saharon shelah- the pcf theorem revisited

8/3/2019 Saharon Shelah- The PCF Theorem Revisited

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THE PCF THEOREM REVISITED

SH506

DEDICATED TO PAUL ERDOS

Saharon Shelah

Institute of MathematicsThe Hebrew University

Jerusalem, Israel

Rutgers UniversityMathematics DepartmentNew Brunswick, NJ USA

Abstract. The pcf theorem (of the possible cofinability theory) was proved forreduced products

Qi


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0 Introduction

An aim of the pcf theory is to answer the question, what are the possible cofi-

nalities (pcf) of the partial ordersi, for different ideals

I on . For a quick introduction to the pcf theory see [Sh 400a], and for a detailedexposition, see [Sh:g] and more history. In 1 and 2 we generalize the basic theo-rem of this theory by weakening the assumption < mini


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THE PCF THEOREM REVISITED SH506 DEDICATED TO PAUL ERDOS 3

normality: J[] = J


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1 Basic pcf

1.1 Notation 1) I, J denote ideals on a set Dom(I), Dom(J) resp., called its domain

(possiblyAI

A Dom(I). If not said otherwise the domain is an infinite cardinal

denoted by and also the ideal is proper i.e. Dom(I) / I. Similarly D denotes afilter on a set Dom(D); we do not always distinguish strictly between an ideal on and the dual filter on .2) Let denote a sequence of the form i : i < . We say is regular if everyi is regular, Min() = Min{i : i < } (of course also in we can replace by another set), and let =

i }.

But we can replace by any set (in the definitions and claims). Let I

denote afixed ideal on .3) For I a filter on let I+ = P() \ I (similarly D+ = {A : \ A / D}), let

lim infI = min{ : {i < : i } I+} and

lim supI = Min{ : {i < : i > } I} and

atomI = { : {i : i = } I+}.

4) For a set A of ordinals with no last element, JbdA = {B A : sup(B) < sup(A)},i.e. the ideal of bounded subsets.5) Generally, if inv(X) = sup{|y| : (X, y)} then inv+(X) = sup{|y|+ : [X, y]},and any y such that [X, y] is a wittness for |y| inv(X) (and |y| < inv+(X)),and it exemplifies this.6) Let Ord be the class of ordinals.

7) Consideringi


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(b) P has true cofinality if there is a sequence p = p : < increasing

and cofinal in P, i.e.:


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6 SAHARON SHELAH

1.3 Observation. 1) Concerning 1.2(4), note: g = Max{g, 1} means g(i) =Max{g(i), 1} for each i < ; if for every f F, {i < : f(i) = 0} I we

can replace Max{g, 1}, Max{f, 1} by g, f respectively in clause () and omit clause

().2) The ideal I on is -weakly saturated iff in the topological space of the ultrafil-ters on the subspace {D : D an ultrafilter on disjoint to I} has spread < , or is a limit ordinal, it has spread but the spread is not obtained (hence 2cf() but it is consistently singular, see [Sh 233], [JuSh 231]).

1.4 Definition. Below if is the ultrafilters disjoint to I, we write I instead of. Recall that we can replace by any set.1) For a property of ultrafilters (if is the empty condition, we omit it):

pcf() = pcf(, ) = {tcf(/D) : D is an ultrafilter on satisfying }

(so is a sequence of ordinals, usually of regular cardinals, note: as D is anultrafilter, /D is linearly ordered hence has true cofinality).1A) More generally, for a property of ideals on we let pcf() = {tcf(/J) : Jis an ideal on satisfying such that /J has true cofinality}; we call closedwhen if I and A, B I+ are disjoint then I + A is a maximal ideal.Similarly below.2) J


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(ii) if I J then pcfJ() pcfI(); and

(iii) for B1, B2 we have pcfI( (B1 B2)) = pcfI( B1)

pcfI( B2).Also

(iv) A J


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8 SAHARON SHELAH

In particular, if one of them is well defined, then so are the others. This is trueeven if we replace s by linear orders or even partial orders with true confinality.

7) If D is an ultrafilter on a set S, s a regular cardinal, then =: tcf(sS

s, |S| or at least s > |{t : t = s}| for any s S then:

(i) E is a filter on S, and if D is an ultrafilter on S then E is an ultrafilter

on S

(ii) S is a set of regular cardinals andif s S s > |S| then ( S) > |S|,

(iii) F = {f sS

s : s = t f(s) = f(t)} is a cover ofsS

s,

(iv) cf(sS

s/D) = cf(S/E) and tcf (

sS

s/D) = tcf(S/E).

9) Assume I is an ideal on , F Ord and g Ord. If g is a I-eub of F theng is a I-lub of F.10) sup pcfI() |/I|.11) If I is an ideal on S and (

sS

s,


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1.6 Definition. 1) Let I is not almost normal for (, ) mean: for some h ,for no j < is {i < : i h(i) < j} = mod I.2) Let I is not lowly normal for (, ) mean: for some h , for no < , is

{i < : i h(i) < } I+

.

Remark. Note that weakly normals implies lowly normal.

Proof. They are all very easy, e.g.0) We shall show (, h(i) . So assume that f for

< . We now define a function f with domain by

f(i) = {f(i) : < h(i)(and < )}.

Now first i < f(i) < i because i is regular, h(i) < i and < f(i) (e.g., Min(E\( + 1)), the set {i < : h(i) < } is included in

{i < : i = } {i < : i E} {i < : i }.]Now the first and second belong to I by an assumption and the third as < =liminfI(), so we are done.5) Let =: lim supI() and define

J =: {A : for some < the set {i < : i > and i A} belongs to I}.


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Clearly J is an ideal on extending I (and / J) and lim supJ() = lim infJ() =.

Case 1: is 0.We do not use the J above. Now the desired conclusion fails then every ultrafilter

on disjoint to I is 1-complete. Now if{i < : i > 0} J+ the construction

is immediate so without loss of generality i < i = 0. But not weaklynormal for (, ) then j < Aj =: {i < : h(i) < j} = mod I but{Aj : j < } = . There is an ultrafilter D on disjoint to J {Aj : j < } soAj : j < exemplifies D is not 1-complete.

Case 2: is singular.By part (4), clause (ii), /I is ()+-directed and by part (4) clause (v) we get

the desired conclusion.

Case 3: is regular > 0.Let h witness that I is not weakly normal for (, ) and let

J = {A : for every h , for some j < we have {i A : h(i) < j} = A mod I}.

Note that if A J then for some < the set A =: {i A : i > } I

hence for every h , the choice j =: witness A J. So J J. AlsoJ P() by its definition. J is closed under subsets (trivial) and under union[why? assume A0, A1 J, A = A0 A1; for every h , choose j0, j1 <

such that A =: {i A : h(i) < j} = A mod I, so j =: max{j0, j1} < and

A = {i A : h(i) < j} = A mod I; so A J]. Also / J [why? as h witnessthat I is not weakly normal for (, )]. So together J is an ideal on extendingI. Now J is not medium normal for (, ), as witnessed by h.[Why? Let us check Definition 1.6(2), so let < . We should prove that A ={i < : i h(i) < } / J

+; now A1 = {i < : i > } J J and

A2 = {i < : h(i) < j} J hence A1 A

2 J but it includes A , so we are

done.]Lastly, /J is ()+-directed (by part (4) clause (iii)), and so pcfJ() is disjointto ()+.9) Let us prove g is a I-lub of F in (

Ord, I). As we can deal separately withI + A, I + ( \ A) where A =: {i : g(i) = 0}, and the later case is trivial we canassume A = . So assume g is not a I-lub, so there is an upper bound g of F,but not g I g. Define g Ord:


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g(i) =

0 & if g(i) g(i)

g(i) & if g(i) < g(i).

Clearly g


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12 SAHARON SHELAH

If is a cardinal , and / J


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For limit let g(i) = for i A and is a sequence of regular cardinals) and let = 0.

For = + 1, suppose that g hence B: < are defined. If B

J


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1.12 Lemma. Suppose () of 1.9, cf() > , f , f < f mod J


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For = + 1, if { < : B J


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1.13 Lemma. Suppose () of 1.9 and < .1) For every B J[] \ J


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that D J


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1.15 Claim. Suppose () of 1.9 holds. Assume for j < , Dj is a filter on extending { \ A : A I}, E a filter on and D = {B : {j < : B Dj} E} (a filter on ). Let j =: tcf(, + .Let

= tcf(,


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[Why? Note that min{j : j < } + + and J[] J[]. By 1.9 we

know (,


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(i) f

(ii) g


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2 Normality of pcf() for

Having found those ideals J


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exemplified by f : < . By 1.9 we can choose by induction on < a function

f such that: , f J


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2.3 Definition. Let there be given regular , < < , possibly an ordinal,S , sup(S) = and for simplicity S is a set of limit ordinals or at least have notwo successive members.

1) We call a = a : < a special continuity condition for (S,,) (or is an(S,,)-continuity condition) if: S is an unbounded subset of, a , otp(a) +r . Then for

some stationary S { < : cf() = r}, there is a continuity condition a for(S, r); moreover, it is continuous in S and S otp(a) = r; so for every

5Note: if otp(a) = and = sup(a) (holds if S, = + 1 and a continuous in S (seebelow)) then E.


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club E of for some S, , [ < & a & a (, ) E = }].2) Assume = ++, then for some stationary S { < : cf() = cf()} there isa continuity condition for (S, + 1, ).

3) If a is a (,,1)-continuity condition and 1 then there is a (, + 1, )-continuity condition.

Proof. 1) By [Sh 420, 1].2) By [Sh 351, 4.4](2) and6.3) Check. 2.5

2.6 Remark. Of course also if = + the conclusion of 2.5(2) may well hold. Wesuspect but do not know that the negation is consistent with ZFC.

2.7 Fact. Suppose () of 1.9, f for < , = cf() (of course = dom()) and A = A[] is as in the proof of 1.9 (i.e., A = {i < : i > }).Then1) Assume a is a -weak continuity condition for (S, ), = sup(S), then we canfind f = f : < such that:

(i) f ,

(ii) for < we have f f(iii) for < < we have f


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4) If < , for < we have f = f : < , where f , then in 2.7(1)

(and 2.7(2)) we can find f as there for all f simultaneously. Only in clause (ii)we replace f f

by f A

f

A

(and f f

by f A

f

A

.

Proof. Easy (using 1.9 of course).

2.8 Claim. In 2.7 we can replace () from 1.9 by /J


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2.10 Claim. Suppose () of 1.9 and

(a) f for < , pcf() and f = f : < is


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E A1, J.

If


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For = : Choose f by induction of satisfying I(i), (ii), (iii) (possible by 1.9).

For = + 1: Use 1.11 to choose B D J[] \ J


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2.12 Definition. 1) We say B = B : c is a generating sequence for if:

(i) B and c pcf()

(ii) J[] = J


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We prove by induction on that for every B (I)+, cf(,


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But g()+1 B()

iB()

i, and B() J


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1B) We say that I satisfies the weaker pcf-th for (, ) if:

(a) (,


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2.17 Claim. 1) If () of 1.9 then I satisfies the weak pcf-th for (, +).2) If () of 1.9 holds, and /I is ++-directed (e.g., + < min ) or just thereis a continuity condition for (+, )) then I satisfies the pcf-th for (, +).

3) If I

satisfy the pcf-th for (, ) then I

satisfy the weak pcf-th for (, ) whichimplies that I satisfies the weaker pcf-th for (, ), which implies that I satisfiesthe weakest pcf-th for (, ).

Proof. 1) Let appropriate be given. By 1.9, 1.13 most demands holds, but weare left with normality. By 2.11, if pcf(), then is semi normal for . Thisfinishing the proof of (1).2) Let pcf() and let f, B be as in 2.2(4). By 2.5(1)+(2) there is a, a (, )-continuity condition; by 2.7(1) without loss of generality f obeys a, by 2.9(1) therelevant B/I

are eventually constant which suffices by 2.2(2).3) Should be clear. 2.17

2.18 Claim. Assume (,


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Proof of 1.8(3)(i). Let J =:


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7) For pcf() let Bi : i < be such that J[] = J


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3 Reduced products of cardinals

We characterize here the cardinalities i


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and

reg(D) =: { : there are no A D

+

for < such that : < A A mod D and no i <

belongs to infinitely many As}.

4) reg(D) = min{ : D is not (, )-regular} where D is (, )-regular meansthat there are A D for < such that the intersection of any of them isempty.

Lastly, reg (D), reg(D) are defined similarly using A D

+. Of course, reg(I),etc., means reg(D) where D is the dual filter.

3.2 Definition. 1) Let

htcfD,(i) = sup{tcf(i


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3.3 Claim. 1) reg(D) is always regular.2) If < reg(D) then some filter extending D is -regular.3) wsat(D) reg(D).

4) reg(D) reg(D) reg(D).5) reg(D) = min{: no ultrafilter D1 on extending D is -regular}.6) If D E are filters on then:

(a) reg(D) reg(E)

(b) reg(D) reg(E).

Proof. Should be clear. E.g.2) Let u : < list the finite subsets of , and let {A : < } D

+ exemplify

< reg(D). Now let D =: {A : for some finite u , for every < wehave: u u A A mod D}, and let A =

{A : u}. Now D is a

filter on extending D and for < we have A D.Finally, the intersection of A0 A

1

. . . for distinct n < is empty, becausefor any memeber j of it we can find n < such that j An and n un . Now if{n : n < } is infinite then there is no such j by the choice of A : < , and if

{n : n < } is finite then without loss of generalityn


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5)i


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So in M, every -sequence of members is coded by at least one member so M =

M, but M = |i


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3.10 Lemma. |i


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So assume everything is defined for every < . If = 0, let Ai = {i}, if limit

Ai =


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y1 y2 = mod D hence X D

+. Also y1 y2 : < is -decreasing hence

X/D : < is -decreasing.Also ifi X1 X2 and 1 < 2 then f

02

(i) f01+1(i) < f01

(i) (first inequality:

as A1+1i A2i and clause (e) above, second inequality by the definition of X1),hence for each ordinal i the set { < : i X} is finite. So < reg(D),contradiction to the assumption (). 3.10

Note we can conclude

3.12 Claim.

i


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(C) for some club C of and some 1 < and i < +1 and wi Ord of order

type i for i < , there are f i


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Let () =


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46 SAHARON SHELAH

any +1 of the sets A is empty (equivalently i < (1)[i A] (reminds

(, +1 )-regularity of ultrafilters).2) We can in 3.13(1) weaken the assumption () to () below if in the conclusion

we weaken clause (A) to (A)

where

() cf() reg(D)

(A) there is a D-upper bound f of {g : < } such thatno f


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2) Let I be the ideal dual to D, and assume () above. If ()() of 1.9 holds and is semi-normal (for (, I)) then it is normal.

Proof.

Case 1 < lim infD().We let

=

if < i

1 if i

and we are done.

Case 2: lim infD() reg(D), > , and ( < reg(D))[reg (D) < ].

Let =: reg(D). There is an unbounded S and an (S, )-continuity systema (see 2.5). As /D has true cofinality , > clearly there are g for < such that g = g : < obeys a (exists as lim infD()).

Now if in claim 3.13(1) for g possibility (A) holds, we are done. By 3.14(1) weget that for some < reg(D), reg

(I) , contradiction.

Case 3: lim infD() reg(D), , and ( < reg(D))[reg (D) < ].

Like the proof of [Sh:g, Ch.II,1.5B] using the silly square.

We turn to other measures of /D.

3.16 Definition.

(a) T0D() = sup{|F| : F and f1 = f2 F f1 =D f2}

(b)

T1D() = Min{|F| :(i) F

(ii) f1 = f2 F f1 =D f2

(iii) F maximal under (i) + (ii)}

(c) T2D() = Min{|F| : F and for every f1 , for some f2 F wehave f1 =D f2}

(d) If T0D() = T1D() = T

2D() then let TD() = T

lD() for l < 3

(e) for f Ord and < 3 let TlD(f) means TlD(f() : < ).


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3.17 Theorem. 0) If D0 D1 are filters on then TD0() TD1

() for = 0, 2.

Also if = A0A1, A0 D+, andA1 D

+ thenTD() = min{TD+A0

(), TD+A1()}for = 0, 2.

1) htcfD() T2D() T1D() T0D().2) If T0D() > |P()/D| or just T

0D() > , andP()/D satisfies the

+-c.c.then T0D() = T

1D() = T

2D() so the supremum in 3.16(a) is obtained (so, e.g.,

T0D() > 2 suffice).

3) T0D()


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> T2D(f) + . Now for each f F let Af = {i < : f(i) = g(i)} clearly Af D

+.Now f Af/D is a function from F

into P()/D, hence, if = |P()/D|, it isnot one to one (by cardinality consideration) so for some f = f from F (hence

form F0) we have Af/D = Af/D; but so

{i < : f(i) = f(i)} {i < : f(i) = g(i)} {i < : f(i) = g(i)} = Af/D

hence is = mod D, so f =D f, contradition the choice ofF0. If = |P()/D|

(as F F0 by the choice of F0) we have:

f1 = f2 F Af1 Af2 = mod D

so {Af : f F

} contradicts the Boolean algebra P()/D satisfies the +

-c.c..3) Assume that < reg(D) and7 + T0D(). As

+ T0D() we can findf for < such that [ < f =D f]. Also (as < reg(D)) we canfind {A : < } D such that for every i < the set wi =: { < : i A} isfinite. Now for every function h : we define gh, a function with domain :

gh(i) = {(, fh()(i)) : wi}.

So |{gh(i) : h }| (i)|wi| = i, and ifh1 = h2 are from then for some < ,h1() = h2() so Bh1,h2 = {i : fh1()(i) = fh2()(i)} D that is Bh1,h2 A D so

1 if i Bh1,h2 A then wi, so gh1(i) = gh2(i)

2 Bh1,h2 A D.

So gh : h exemplifies T0D()

. If the supremum in the definition ofT0D() is obtained we are done. If not then T

0D() is a limit cardinal, and by the

proof above:

[ < T0D() & < reg(D) < T0D()].

So if T0D() has cofinality reg(D) we are done; otherwise let it be


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4) For the first inequality assume it fails so =: T2D(f) < htcfD(i


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() there are i,n : n < ni : i < , i,n = cf(i,n) < f(i) and a filter

D oni


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3.19 Conclusion Suppose D is an 1-complete filter on . If i 2 for i < andsupAD+ TD+A() >

0 then for some i = cf(i) i we have

supAD+ tcfD+A(

i .

3.20 Conclusion Let D be an 1-complete filter on . If for i < , Bi is a Booleanalgebra and i < Depth

+(Bi) (see below) and

2 < 0 < supAD+

TD+A()

then + < Depth+(i


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induction on , < + as follows: is the minimal ordinal <

+ such thatE, D where E, = the -complete filter generated by

{i < : f(i) = f(i)} : <

(note: each generator of E, is in D but not necessarly E, D!).

Let be well defined if < , clearly < < . Now if

< +, then

clearly =


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54 SAHARON SHELAH

with limit ,

< cf()

/Jbdcf() has true confinality . Let = sup{ : < } + 2

,

let i : cf() be i(i) = sup{ + 1 : i }. If there is a function h i


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Let (i) = max{ < 2n(i) : (i, ) A} and let B = {i : (i) well defined}. ClearlyB D+ (otherwise we can find < < such that f/D = f/D, contradiction).For (i, ) A define i, by

i, = sup{

j,m + 1 : (j,m) A

and j,m < i,}. Now

i, < i, as cf(

i,) > 2

. Let

Y = { < : if(i, ) A \ A then j,i, = i,

and if (i, ) A then i, < j,,i < ,i}.

Let B1 = {i B : (i) is odd}. Clearly B1 B and B \ B1 = mod D (otherwiseas in ()1, ()2 below get contradiction) hence B1 D

+. Now

()1 for < from Y we havej,i,(i) : i B1 j,i,(i) : i B1 mod (D B1)

[Why? as f/D was non decreasing ini


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56 SAHARON SHELAH

Proof. Check.

3.29 Claim. Let D be a filter on , i : i < a sequence of cardinals and

2 < = cf(). Then () () () (), and if ( < )(0 < ) we alsohave () () where

() if Bi is a Boolean algebra, i < Depth+(Bi) then < Depth

+(i


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at least when i > 2;

for D a filter on , does i < Depth+(Bi) for i < implies, assuming

i > 2 for simplicity,

TD(i : i < ) < Depth+(i


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58 SAHARON SHELAH

4 Remarks on the conditions for the pcf analysis

We consider a generalization whose interest is not so clear.

4.1 Claim. Suppose = i : i < is a sequence of regular cardinals, and is acardinal and I is an ideal on ; and H is a function with domain . We considerthe following statements:

()H lim infI() wsat(I) and H is a function from to P() such that:

(a) for every < we have {i < : H(i)} = mod I

(b) for i < we have otp(H(i)) i or at least {i < : |H(i)| i} I

()+ similarly but(b)+ for i < we have otp(H(i)) < i.

1) In 1.9 we can replace the assumption () by ()H above.2) Also in 1.11, 1.12, 1.13, 1.14, 1.15, ? we can replace 1.9() by ()H.

> scite{1.11} undefined3) Suppose in Definition 2.3(2) we say f obeys a for H (instead of for A) if

(i) for a such that =: otp(a) < we have

otp(a), otp(a) H(i) f(i) f(i)

and in 2.3(2A), f(i) = sup{f(i) : a and otp(a), otp(a) H(i)}.

Then we can replace 1.9() by ()H in 2.7, 2.8, ?; and replace 1.9() by ()+H in

> scite{2,6} undefined2.10 (with the natural changes).

Proof. 1) Like the proof of 1.9, but defining the gs by induction on we change

requirement (ii) to

(ii) if < , and i H() H() then g(i) < g(i).

We can not succeded as

(B() \ B+1()) {i < : , + 1 H(i)} : <


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is a sequence of pairwise disjoint member of (I)+.In the induction, for limit let g(i) < {g(i) : H(i) and H(i)} (so this

is a union at most otp(H(i) ) but only when H(i) hence is otp(H(i)) i).

2) The proof of 1.11 is the same, in the proof of 1.12 we again replace (ii) by (ii)

.Also the proof of the rest is the same.3) Left to the reader. 4.1

We want to see how much weakening () of 1.9 to lim infI() wsat(I)suffices. If singular or lim infI() > or just (, h(i) and < {i : h(i) < } J}

(c) for < () and A J+1 \ J, the pair (, J + ( \ A)) (equivalently A, J A)) satisfies condition 1.9() (case ()) hence its consequences,(in particular it satisfies the weak pcf-th for )

(d) if


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60 SAHARON SHELAH

4.3 Conclusion. Under the assumptions of 4.2, I satisfies the pseudo pcf-th (seeDefinition 2.16(4)), hence cf(,


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regular cardinals and cardinal invariants of Boolean Algebra,


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62 SAHARON SHELAH

REFERENCES.

[JuSh 231] Istvan Juhasz and Saharon Shelah. How large can a hereditarily sepa-rable or hereditarily Lindelof space be? Israel Journal of Mathematics,53:355364, 1986.

[Kn] Akihiro Kanamori. Weakly normal filters and irregular ultra-filters.Trans. of A.M.S., 220:393396, 1976.

[Kt] Jussi Ketonen. Some combinatorial properties of ultra-filters. Fund.Math., CVII:225235, 1980.

[Ko] Peter Komjath. On second-category sets. Proc. Amer. Math. Soc.,107:653654, 1989.

[MgSh 433] Menachem Magidor and Saharon Shelah. Length of Boolean alge-bras and ultraproducts. Mathematica Japonica, 48(2):301307, 1998.math.LO/9805145.

[M] J. Donald Monk. Cardinal functions of Boolean algebras. circulatednotes.

[Sh 7] Saharon Shelah. On the cardinality of ultraproduct of finite sets. Jour-nal of Symbolic Logic, 35:8384, 1970.

[Sh:a] Saharon Shelah. Classification theory and the number of nonisomor-phic models, volume 92 of Studies in Logic and the Foundations of

Mathematics. North-Holland Publishing Co., Amsterdam-New York,xvi+544 pp, $62.25, 1978.

[Sh 233] Saharon Shelah. Remarks on the numbers of ideals of Boolean algebraand open sets of a topology. In Around classification theory of models,volume 1182 ofLecture Notes in Mathematics, pages 151187. Springer,Berlin, 1986.

[Sh:c] Saharon Shelah. Classification theory and the number of nonisomor-phic models, volume 92 of Studies in Logic and the Foundations ofMathematics. North-Holland Publishing Co., Amsterdam, xxxiv+705

pp, 1990.[Sh 351] Saharon Shelah. Reflecting stationary sets and successors of singular

cardinals. Archive for Mathematical Logic, 31:2553, 1991.

[Sh 400a] Saharon Shelah. Cardinal arithmetic for skeptics. Ameri-can Mathematical Society. Bulletin. New Series, 26:197210, 1992.math.LO/9201251.


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[Sh 420] Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and Infi-nite Combinatorics in Sets and Logic, pages 355383. Kluwer AcademicPublishers, 1993. N.W. Sauer et al (eds.). 0708.1979.

[Sh 410] Saharon Shelah. More on Cardinal Arithmetic. Archive for Mathemat-ical Logic, 32:399428, 1993. math.LO/0406550.

[Sh 345a] Saharon Shelah. Basic: Cofinalities of small reduced products. In Car-dinal Arithmetic, volume 29 ofOxford Logic Guides, chapter I. OxfordUniversity Press, 1994.

[Sh:g] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford LogicGuides. Oxford University Press, 1994.

[Sh 430] Saharon Shelah. Further cardinal arithmetic. Israel Journal of Math-

ematics,95

:61114, 1996. math.LO/9610226.[Sh 580] Saharon Shelah. Strong covering without squares. Fundamenta Math-

ematicae, 166:87107, 2000. math.LO/9604243.

[Sh 513] Saharon Shelah. PCF and infinite free subsets in an algebra. Archivefor Mathematical Logic, 41:321359, 2002. math.LO/9807177.

saharon shelah- the pcf theorem revisited

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