Q-names is always a subclass of the class of IP-names. (Of course, a Q-name - might have very different properties when
regarded as a P-name. For example, it is possible that 1FQ - is a function, whilst
IWp - is a function.) However, we will not always have that VQ C VP; where this means that V[G n Q] c V[G] for some unspecified generic filter G C P.
DEFINITION 3.1. Let Q be a suborder of P. Q is a complete suborder of P, written Q < P, if the following two conditions hold.
1. If ql, q2 E Q and there exists p E P such that p < qi, q2, then there exists
r e Qsuchthatr < qiq2.
2. For all p e P, there exists q e Q such that whenever q' e Q satisfies q' < q, then q' and p are compatible in P. (We say that q is a reduction of p to Q.)
It is wellknown that if Q < P, then VQ C VP; and we shall only write VQ C VP when Q < P.
We are now ready to explain the idea behind the proof of Theorem 1.3. Let V k GCH, and let C be a set of regular uncountable cardinals which contains a maximum element r,. We seek a c.c.c. P such that VP 1= 2' = X A CF(S) = C. The easiest part of our task is to ensure that VP l= C C CF(S). We shall accomplish this by constructing IP so that the following property holds for each A E C.
(3.2) There exists a sequence (IP: < A) E V of suborders of P such that
(a) if < < q A,then P < P' <P1P;
(b) for each 7i E Sym(co) v, there exists 4 < A such that i E Sym(co) v (c) for each 4 < A, there exists ir E Sym(CO)V V, Sym(CO) V
The harder part is to ensure that VP l= CF(S) C C. This includes the requirement
that (3.2); fails for every A V C. So, roughly speaking, we are seeking a c.c.c. IP which can be regarded as a "kind of iteration" of length A precisely when A E C. We shall use the technique of [10, Section 3] to construct such a notion of forcing
P.
DEFINITION 3.3. Let (ai I i < a) be a sequence of subsets of a. We say that b C a is closed for (ai I i < a) if ai C b for all i E b.
DEFINITION 3.4. Let F be the class of all sequences
Q = (Pi, Q,aj I i < acj < a)
for some a which satisfy the following conditions. (We say that Q has length a and write a = lg (Q).)
(a) ai C i.
(b) ai is closed for (a1 j < i).
(c) Pi is a notion of forcing and ?2i is a Pj-name such that lFpj Q? is a c.c.c. partial order.
(d) (Pi, Q1(j I i < a, j < a) is a finite support iteration.
(e) For each j < a, define the suborder P* of Pj inductively by
IP = {p I P1 supt(p) C aj and p(k) is a P- name for all k E supt(p)}.
Then ?2i is a Pa. -name. (At this stage, we do not know whether Pa*. is a complete suborder of P1. It is for this reason that we are being careful with our notation. However, we shall soon see that P*. < Pi, and then we can relax again.)
DEFINITION 3.5. Let Q E &' be as above, so that a = lg (Q).
(a) We say that b C a is closed for Q if b is closed for (a1 j < a).
(b) If b C a is closed for Q, then we define Pb* = {P E I I supt(p) C b and p(k) is a Pk -name for all k E supt(p)}.
If f < a, then we identify PPp with the corresponding complete suborder of Pa in the usual way. If b C a, then p [ b denotes the a-sequence defined by
(p [ b)(4)= p() if b
LEMMA 3.6. Let Q E F and let a = lg(Q). Suppose that b C c C f, < a, and that b and c are closedfor Q.
(1) /3 is closedfor Q, and p = IP.
(2) If p E Pp and i E supt(p), then p [ ai 1- p(i) E Ci.
(3) Suppose that p, q cE Pp and p < q. If i E supt(q), then p [ ai I- p(i) < q(i). (4) If p E IP, then p [ b E P*.
(5) Suppose that E P, q E cP and p < q. Then p [ b < q.
(6) Suppose that p c P*, q E P# and p < q c. Define the a -sequence r by
r(G) =p() if E c
- qG() otherwise. Then r E Pp and r < p, q.
(7) P* <i Pf
PROOF. This is left as a straightforward exercise for the reader. LEMMA 3.7. Let Q E F and let a = Ig(Q). Suppose that b c a is closedfor Q and thati ECa-b.
(1) c = b U i and c U {i} are closedfor Q.
(2) P* < P* < P*Ui < P'
(3) P1u} is isomorphic to Pc * ti.
PROOF. Once again left to the reader. H Now we are ready to begin the proof of Theorem 1.3. Suppose that V l= GCH, and let C be a set of regular uncountable cardinals which satisfies the following conditions.
(1.4)
(a) C contains a maximum element, say ,.
(b) If u is an inaccessible cardinal such that ,u = sup(C n u), then ,u E C. (c) If u is a singular cardinal such that ,I = sup(C n ,), then ,u+ E C. DEFINITION 3.8. (a) EIC denotes the set of all functions f such that dom f = C andf(A) E AforallA E C.
(b) Yc is the set of all functions f C EIC which satisfy the following condition.
(*) If ,u is an inaccessible cardinal such that ,u = sup(C n u), then there exists
A <,u suchthatf(0) = OforallA < 0 E C npu.
DEFINITION 3.9. In V, we define a sequence
(PiQDf1j I ?i < j <)
such that the following conditions are satisfied. (a) f C E c.
(b) Let ai = { j < i I fj < f i . Then Q=(P~i, (0j, aj I i < r, j < r) c F.
(c) For each f E Sc, there exists a cofinal set of ordinals j < r, such that
fi = f
(d) Suppose that i < r, and that Q is a P*.-name with Q1 < ,. Then there exists i < j < r, such that
(1) fj = fi, and so ai C aj;
(2) if IFV Q is c.c.c., then Qj = Q.
We shall prove that Vet l= CF(S) = C. From now on, we shall work inside VP-.
DEFINITION 3.10. If b C K is closed for Q, then Sb = Sym(co)Vvb First we shall show that C C CF(S). Fix some ,u E C. For each 4 < u, let
be = {i < X'z f i (,u) < ,}. Clearly be is closed for Q; and if 4 < q < a, then be C b,1.
Thus (s5 b < u) is a chain of subgroups of S.
LEMMA 3.1 1. For each 4 <a, 5bC is a proper subgroup of S.
PROOF. Let 4 < ,u and let i < K' satisfy fi (u) > 4. Let Q be the partial order
of finite invective functions q: co -+ co, and let Q be the canonical IPa -name for Q.
Then there exists i < j < X, such that fj = f i and Qb = (Q. Clearly j, be. Let
c = be U j. By Lemma 3.7, Qj adjoins a permutation 7r of co such that r , Ve. It follows that 7 r H
LEMMA 3.12. S =U sbx
4<'u
PROOF. Let 7r E S. Let g be a nice IPZ-name for 7U. (Remember that IP,< = IP*.) Thus there exist antichains Aem of IP* for each (E, m) E co x co such that
g LUf,m{(em)} x Aim. Let U{ supt(p) I p E ULm Am} = {& k I k < co}. Let
sup{f~k (8u)Ik <c }. Then p E Pc for each p E Ue'm Aem, and so g is a nice
P* -name. Hence r E SbH
This completes the proof of the following result. LEMMA 3.13. If ,u e C, then E CF(S).
To complete the proof of Theorem 1.3, we must show that if ,u , C, then ,u V CF(S). We shall make use of the following easy observation.
LEMMA 3.14. Let M - ZFC, and let (gs I ,6 < a) C M be a generic sequence
of elements of Sym(co). Let Q be the partial order of finite infective functions
q: co -+ co, and let 7r E MQ be the Q-generic permutation. Then for all h E
Sym(co)M, (gfi I fi < a)Th7r is generic.
PROOF. For each finite subsequence /I <. </3n <a, the set C(al,..., an)=
{q E Sym(wo) I (ga, 5 ... , ga,, )>^ is generic} is comeagre in Sym(co). Hence h - I C (as,
5 an) is also comeagre for each h E Sym(co). So for each h E Sym(co)M, 7r E h-l C (al,... , an). The result follows. H
LEMMA 3.15. Suppose that a < K and that (gjs | fi < a) is a generic sequence of elements of Sym(co). If H is any proper subgroups of Sym(co), then there exists a permutation q V H such that (gjl I fi < a)-q$ is generic.
PROOF. Let h E Sym(co) x H. Then there exists i < K, such that h, (gf fi <
a) E VPi. There exists i < j < X, such that Q1 is the canonical PE. -name for the
a permutation TC E VPi+1 such that both (g# |ll < a)-)7r and (gf 1# < a)<^hrr are
generic. Clearly either Xr V H or h7r V H. -H
Now fix some ,u V C, and suppose that ,u e CF(S). It is easily checked that
2"O = ,., and so we can suppose that ,u is a regular uncountable cardinal such that
,u < a. Express S = Ua<u Ga as the union of a chain of ,u proper subgroups. We can suppose that Fin(co) < G,. Using Lemma 3.15, we can inductively construct a
generic sequence of elements of S
(go ~g, . agat ... )a',
such that for each a < u, there exists a < ye < u such that g, E Gy and g Gy .
LEMMA 3.16. There exists a subset X E [,u}u and an ordinal < Es such that
(gaogala E X) E VP*
PROOF. For each a < ,u and z E {O, 1}, let go be a nice P* -name for gT. Thus there exist antichains AaTm of 1P* for each (e, m) E cl x cl such that
ga = Ux(i, m) A X Aa.m
elm
For each a < yu, letU{ supt(p) I p E Ujm Aao UUjm Aa,'} = {fl 1 k < ad. Define ho E YIc byha(A) = sup{ffl (A) Ik < co} for each A E C.
It is easily checked that there are less than 4u possibilities for the restriction ha [ C n Mu. (This calculation is the only point in the proof of Theorem 1.3 where we make use of the hypothesis that C satisfies conditions (1.4)(b) and (1.4)(c).) Hence there exists X E [yu]y such that ha [ C n u = hfl r C nfu for all a,fl E X.
Define the function f E JIC by f [ C nfu = ha C nflf, where a E X, and
f (A) = sup{ha (A) I a E X} for each A E C X u. Then it is easily checked that
f E Yc; and clearly fTV < hot < for all a E X and k < co. Now choose
4 > sup{ffk I a E X, k < co} such that fo = f. Ifa E X and E {O.,1}, then
p E IP~a for each p E Ue m A`m; and hence ga is a nice aI -name. It follows that
(gao,ga a E X) e VP
By Lemma 2.13, there exists a o-centred Q E Vp' such that
U- There exists 7t E Sym(co) such that nrgyf-' = * gfor all a e X.
Q a g
Let Q be a PaP~ -name for Q. Then there exists 4 < q < ar such that fj7 = fi and
'= Q2 Hence there exists 7r E S such that izgZir-l = *gl for all a E X. But this
implies that 7 V Ua<, Ga, which is a contradiction. This completes the proof of Theorem 1.3.
By modifying the choice of the set Yic of functions, we can obtain some interesting variants of Theorem 1.3. For example, the following theorem shows that Theorem
2.2 cannot be proved in ZFC. (Of course, it also shows that (1.4)(c) is not a
THEOREM 3.17. Suppose that V t GCH and that K. > ,,, +1 is regular Let C = aN,+l I a < co} U {K}. Then there exists a c.c.c. notion offorcing P such that
VP t CF(S) = C.
PROOF. The proof is almost identical to that of Theorem 1.3. The only change is that we use the set of functions
5rc = {f E IC I There exists a < co such that f (Nl+l) = O for all a < fl < c1o}
in the definition of P,C. This ensures that the counting argument in the analogue of Lemma 3.16 goes through. H
Using some more pcf theory, we can prove the following result. THEOREM 3.18. Suppose that V satisfies the following statements.
(a) 2Rn = Nn+l for all n < co.
(b) 28 = ?+1 for some o < < co,. (c) 2" 1 = Nq1+1 for all q > A,.
Let T E [co] and let K. be a regular cardinal such that K. > ? +?. Let C =
pCf(HfLET Nn) U {i-}. Then there exists a c.c.c. notion offorcing P such that VP t CF(S) = C.
PROOF. Again we argue as in the proof of Theorem 1.3. This time we use the set of functions, SC = Jln ET Nn, in the definition of P,C. Examining the proof of Lemma 3.16, we see that it is enough to prove that the following statement holds for each regular uncountable yu f C.
(3. 19),u
If (hog I a < 4u) is a sequence in I| Nn, then there exists X E nET
and an f E H en such that hog < f for all a E X.
nET
This is easy if u < c, If yu > c,,, then (3.19), is a consequence of the following result. H
THEOREM 3.20. Let {,i I i E I} be a set of regular cardinals such that min{Ii I i E
I } > II 1. Let 4u be a regular cardinal such that 4u > 2!'! and 4u 5 pcf (H~iE , i). If (hog I a < 4u) is a sequence in fJE.I Ai, then there exists X E [yu]" and f E REI i
such that hog < f for all a E X.
PROOF This is included in the proof of [13, II 3.1]. (More information on this topic is given in [9, Section 5]. Also [8, 6.6D] gives even more information under
the hypothesis that 21I < min{ji I i E I}.) Alternatively, argue as in the proof of [1, 7.11]. H
It is known that, assuming the consistency of a suitable large cardinal hypothesis,
for each co < 4 < co, there exists a universe which satisfies the hypotheses of
Theorem 3.18. (See [2].) Thus the following result shows that Theorem 1.2 cannot be substantially improved in ZFC.
COROLLARY 3.21. Suppose that V satisfies the hypotheses of Theorem 3.18 with respect to some ao < A < w1,. Then for each co < a < A and X > + there exists a set T E [cw]' and a c. c. c. notion offorcing EP such that
VP = CF(S) {` n I n E T} U {R,+I } U {X}.
In particular, if co < a < A, then
VP F- NC,+1 , CF(S).
PROOF. With the above hypotheses, [13, VIII] implies that there exists T E [CO]' such that tcf (HIfLT 8n/Jbd) = ,+ - It follows that pCf(lHnJ T N') " {& In E
T} U f Na+1 } So the result is a consequence of Theorem 3.18. A
Finally we shall show that (1.4) (a) is not a necessary condition in Theorem 1.3, and that 21o cannot be bounded in terms of the set CF(S).
THEOREM 3.22. Suppose that V - GCH and that C -={ +j I a < co, }. If K' is
any singular cardinal such that cf (X) E C, then there exists a c.c.c notion offorcing P such that VP t CF(S) = C and 28 = -.
PROOF. Let K. be a singular cardinal such that cf(K) E C. Let (i: fi < cf(K))
be a strictly increasing sequence of regular cardinals such that A0 = +I and SUpl<cf(ro) i{/n -= i. Let
{f ENC I There exists a < co, such that f (tR?+') = O for all a < fl< w)}. In V, we define a sequence (IPx, Qj, fj I i < es, j < a) such that the following conditions are satisfied.
(a) f E E Tic
(b) Let ai = f j < i I fj ~< f i 1. Then Q= (PEi, Qp aj I i < es, j< ,) E W
(c) For each f E Tic and ,8 < cf (i), there exists a cofinal set of ordinals
j < Ap such that fj = f.
(d) Suppose that P < cf (0), i < Ap and that (Q is a P.,*-name with QI< ap.
Then there exists i < j < A: such that
(1) fj = fi, and so ai C aj;
(2) if pj[- Q is c.c.c., then Q(j Q.
Clearly V1P k 28o?- =a;. Arguing as in the proof of Lemma 3.13, we see that
VP, k C C CF(S). From now on, we shall work inside VP-. Let yu be a regular
cardinal such that +I < u < ta. Suppose that we can express S = U,< Go as
the union of a chain of yu proper subgroups. For each a < ju, choose an element hog c G - Ga. Then there exists a subset I E [,u]/i and an ordinal P < cf (K) such
that (ha I acC I) E Vp'i3 and yu < %d. In VP,, we can inductively construct a generic sequence of elements of S
(go Ig0I . gal gal -)
such that for each a < y
(1) there exists a < Ya < u such that go E G and gl f Grad and (2) there exists Ap < i, < AX+ such that (g?, g~l I < a) C V1po.
For suppose that (g9, 1 5 < a) has been defined. By Lemma 3.14, there exists i, < j < A#+1 and g? E VP' such that (g 9 gl 1 < a)<gg is generic. Choose ya E I such that a < ya < u and g Y E e By a second application of Lemma 3.14, there exists j < iA-1 < AX+, and mr E Vpic+ such that both (g9,gJ 1 < a)-g% > and
(g%,gj I < a)<g%0ho mr are generic. Clearly either m V Gyo or hear V Gya. Hence we can also find a suitable ga.
There exists a subset J E [,u]' and an ordinal 5 < cf (K) such that (go, gl | a E J) E Vp')? and ,u < i,. Arguing as in the proofs of Theorems 1.3 and 3.17, there
exists mr E Vlpih? such that 7mgg 1 = * gI for all a E J. This is a contradiction.
REFERENCES
[1] M. R. BURKE and M. MAGIDOR, Shelah's pcf theory and its applications, Annals of Pure and
Applied Logic, vol. 50 (1990), pp. 207-254.
[2] M. GITIK and M. MAGIDOR, The singular cardinal hypothesis revisited, Set theory of the continuum
(H. Judah, W Just, and H. Woodin, editors), Mathematical Sciences Research Institute Publications,
vol. 26, Springer-Verlag, 1992, pp. 243-279.
[3] W HODGES, I. HODKINSON, D. LASCAR, and S. SHELAH, The small index property for co-stable co-categorical structures andfor the random graph, Journal of the London Mathematical Society, vol. 48
(1993), no. 2, pp. 204-218.
[4] K. KUNEN, Set theory. an introduction to independence proofs, North Holland, Amsterdam, 1980.
[5] H. D. MACPHERSON and P. M. NEUMANN, Subgroups of infinite symmetric groups, Journal of the
London Mathematical Society, vol. 42 (1990), no. 2, pp. 64-84.
[6] J. D. SHARP and SIMON THOMAS, Uniformisation problems and the cofinality of the infinite symmetric
group, Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 328-345.
[7] J. D. SHARP and SIMON THOMAS, Unbounded families and the cofinality of the infinite symmetric
group, Arch. Math. Logic, vol. 34 (1995), pp. 33-45.
[8] S. SHELAH, Further cardinal arithmetic, to appear in Israel Journal of Mathematics.
[9] , PCF and infinite free subsets, submitted to Archive for Mathematical Logic.
[10] , Strong partition relations below the power set. Consistency. Was Sierpinski right? II, in Proceedings of the Conference on Set Theory and its Applications in honor of A. Hajnal and V T Sos, Budapest, Sets, Graphs and Numbers, vol. 60 of Colloquia Mathematica Societatis Janos Bolyai (1991),
pp. 637-668.
[11] ' Cardinal arithmetic for skeptics, American Mathematical Society Bulletin, New Series,
vol. 26 (1992), pp. 197-210.
[12] , Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic (N. W Sauer, R. E. Woodrow, and B. Sands, editors), Kluwer Academic Publishers, 1993, pp. 355-383.
[13] , Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.
MATHEMATICS DEPARTMENT
BILKENT UNIVERSITY ANKARA, TURKEY
MATHEMATICS DEPARTMENT THE HEBREW UNIVERSITY
JERUSALEM, ISRAEL and
MATHEMATICS DEPARTMENT
RUTGERS UNIVERSITY
