A SHORT PROOF OF THE PCF THEOREM 1

A SHORT PROOF OF THE PCF THEOREM
MENACHEM KOJMAN
Abstract. It is shown that every proper ideal over a small set of
regular cardinals has an immediate successor (Theorem 19 below).
From this, a short proof of Shelah’s pcf theorem follows.
1. preliminaries
We shall need to know that for every regular κ and regular λ > κ+ +
there is a stationary set in I[λ]. We give below the definition of I[λ]
and the proof of this fact. The proof uses club-guessing, which should
be well known.
Theorem 1 (Club guessing). If k and λ are regular uncountable cardinals and κ+ < λ then there is a club guessing sequence on Sκλ := {δ <
λ : cfδ = κ}. That is, there is a sequence hcδ : δ ∈ Sδλ i so that:
(1) cδ ⊆ δ = sup cδ is club of δ and otpcδ = κ.
(2) for every club E ⊆ λ the set {δ ∈ Sκλ : cδ ⊆ E} is stationary in
λ.
Proof. Suppose that hcδ : δ ∈ Sκλ i is any sequence that satisfies condition (1) and we shall find a club E ⊆ λ so that hcδ ∩ E : δ ∈ Sκλ ∩ lim Ei
+
satisfies conditions (1) and (2). By induction on
T i < κ define a club
+
Ei .Let E0 = λ and for limit i < κ let Ei = j<i Ej . Suppose Ei is
define. The sequence hcδ ∩ E : δ ∈ Sκλ ∩ lim Ei must satisfy condition
(1) since E ∩ δ and cδ are clubs of δ for δ
in lim E and hence their intersection is a club of δ of ordertype κ. Assume then that it does not satisfy condition (2). This means there
is a club C1i ⊆ λ such that {δ ∈ Sκλ ∩ E : cδ ∩ E ⊆ C1i } is not stationary; namely, there is a club C2i ⊆ λ which avoids this set. Let
Ei+1 = Ei ∩ C1i ∩ C2i .
T
Suppose the induction goes on for all i < κ+ and let E ∗ = i<κ+ Ei .
Since κ+ < λ, E ∗ is a club of λ. Fix δ ∈ lim E ∗ ∩ Sκλ . For each i < κ+
it holds that (a): i ∈ lim Ei ∩ C2i hence that cδ ∩ Ei 6⊆ C1i . However,
since Ei+1 = Ei ∩ C1i ∩ C2i , it holds that (b): cδ ∩ Ei+1 ⊆ C2i . From (a)
and (b) we get that cδ ∩ Ei+1 $ cδ ∩ Ei for all i < κ+ . This makes the
1
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MENACHEM KOJMAN
sequence hcδ ∩ Ei : k < κ+ i a strictly decreasing sequence of subsets of
cδ of length κ+ which is of course absurd, since otpcδ = κ.
We conclude that for some i < κ+ condition (2) holds for hcδ ∩ Ei :
δ ∈ lim Ei i.
Definition 2. Suppose λ is regular. We define an ideal I[λ] over λ.
For S ⊆ λ, S ∈ I [λ] if and only if there exists a sequence P̄ =
hPα : α < λi and a closed and unbounded set E ⊆ λ such that:
(1) Pα ⊆ P (α) and |Pα | < λ
(2) If δ ∈ E ∩ S then there exists a set c S
⊆ δ such that δ = sup c,
otpc = cf(δ) < δ and (∀β ∈ c)c ∩ β ∈ γ<δ Pγ
Theorem 3. If κ, λ are regular cardinals and κ++ < λ then there is a
stationary set S ⊆ Sκλ in I [λ].
++
Proof. Fix a club guessing sequence C = hcα : α ∈ Sκκ i as in Theorem
1 in the previous section.
Description of the proof.
The proof will involve two elementary chains of models of H(χ), for
some large enough regular χ. The first chain will be used to define
hPi : i < λi; the other, to prove that this choice works. The first chain
is going to be an element of every member in the second. At some
point of the proof, though, we shall need some set which is definable
in the second chain to belong to some member of the first chain. We
use the prediction, or “guessing”, property of a club guessing sequence
to obtain this.
Fix an elementary chain M := hMi : i ≤ λi of submodels of hH(χ), ∈ i
(for a large enough regular χ) satisfying:
• ||Mi || < λ
• hMj : j ≤ ii ∈ Mi+1 , i ∈ Mi and i ⊆ Mi
• C, λ ∈ M0 and κ++ + 1 ⊆ M0
Let us define Pi := Mi ∩ P(i). By condition (1) we see that |Pi | < λ.
Let S ⊆ λ be the set
(
S :=
)
[
α < λ : cfα = κ ∧ ∃c club of α (∀γ < α)(c ∩ γ ∈
Pi )
i<α
The sequence hPi : i < λi witnesses that S ∈ I[λ] according to the
definition of I[λ]. All that is left to be shown is that S is stationary.
A SHORT PROOF OF THE PCF THEOREM
3
Explanation If indeed there is a stationary S ⊆ Sκλ in I[λ] them M0
should know about it and about some witness P , by elementarity, and
therefore Pi as chosen here must suffice.
We prove now that S is stationary. Fix a club E ⊆ λ and we will
show that E meets S. By shrinking E we may assume that Mi ∩ λ = i
for all i ∈ E; this follows from the fact that {i < λ : Mi ∩ λ = i} is a
club of λ.
Define a second elementary chain of models of H(χ), N := hNζ : ζ ≤
κ++ i satisfying:
• M , E, λ, C ∈ N0 and κ++ + 1 ⊆ N0
• ||Nζ || = κ++
• hNξ : ξ ≤ ζi ∈ Nζ+1 for ζ < κ++
Define f (ζ) := sup Nζ ∩ λ. The function f : (κ++ + 1) → θ is
increasing and continuous. Denote θ = f (κ++ . By condition 3 in the
choice of N , we see that f ζ ∈ Nζ+1 for every ζ < κ++ .
We observe that for every ζ ≤ κ++ the ordinal f (ζ) belongs to
E. This is true by elementarity and the fact that E ∈ Nζ for all
ζ: if β ∈ Nζ ∩ λ is arbitrary, then Nζ |= (∃γ ∈ λ) [γ ∈ E ∧ γ > β].
Therefore there exists β < γ ∈ E ∩ N is and thus E unbounded below
f (ζ), implying f (ζ) ∈ E. Thus ranf ⊆ E. Turn now to the chain
M and work in Mθ+1 . Use the fact that θ ∈ Mθ+1 (condition (2) in
the choice of M ) and the elementarity of Mθ+1 and choose a function
g ∈ Mθ+1 such that g : κ++ → θ increasing and continuous with
θ = sup rang.
Since both f and g are increasing continuous on κ++ with ranges
cofinally contained in θ, a standard argument allows us to fix a clubE ⊆
κ++ such that f E = gE.
++
Use the club guessing property of C to fix δ < Sκκ such that cδ ⊆ E,
δ = sup cδ and otpcδ = κ. Define c := f [cδ ] = g[cδ ].
Thus c ⊆ f (δ) is a club of f (δ) of order type κ. We already know
S that
f (δ) ∈ E; we will show that f (δ) ∈ S by showing that c ∩ γ ∈ i<θδ Pi
for every γ < f (δ).
Let, then, X = cδ ∩ ζ for some ζ ∈ cδ be an initial segment of cδ
and let Y := f [X] be the corresponding initial segment of c. As cδ and
ζ belong to N0 , we have X ∈ N0 ⊆ Nζ+1 ; and since f ζ ∈ Nζ+1 by
condition 3 in the choice of N we conclude that Y = f [X] ∈ Nζ+1 .
If we knew that Y ∈ Mi for some i < λ we would be done: Suppose
that there were some i < λ such that
H(χ) |= ∃i < λ [Y ∈ Mi )]
Since Y, M , λ ∈ Mζ+1 and Nζ+1 ≺ H(χ)
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MENACHEM KOJMAN
Nζ+1 |= ∃i < λ[Y ∈ Mi ]
by elementarity. Thus there is some i < f (ζ + 1) < f (δ) such that
Y ∈ Pi , as required.
But why is there such i < λ at all? The reason is the club guessing
sequence, which enables M to “predict” the set Y . Recall that Y =
f [X]. The parameter X in this definition belongs to M0 . But f may
not belong to any Mi ∈ M . However, f cδ = gcδ and so we can
define Y using g instead. The function g belongs to Mθ+1 , and the
proof is now complete, as Y = g[X] is definable in Mθ+1 and hence
Y ∈ Mθ+1 .
2. Exact upper bounds modulo an ideal
The basic object we are examining is the following: let A be some
infinite set and I be an ideal over A. Let f = hfα : α < δi be a sequence
of functions from A to the ordinals which is increasing modulo I, where
δ is some limit ordinal. The question we address is the existence and
structure of an exact upper bound modulo I for this sequence. In this
Section we establish notation and prove a few basic facts about exact
upper bounds modulo an ideal which are needed to facilitate the rest
of the discussion.
2.1. Basics. Let A be a fixed infinite set. By OnA we denote the
class of all functions from A to the ordinal numbers. Given an ideal
I over A, we quasi order OnA by defining f ≤I g for f, g ∈ OnA iff
{a ∈ A : f (a) > g(a)} ∈ I. Similarly, =I and <I are defined.
In the special case that I = {∅}, the relation <I is the relation of
domination everywhere and is denoted by <.
For subsets F1 , F2 ⊆ OnA write F1 ≡I F2 if for every f ∈ F1 there is
g ∈ F2 such that f ≤I g and for every g ∈ F2 there is f ∈ F1 such that
g ≤I F . The relation ≡I is an equivalence relation.
We shall investigate the relation between the following two properties
of subsets of OnA :
Definition 4.
(1) F ⊆ OnA has an exact upper bound iff there
exists g ∈ OnA such that:
(a) (∀f ∈ F )(f ≤I g)
(b) g 0 <I g ⇒ (∃f ∈ F )(g 0 <I f )
A function g which satisfies (a) and (b) is an exact upper bound
(an eub) of F .
A SHORT PROOF OF THE PCF THEOREM
5
(2) F ⊆ OnA has true cofinality iff there exists some F 0 ⊆ OnA
such that F 0 ≡I F and F 0 is linearly ordered by <I and has no
last element.
If F has true cofinality then the true cofinality of F is denoted
by tcf F and is the cofinality of the order type of some (of every)
linearly ordered F 0 that is equivalent to F .
The following points should be noticed: first, each of the properties
defined below is invariant under ≡I . Second, neither property implies
the other. Third, eubs are also least upper bounds, except in the
trivial case where F has an upper bound which assumes the value 0 on
a positive set, and is therefore an eub vacuously.
Fact 5.
(1) A set F ⊆ OnA without a maximum with respect to ≤I
has an eub f iff F is equivalent to a copy of a product of regular
cardinals, namely there exists sets S(a)
Q ⊆ f (a) for a ∈ A such
that otpS(a) = cff (a) is and F ≡I a∈A S(a).
(2) A set F ⊆ OnA has true cofinality λ iff it is equivalent to a <I
increasing sequence hfα : α < λi.
Both properties above are preserved when the ideal I is extended.
We shall be using the following fact freely:
Fact 6. Suppose I1 ⊆ I2 are ideals over an infinite set A and F ⊆ OnA .
Then:
(1) If g is an eub of F modulo I1 and 0 <I1 g then g is also an eub
of F modulo I2 .
(2) If F has true cofinality λ modulo I1 than F has true cofinality
λ modulo I2 .
A particular instance of this fact is when I2 = I1 B := hI ∪ (A \ B)i,
the ideal generated by joining A \ B to I, for some B ∈ I1+ .
The following is a simple, yet important example of a set of functions
which has both an eub and true cofinality regardless to which ideal I
over A is involved:
Fact 7. Suppose λ is regular and λ > |A|. Then tcf(λA , <) = λ.
Consequently, tcf(λA , <I ) = λ for every ideal I over A, by Fact 6.
Proof. Let gi (a) = i for γ < λ and a ∈ A. The sequence g = hgγ :
γ < λi is <-increasing. It is also cofinal in (λA , <) by the following
“rectangle argument”: Let g ∈ λA be arbitrary. Since λ > |A| is
regular, γ := sup{g(a) : a ∈ A} < λ and therefore g ≤ gγ .
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MENACHEM KOJMAN
Claim 8. Suppose that λ > |A| is regular and f = hfα : α < δi ⊆ OnA
is <I -increasing. The following are equivalent:
(1) There is an eub f of f such that cff (a) = λ for all a ∈ A.
(2) ThereQare sets S(a) for a ∈ A with otpS(a) = λ such that
f ≡I S(a).
(3) There is some <-increasing g = hgγ : γ < λi and some increasing, continuous and cofinal subsequence hα(γ) : γ < λi ⊆ λ such
that fα(γ) <I gγ+1 <I fα(γ+1) .
(4) There is some ≤-increasing g = hgγ : γ < λi and some increasing, continuous and cofinal subsequence hα(γ) : γ < λi ⊆ λ such
that fα(γ) <I gγ+1 <I fα(γ+1) .
Definition 9.
(1) A <I -increasing f = hfα : α < δi ⊆ OnA is
flat (of cofinality λ) mod I if and only if one of the equivalent
conditions in 8 holds for f .
(2) α < δ is a flat point in a <I -increasing f = hfβ : β < δi if and
only if f α is flat.
The next theorem gives a necessary and sufficient condition for the
existence of an eub of a sequence with large cofinalities, in terms of
flatness of initial segments of the sequence.
Theorem 10. Suppose f = hfα : α < λi ⊆ OnA is a |A|+ . For every regular κ ∈ (|A|, λ) the following are
equivalent:
• f has an eub g so that {a : cfg(a) ≤ κ} ∈ I
• The set {α < cfα = κ and f α is flat } is stationary in λ.
Proof. Suppose |A| < κ < λ and κ is regular. Suppose first that f
has an eub g with cfg(a) > κ for all a ∈ A except for a set in I. By
changing g on a set in I we may assume that cfg(a) > κ for all a ∈ A.
Let C ⊆ λ be a given club. By induction on i < κ find α(i) ∈ C as
0
0
follows. At stage i let gi (a) = supj κ, gi < g. Hence there is some α < λ
so that g <I fα . Let α(i) = min(C \ {α + 1} ∪ {α(j) + 1 : j < i}.
Since hα(i) : i < κi is astrictly increasing sequence of points in C,
α = sup{α(i) : i < κ} belongs to C and has cofinality κ. We argue
that f α is flat. Indeed, f α is equivalent to hgi : i < κi which is
≤-increasing, and is hence flat.
Now for the other direction, suppose that S = {α < λ : cfα = κ}
is stationary in λ. The trichotomy theorem applies to f , thus to show
A SHORT PROOF OF THE PCF THEOREM
7
the existence of an eub g it suffices to rule out Bad and Ugly. If either
condition o the two holds for f , then the set of α < λ such that the
same condition holds for f α is a club of λ, and hence meets S. This
leeds to a contradition.
Fix an eub g of f . The fact that {a ∈ A : cfg(a) < κ} ∈ I is
obvious.
3. The possible true cofinalities of small sets of regular
cardinals
Definition 11. Suppose (Q, ≤) is a quasi-ordered set with no maximal
elements. Let b(Q, ≤) be the lest cardinality of an unbounded subset of
Q. If (Q, ≤) has a maximum put b(Q, ≤) = ∞, where ∞ is greater
than all cardinals.
Let d(Q, ≤) be the smallest cardinality of a dominating subset of Q.
Claim 12. Suppose that (Q, ≤) is a quasi-ordered set with no maximum. Then b(Q, ≤) is either 2 or an infinite regular cardinal.
Proof. Suppose first that b(Q, ≤) = n is finite, and fix an unbounded
set {qi : i < n} of cardinality n. Since all singletons in Q are bounded
subsets of Q, n > 1.
By minimality of n, the set {qi : i < n − 1} is bounded by some
p ∈ Q. Now the set {p, qn−1 } is unbounded, since {qi : i < n} is
unbounded, and is of cardinality 2.
Suppose now b(Q, ≤) = λ is infinite, and fix an unbounded set {qα :
α < λ} ⊆ Q of cardinality λ. Let hpα : α < λi be chosen recursively
as follows: pα is an upper bound of {qβ : β < α} ∪ {pβ : β < α}.
Since every subset of Q of cardinality smaller than λ is bounded, this
recursive definition is good.
Now {pα : α < λ} is a well-ordered chain in Q or ordertype λ, and
for every α < λ, qα is dominated by some member of the chain (e.g.
pα+1 ). Thus, {pα : α < λ} is not bounded.
If λ were singular with cfλ = κ < λ there would be a cofinal A ⊆ λ
with otpA = κ. By minimality of λ, there would be some p ∈ Q
which bounds A, and therefore bounds all of {pα : α < λ}, contrary to
{pα : α < λ} being unbounded in Q. We conclude that λ is regular. Claim 13. Suppose (Q, ≤) is a quasi-ordered set with no maximum.
Then b(Q, ≤) ≤ d(Q, ≤).
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MENACHEM KOJMAN
Proof. Since (Q, ≤) has no maximum, d(Q, ≤) ≥ 2. So in the case
b(Q, ≤) = 2 it is hods that b(Q, ≤) ≤ d(Q, ≤) (but it could be that
b(Q, ≤) = 2 and d(Q, ≤) is infinite).
Suppose then that b(Q, ≤) = λ is an infinite regular cardinal, and
fix an unbounded wellordered chain B ⊆ Q of ordertype λ. If D ⊆
Q is dominating, then every element of D dominates a proper initial
segment of B and those initial segments are unbounded in B. Thus,
by regularity of λ, |D| ≥ λ. thus, b(Q, ≤) ≤ d(Q, ≤).
The cardinal d(Q, ≤) may be singular.
Definition 14. A quasi-ordered set (Q, ≤) with no maximum has true
cofinality λ if λ = b(Q, ≤) = d(Q, ≤) and λ is infinite.
Definition 15. A set A of cardinals is small if |A| < min A.
Suppose that A is a small set of regular cardinals
and that I ⊆ P(S)
Q
is a proper ideal.QThe quasi ordered set ( A, <I ) has no maximum
and therefore b( A, <I ) < ∞. We denote this number by b(I) for
short.
Definition 16. For a small set A of regular cardinals let
PCF A = {b(I) : I ⊆ P(A) is a proper ideal }
We show that in a product of a small set of regular cardinals modulo
an ideal I, above every unbounded <I -increasin sequence there is a
<I -increasing sequence with an exact upper bound.
Theorem 17. Suppose A is a small set of regular cardinals. Let I be
an ideal over A and assume
that b(I) = λ. Then for every sequence
Q
h = hhα : α < λi ⊆ A there exists a |A| for all a ∈ A.
Q
Proof. Suppose that h = hhα : α < λi ⊆ A is given.
If A ∩ λ ∈ I, set B = A \ (λ ∩ A) and by a straightforward induction
on α < λ construct a sequence f = hfα : α < κi so that hα ≤ fα and
hfα B : α < λi is <-increasing. Such a sequence is also <I -increasing
and has an eub g mod I with cfg(a) = λ for all a ∈ B (for a /∈ B g(a)
can be defined to be λ).
We assume then that A ∩ λ /∈ I. The set A ∩ λ cannot be finite, for
then λ = b(I) ≤ b(I(A ∩ λ)) ≤ max{A ∩ λ} < λ which is impossible.
We are left with the case that A ∩ λ /∈ I and λ > |A|+ω . This is the
A SHORT PROOF OF THE PCF THEOREM
9
interesting case, which we have if, e.g., A = {ωn : 0 < n < ωi and I
contains all finite subsets of A.
Let κ = |A|+ . Since λ > κ++ , there is a stationary S ⊆ Sκλ in I[λ] by
Theorem 3. Fix a stationary S ⊆ Sκλ in I[λ] together with hPα : α < λi
which demonstrates that S ∈ I[λ], that is, it satisfies condition (1) in
Definition 2 and condition (2) holds for all δ ∈ S. We may assume that
Pα is ⊆-increasing with α.
Define fα ≥ hα by induction on α < λ as follows. At step α <
λ define, for each c ∈ Pα with otpc < κ, a function gc by gc (a)
Q =
sup{fβ (a) : β ∈ c}. Since a > κ+ is regular and otpc < κ, gc ∈ A.
The set {gc : c ∈ Pα }Q∪ {fβ : β < α} has
Q cardinality < λ and is
therefore bounded in ( A, <I ). Let fα0 ∈ A be a bound of this set
and now set fα = max{fα0 , hα }.
Clearly, hα ≤ fα for all α < λ and f is <I -increasing.
Suppose δ ∈ SS∈ I[λ] and fix c ⊆ δ = sup c with otpc = κ so that
(∀β ∈ c) c ∩ β ∈ γ∈c Pγ .
For each β ∈ c let gβ = sup{fξ : ξ ∈ cδ ∩ β}. Clearly, g = hgβ : β ∈ ci
is ≤-increasing of length κ and f δ ≤I g.
For every β ∈ c there is γ ∈ c such that c ∩ β ∈ Pγ , so by the
construction of f it holds that gβ <I fγ . This shows that also f ≤I g
holds, and thus f δ ∼I g, that is, f α is equivalent to a ≤-increasing
sequence of length κ, and is therefore flat.
We have thus shown that f has a stationary set S ⊆ Sκλ of flat points.
By Theorem 10, f has an eub g with cfg(a) > |A|+ for all a ∈ A, as
required.
Corollary 18. If A is a small set of regular cardinals then
PCF A = {tcf(I) : I ⊆ P(A) is a proper ideal and tcf(I) exists}
Proof. Suppose λ = b(I) ∈ PCF
Q A. Let h = hhα : α < λi be <I increasing and unbounded in ( A, <I ). By the previous theorem there
is f = hfα : α < λi with hα ≤ fα for all α < λ so that f has an eub
g. We may assume that g(a) ≤ a for all a ∈ A.
Q Let B = {a ∈ A :
g(a) = a}. If B ∈ I then f is bounded in ( A, <I ) by g 0 where
g 0 (a) = 0 if a ∈ B and g 0 (a) = g(a) otherwise. Since h and hence
also f is unbounded, B /∈ I. Thus, f is both <IB
Q-increasing and
<IB -dominating, and thus demonstrates that tcf( A, <IB ) = λ.
This shows the inclusion ⊆. The converse inclusion is immediate. Theorem 19. Suppose A is a small set of regular cardinals. Then for
every proper ideal I over A there is an ideal I s ⊇ I over A so that:
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MENACHEM KOJMAN
(1) b(I) < b(I s )
(2) I s ⊆ J for every J ⊇ I that satisfies b(J) > b(I).
(3) There is a single set BI ∈ I + so that I s = hI ∪ {BI }i, that is,
I s is generated from I by a single set.
Q
+
Proof. Let I + = hI
Q∪ {B ∈ I : b(I) = tcf( B, <I )}i.
Suppose F ⊆
A has cardinality b(I) and we shall show that F
s
is bounded mod I . We may assume F is a <I -increasing sequence
f = hfα : α < λi where λ = b(I). If f is bounded mod I, it is also
bounded mod I s . So assume f is not bounded mod I. By Theorem 17
we may assume that f has an eub g. If B = {a ∈ a : g(a) ≥ a} ∈ I
then f is bounded mod I. Otherwise, B ∈ I s . In each case, f is
bounded mod I s .
The fact that b(I s ) > b(I) implies in particular that I s \ I 6= ∅.
Suppose
now that I ⊆ J and thatQb(J) > λ. If B /∈QJ then
Q
tcf( A, <JB ) = λ, in particular b( A, <JB ) = λ ≥ ( A, <J )
— contradiction.
Now we prove that I s is generated from I by a single set B ∈ I s \ I.
Suppose that for some B ∈ I s \ I, hI ∪ {B}i =
6 I s ; this means that
0
2
0
thare is some B ∈ I so that B 6⊆I B and putting B 00 = B ∪ B 0 we see
that this is equivalent to the existence of a B 00 ∈ I s so that B ⊆I B 00
but B 00 6⊆I B. (Recall that B ⊆I B 00 if B \ B 00 ∈ I.)
Assume now that for all B ∈ I s \ I it holds that I s 6= hI ∪ {B}i
and we shall reach a contradiction. By induction on i < |A|+ we define
a ⊆I -increasing sequence hBi : i < |A|+ i, and for each i we define a
Q
i
i
<IB -increasing and <IB cofinal
f
=
hf
:
i
<
(I)i
⊆
B
i
Q
Qi . We
agree to view a function f ∈ Bi , for B ⊆ A, as a member of
A by
putting f (a) = 0 for all a ∈ A \ B.
We require that:
• fαj ≤ fαi (everywhere) for all i < j < |A|+ and all α < λ.
At successor stage i + 1 < |A|+ find Bi+1 ∈QI + so that Bi ⊆I Bi+1
and Bi+1 6⊆I Bi and, using the fact that tcf( Bi+1 , |A| is regular for each a ∈ A, this supremum belongs to
A. By
Theorem 17 there is a |A| for all a ∈ A and so that hα ≤ fα for all
i
α < λ. Define this sequence to be f and Let Bi := {a ∈ A : g(a) ≥ a}.
We verify now that Bj ⊆I Bi and Bi 6⊆ Bj for all i < j. Suppose j < i
and let C := Bj \ Bi = {a ∈ Bj : g(a) < a}. If C /∈ I then for some
A SHORT PROOF OF THE PCF THEOREM
11
j
α < λ it holds that gC <I fαj — since f is in this case <IC -increasing
and <IC -cofinal. But fαj ≤ fαi , so gC <I fαj C ≤ fαi C <I gC,
which is impossible; thus Bj ⊆I BI for all j 0}. Since the sequence hf1i :
i < |A|+ i is ≤-increasing by the construction, i < i < |A|+ implies
that Ci ⊆ Cj . Given i < j < |A|+ we have Bj \ Bi /∈ I, and since
f1i (Bj \ Bi ) = 0 while 0 <IBj f1j , there exists some a ∈ Cj \ Ci . We
conclude that hCi : i < |A|i is a strictly increasing sequence of subsets
of A of length |A|+ — a contradiction.
3.1. The inductive definiton of PCF. Let A be a small set of regular cardinals. We define recursively a strictly increasing and continuous sequence of proper ideals hIα : α ≤ α(∗)i with a strictly increasing sequence of bounding numbers
λα = b(Iα ). Let I0 = {∅} (so
S
λ0 = min A). For limit α let Iα = β<α Iβ . Since an increasing union
of proper ideals over A is a proper ideal over A, Iα is proper, and clearly
λα = b(Iα ) > λβ for all β < α.
If Iα is defined and is proper, let Iα+1 = Iαs . Let Bα ∈ Iα+1 \ Iα be
so that of Iα+1 = hIα ∪ {Bα }i from Iα .
Since a strictly increasing sequence of proper ideals over A has to be
of bounded length, there is some ordinal α(∗) so that Iα(∗) is proper
s
= hIα(∗) ∪ {Bα(∗) }i = P(A).
but Iα(∗)
Claim 20. For every proper ideal I ⊆ P(A),
b(I) = min{λα : α ≤ α(∗) ∧ Bα /∈ I}
s
Proof. Given a proper ideal I, clearly Iα(∗)
= P(A) 6⊆ I. By the
continuity of hIα : α ≤ α(∗)i thus there is a largest α ≤ α(∗) so that
Iα ⊆ I. Therefore
λα = b(Iα ) ≤ b(I). Conversely,
Q
Q b(I) ≤ b(IBα )
and sicne tcf( A, <λBα ) = λα and Bα /∈ I, tcf( A, <IBα ) = λα as
well.
Corollary 21. PCF A = {λα : α ≤ α(∗)}
Theorem 22 (The pcf theorem). For every small set A of regular
cardinal there is a sequence of sets hBλ : λ ∈ PCF Ai ⊆ P(A) so that
letting I<λ = h{Bθ : θ < λ ∧ θ ∈ PCF A}i, the following holds for all
λ ∈ PCF A:
• PCF
Q A has a maximum
Q and Bmax PCF A =I<max PCF A A.
• b( A, <I<λ ) = tcf( A, <I<λ Bλ ) = λ.
12
MENACHEM KOJMAN
• For every proper ideal I over A
(1) b(I) = λ for the first λ so that Bλ /∈ I.
(2) {b(J) : I ⊆ J and J is a proper ideal over A} = {λ ∈
PCF
Q A : Bλ /∈ I}.
(3)
A, <I has true cofinality if and only if I omits exactly
one Bλ ;
Q
Corollary 23. For every T ⊆ PCF A the set {U : U is an ultrafilter over A and ( A/U ) ∈
T } is a closed subset of the space of ultrafilters over A.
Q
cf( A/U ) = λ iff Bλ ∈ U . Unique λ such.
Claim 24. Suppose that µ is singular of cofinality κ > ℵ0 and that
C ⊆ µ is closed unbounded with otpC = κ. The for some club C 0 ⊆ C
of µ it holds that max PCF{λ+ : λ ∈ C 0 } = µ+ .
Proof. First assume, by thinning C out, that every λ ∈ C is singular
and that min C > κ.
Show that for every stationary S ⊆ C there is stationary S 0 ⊆ S so
that µ+ ∈ PCF{λ+
i : i ∈ S}. This shows:
+
(1) µ ∈ PCF{λ+ : λ ∈ C}, hence Bµ+ [C + ] exists.
(2) Bµ+ [C + ] contains C 0+ for some club C 0 ⊆ C.
Q
And (2) is certainly enough for the Lemma, as µ+ ≤ b( C 0+ , <Jbd
) ≤ max PCF C 0+ ≤ max PCF Bµ+ = µ+ .
Given
Qa stationary S ⊆ C construct a <Jbd increasing f = fα : α <
µ+ i ⊆ S + with an eub g for which lim inf bd cfg(a) = µ. Now argue
that except for a non-stationary subset of C, it holds that cfg(λ) = λ+ ;
for else by Fodor’s Lemma on an unbounded set cofinality is small! Theorem 25. For each λ ∈ N , {a : χn (a) = fχλn (λ) (a)} =J<λ Bλ .
Theorem 26. If |A| < θ ≤ min A is regular, then for every D ∈
[PCF A]θ there is a sequence hBλ : λ ∈ Di of PCF generators which
satisfies:
(i)
PCF Bλ ∩ D = Bλ
(ii)
µ ∈ Bλ ⇒ Bµ ⊆ Bλ
(Bλ is “closed” in D)
(“transitivity”)
for all λ, µ ∈ D.
Proof. First let us see that from any sequence hBλ : λ ∈ Di that
satisfies the transitivity condition (2) above a sequence hBλ0 : λ ∈ Di
can be gotten that satisfies (1) and (2). Suppose that hBλ : λ ∈ Di
satisfies (2).
A SHORT PROOF OF THE PCF THEOREM
13
Now let N be a θ-i.a. model.
Suppose
Q
cf( A, <) = max pcf . Even better:
Normal scales. The characteristic function theorem.
N ∩ µ determined by χN (|N |, µ).
Theorem 27 (The characteristic function theorem). Suppose that A ⊆
Reg, and hBλ : λ ∈ Ai is a sequence of PCF generators for A. Fix some
system hfαλ : α < λ ∈ Ai of normal scales. Suppose |A| < κ < min A
and κ regular. Suppose N = Nκ is a κ-i.a. model with hfαλ : α < λ ∈
Ai, A ∈ N0 , A ⊆ N0 . Then:
(1) for every λ ∈ A there is some club Eλ ⊆ κ so that for all i < j
in Ek it holds that
{θ ∈ A : χNi (λ)(θ) < fχλN
j
(λ) (θ)}
= {θ ∈ A : θ ≤ λ∧χNκ (θ) = fχλNκ (λ)(θ)}
4. Transitive generators
Theorem 28. Suppose A ⊆ Reg and |A|+ < min A. Then there exists
a sequence hBλ : λ ∈ Ai of pcf generators which is:
(1) J≤ [A] = J<λ [A] + Bλ
(i) Bλ is closed in A: PCF Bλ ∩ A = Bλ
(ii) ”Smoothness”, or ”Transitivity”: θ ∈ Bλ ⇒ Bθ ⊆ Bλ for all
λ ∈ B.
Proof. We first prove the existece of a sequence of generators which
satisfies (ii). Then it will be fairly easy to obtain from a sequence
which satisfies (ii) a sequence that satisfies (i) and (ii).
Let N be a κ-i.a. model for regular κ ∈ (|A|, min A) so that A ∈ N0
and A ⊆ N0 . Let hfαλ : λ ∈ A, α < λi be a system of normal scales for
all λ ∈ A.
For every λ ∈ A there is a closed unbounded Eλ ⊆ κ with the
property that for every i < j in Eλ it holds that
B λ,i,j :={θ ∈ A ∩ (λ + 1) : χNi (θ) < fχλN (θ)}
j
={θ ∈ A : χNκ (θ) = χNκ (θ)}
=J<λ Bλ∗
T
E = λ∈A Eλ is a club of κ. Fix i < j to be the first (or just any)
two members of E.
14
MENACHEM KOJMAN
For each λ ∈ A define inductively
B λ,n , n < ω as follows: B λ,0 =
S µ,n
λ,n+1
λ,n
B
and B
= B ∪ {B : µ ∈ B λ,n }. Let
[
Bλ =
B λ,n .
λ,i,j
n
Simultaneously, for every λ ∈ A and every α < λ, define fαλ,n with
domain B λ,n and so that fαλ,n ⊆ fαλ,n+1 for all n, by: fαλ,0 = fαλ B λ,0 ; for
n + 1 and θ ∈ B λ,n+1 \ B λ,n let µ ∈ B λ,n be the least for which θ ∈ B λ,n
and put fαλ,n+1 (θ) = fβµ,n (θ) where β = fαλ,n (µ). Let
[
faλ,ω =
fαλ,n .
n
Clearly, the sequence hBλ : λ ∈ Ai satisfies (ii) (if m is the least
such that µ ∈ B λ,n then for all n > m it holds that B µ,n ⊆ B λ,n+1 ).
Since B λ,0 ⊆ B λ and B λ,0 =J<λ Bλ∗ it is clear that J≤λ ⊆ J<λ + Bλ . It
remains only to check that Bλ is not “too large”, that is, to verify that
Bλ ∈ J≤λ . For this we claim that:
(1)
(θ) = χNκ (θ)
θ ∈ Bλ ⇒ fχλ,ω
N (λ)
κ
Given θ ∈ Bn let n be the least for which θ ∈ B λ,n . If n = 0 then
then (1) holds by the definition of B λ,0 . If n = m + 1 let µ ∈ B λ,m be
the least for which θ ∈ B µ,m . Let β = fχλ,m
(µ). By the induction
Nκ (λ)
µ,m
hypothesis, fχNκ (µ) (θ) = χNκ (θ). By the inductive definition,
(2)
fχλ,m+1
(θ) = fβµ,m (θ) = χNκ (θ)
N (λ)
κ
and (1) is proved.
Now observe that since i, j, A and {fαλ : λ ∈ A, α < λ} belong to
Nj+1 , also {fαλ,ω : λ ∈ A, α < λ} ∈ Nj+1 . Thus, {fαλ,ω : α < λ} has an
upper bound g modulo J≤λ (since <J≤λ is λ+ -directed), with g ∈ Nj+1 .
However, g < χNκ , thus, by (1),
Bλ ⊆ {θ ∈ A : fχλ,ω
(θ) > g(θ)} ∈ J≤λ
Nκ (λ)
and we are done.
Now we need to show that from hBλ : λ ∈ Ai which satisfies (ii)
we can obtain hBλ0 : λ ∈ Ai which satisfies (ii) and (i). Let Bλ0 =
Bλ for λ = min A and define Bλ0 by induction on λ ∈ A. For λ =
min A let Bλ0 = Bλ . Suppose that Bθ0 is defined for all θ < λ in A.
Since max PCF Bλ = λ, it holds thatSfor finitely many θ1 , θ2 ,S
. . . , θn ∈
PCF Bλ ∩A we have PCF Bλ ⊆ Bλ ∪ i≤n Bθ0 i . Let Bλ0 =S
Bλ ∪ i≤n Bθ0 i .
Clearly, Bλ0 = J<λ Bλ . Also, PCF Bλ0 = PCF Bλ ∪ PCF( i≤n Bθ0 i ).
A SHORT PROOF OF THE PCF THEOREM
15
To show S
transitivity, argue by double induction. Suppose that µ ∈
Bλ0 = Bλ ∪ i≤n Bθ0 i . If µ ∈ Bθ0 i for some i ≤ n then by the induction
hypothesis (for θi ) Bµ0 ⊆ Bθ0 . Else, µ ∈ Bλ and by transitivity of the
original sequence, Bµ ⊆SBλ , which implies that PCF Bµ ⊆ PCF Bλ ⊆
Bλ0 . Since Bµ0 = Bµ ∪ j≤k Bσ0 j for σj ∈ PCF Bµ ∩ A for j ≤ k , it
follows that σj ∈ Bλ0 and by the induction hypothesis (for µ) it follows
that Bσ0 i ⊆ Bσ0 and hence Bµ0 ⊆ Bλ0 .
Ben-Gurion University of the Negev, Beer-Sheva, Israel

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