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such that a < 3 implies p < p , there is p £ P such that p < p for all a < A. The expressions "countable chain condition" and "countably closed" are generally used instead of "$» -chain condition" and " J£,closed". W e say P is K-directed closed iff for any set X £_P, if X is directed (i.e., if (\/p,q £ X ) ( 3 r £ X) r < p,q) and |x| < K then there exists p £ P such that (\/q £ X) p < q. Note that if P is K-directed closed then P is K-closed, and if K = jy. then the converse is true as well. It is well-known that ir P has the K-chain condition then forcing^P preserves all cardinals > K , while if P is K-closed then forcing with P p r e serves all cardinals < K . A family of sets F is a A-system iff there is a set A, called the kernel of the A-system, such that for all A,B £ F, if A / B then A n B = A. The K-chain condition is often verified using A-systems. If x £ V then we will use x as a term of ^he language of forcing over V to denote x. Some autgors use x to denote x; our choice is simply a matter of typographical convenience. Other terms, i.e., terms not necessarily denoting specific elements of
V, will be denoted by letters with a dot over the top. Thus, for example, it is perfectly possible to have p \\- a £ V without having p ||- a = a for any specific a € VT Of course it is always true that if p ||r a £ V, then (Vq < p) (3r < q) (Ja. e V) r ||- a = a. The term 1 will always be used to denote the maximum element of a partial ordering. We shall assume that the class of terms of the language of forcing with P over V is full, meaning that if p ||~ -Dxc)) (x) , then there is a term x such that p ||^ cj) (x) . This assumption is crucial for many arguments, and is often used withour comment. It requires the axiom of choice in V, which is assumed throughout the paper. Sentences of the language of forcing are generally expressed in mathematical English. If the sentence is more than a few words in length, then it is usually set off from the rest of the text by quotation marks, thus p ||- "f is a function mapping x P into y". Since there are several different definitions of the class of terms of the language of forcing, we assume no particular definition here. Rather we make use of the fact that every element of V [ G ] is definable in V [ G ] from G together with (finitely many) elements of V. Every definition corresponds to a term, and every term corresponds to a definition. Accordingly, when we wish to specify a term x of the language of forcing, we simply specify how the object to be denoted by x is to be defined from G and certain elements of V. If x is a term, then the object denoted or-.represented by x in the generic extension V [ G ] is written x
1. Iterated forcing. It is well-known that a generic extension of a generic extension is expressible as a single generic extension.
Suppose P is
a partial ordering and ||- Q is a partial ordering.
Let
P ® Q = { (p,q) : p e P and ||-p q £ Q} . Let (p^c^) < (p2,q2) iff p.. < p~ and p, ||- q.. < q~ . We identify elements (p-i/q-.) and (P2#q2)
if
(Pi'^n) ^ (P2#<32) - (P1/41) • Then forcing with
P ® Q accomplishes exactly the same thing as forcing first with P and then with Q.
More precisely:
Theorem 1.1. (a) Suppose G is P-generic over V and H is Q generic over V[G]. Let J = {(p,q) £ P ® Q: p e G and q Then J is P ® Q-generic over V.
£ H}.
(J is sometimes denoted by
G 8 H.) (b) Suppose J is P ® Q-generic over V.
Then G =
{p £ P: 3q (pfq) € j} is P-generic over V and H = {q ^P (P/4) e J} is Q Proof.
-generic over V[G].
In each case we check only the denseness condition.
(a) Suppose D is dense in P ® Q. letting q
e E iff 3p e G
name for E . ||-
q £ Q.
(Pj^/q-L) ^
W e claim
choose q
||-
Define E in V [ G ] by
(p,q) e D.
Let E be the canonical
E is dense in Q.
Since D is dense there is
(P/<3) •
{p £ P: p
Then
:
But clearly px
Suppose p e P and
(p.,4-,) £ D such that
||- " q i £ E and q^ < q".
||- 9q ^ q q £ E} is dense in P and we're d o n e . £ E n H and let p £ G be such that p
Thus Now
||- q £ E .
(p,q) £ J n D. (b) If D is dense in P then { (p,q) e P ® Q: p e D} is dense
in P ® Q. dense in Q ||-
Thus G n D =j= 0 and G is P-generic over V. then there is a name E so that E = E
E is dense in Q.
But then { (p,q) £ P ® Q: p
dense in P ® Q and the proof is complete.
If E is and
||- q £ E} is D
6 Remarks. 1. W e will occasionally declare that when we have only checked that p ||- q e Q.
(p,q) e P ® 6
This is permissible
since there is always a term f such that ||- f € Q and p
||- q = r. 2.
If Q e V then P X Q
(with the coordinatewise ordering)
is dense in P ® Q, provided each q e Q is identified with its canonical name.
It is easy to see that in Theorem 1.1(a) and
(b), J n (P x Q ) = G x H .
Thus if G is P-generic over V and H
is Q-generic over V [ G ] then by symmetry we obtain the wellknown result that G is also P-generic over V [ H ] . Now that we have seen how to handle a two-stage iteration, a three-stage iteration presents no problem. with
(P ® Q) ® R.
We simply force
In order to treat iterations uniformly, it is
natural to consider elements of ((p,q),f), but sequences
(P ® Q) ® R not to be pairs
(p,q,f) of length three.
This leads
ultimately to the following inductive definition, which works even for iteration into the transfinite.
iteration iff P
is a set of a-sequences satisfying the following
conditions: (a) If a = 1 then there is a partial ordering Q p e P iff p(0) € Q Q and p < q iff p(0) < q ( 0 ) . (b) If a = 3 + 1/ 3 ^ 1 ,
then P
3-stage iteration and there is Q
So P±
so that = QQ.
= {p|3: p e P^} is a
such that II-
Q is a partial " 5 ordering, and p e P iff pi 3 e P o and I- p($) e Q o . Moreover, ot p p p P
p < q iff p| 3 ^ q U ^ A f|pl|-^fe)^q(3) . (Note by induction that the ordering on P. is uniquely determined) .
Thus P
= P ® Q . a p p (c) If a is a limit ordinal then V$ < a P. = {p|3: P e P } p a P
is a 3-stage iteration, and (i) I € P , where I(y) = 1 the maximal element of Q ) .
for all y < a
(recall that 1 is
(ii)
i f $ < a
/
p e P , q e P
and q < p 3 , t h e n r e P ,
D
p
ot
oc
where r|3 = q and r(y) = p(y) for 3 ^ Y < ex. (iii) for all p,q € P , p < q iff V3 < a p|3 ^ q|3. It is easy to see by induction that if p e P and V3 < a 3 ^ 1 => |hDp(3) e QQ. P
then p(0) e Q
If p,q € P then p < q iff ex
P
p(0) < q(0) and V3 < a 3 ^ 1 => p| 3 ||- p(3) ^ q(3) .
These
observations will be used constantly. Note that P is completely determined by Q , and P g + 1 is completely determined by P D and Qo. P
If a is a limit ordinal,
P
however, then except in certain trivial cases P
is not deter-
mined by
. There are several possibilities: P We say that P is the direct limit of
if p e P iff ^3 < a pi 3 £ P o and Vy 3 < y < a => p(y) = 1. We say that P } is stationary in a* ^ is stationary in a, f is regressive on a stationary subset of K, namely {£: £ is a limit ordinal and P }. whenever 3 ^ K and 3 . Then P is K-closed. Y a Proof. Let , £ < K, be a decreasing sequence in P . t, a We must find p < p for all £ < £ . By induction on M ct, we obtain p 3 e P o such that pi 3 ^ pr I 3 for all E, < C and 3 - (3) : ? < C> is a decreasing sequence in Q . If all P r (3) = 1, let p(3) = i; otherwise let p(3) be any term such that p|3 ||-R (V^<£)p(3) ^ p- (3) . This clearly works. If 3 is a limit ordinal then p|3 is completely determined by the p|y, y < 3, and everything is clear except possibly that p|3 € P . and p|3 € P R by virtue of the fact that p|y € P Corollary 2.6. . Then P is K-directed closed. a Y The proof is left to the reader. , and <£ , let A(p) = {a : n e ca}, and define g 03 by g , a (q) = , is such that p n - p for all n, n n+1 n n then be such that for any n, < , >: £ <—> £ is an order isomorphism, where the set n £ of sequences from £ of length n is ordered by first point of difference (i.e., by < ) . For any f: co -> £ let f': co -> £ w be given by f'(n) = lex ^
P i s the inverse limit of
common.
Perhaps the best known intermediate limit is Jensen's
construction [5] for obtaining the consistency of Souslin's Hypothesis with the Continuum Hypothesis. Thus an a-stage iteration P
will be completely determined
if we specify the initial ordering Q , the orderings Q g to be used at stages 3 + 1 - ex, and the type of limit to be taken at each limit ordinal 6 < a. If p £ P then the support of p is defined by support(p) = (3 < a: p(3) =f= 1} . Note, for example, that if direct limits are taken at every 3 ^ ex such that cf3 > to (and any kind of limit elsewhere) then the support of every p e P
is countable or
finite. Some authors replace our P
by {p|support(p): p e P }.
This is obviously an equivalent approach. Henceforth, when the notation P tion is made that P
is used the tacit assump-
is an a-stage iteration.
When the notation
Qo
is employed in connection with P , it has the meaning in the a definition above. Also, if the notation G is used, then it P
will be assumed that G a
is P -generic over V. a
If G
and G are pa
used together, and 3 < a, then it is assumed in addition that G
= {p|3: p e G }.
This assumption is justified by the following
theorem. Theorem 1.2.
If (3 < a, G
i s P -generic over V and G
=
{pi 3: P e G }, then G o is P -generic over V. a p p Proof.
Again we check only the denseness condition.
Suppose
D c p is dense. For p e P o let p e P be such that pi 3 = P ~~ _P p _ a and p(y) = i if M y < a. Let D = {q e P : (]p 6 D) q < p} . We claim 5 is dense in P . If r e P then r|3 e P o so there is a a ' 3 p £ D such that p < r | 3 . I f q e P is defined so that q|3 = P and q(y) = r (y) for 3 - y < a, then q < p and q e D. Hence D is dense.
Then G
But then p e G
n 5 =j= 0 so there is p e D such that p e G .
n D.
•
One final comment will save considerable time in the proofs to come later.
Part (a) of the definition of a-stage iteration
may be regarded as a special case of part (b) for 3 = 0 . define P
= {0} then Q
If we
is a partial ordering in the (trivial)
generic extension of the universe via forcing with P . This allows us to omit a step in several proofs.
2.
Chain conditions and closure.
Here we investigate the preservation of chain conditions and closure conditions by iterations. Throughout this section, K denotes a regular uncountable cardinal.
Theorem 2.1.
If P has the K-chain condition and
-
Q has the
K-chain condition, then P ® Q has the K-chain condition.
Proof. ||-
Note that since P has the K-chain condition, we have
K is regular and uncountable, so the second hypothesis makes
sense. Suppose by way of contradiction that A ={ (p ,q ) : a < K } is an antichain.
Let G be P-generic over V, and in V [ G ] , let
B = {a: p
If B is the natural term denoting B, then
clearly J
e G}.
II- {q : a e B} is an antichain. " P a
K-chain condition, 'we have ||-
Since
^3 < K B £ 3.
-
Q has the P
Since P has the
K-chain condition, there is y < K such that ||-
B<^_y.
this contradicts the fact that if y < a < K then p a £ B.
But
||-
D
Remarks. 1.
The converse of Theorem 2.1 is also true, i.e., if
P ® Q has the K-chain condition then P has the K-chain condition and ||-
Q has the K-chain condition.
We will not need this
result so the proof is left to the reader. 2. ||-
It should be pointed out that if Q e V then to say that
"Q has the K-chain condition" is stronger than simply to say
that Q has the K-chain condition.
It can happen that P and Q
both have the K-chain condition but P x Q does not.
See [7]
for a discussion of this topic.
Theorem 2.2. Assume (a) P is the direct limit of < P D : 3 < a> , P a (b) if 3 < a then P. has the K-chain condition P
(c) if cf a = K then (3 < a: P
is the direct limit of P
10 Then P
has the K-chain condition.
Proof.
First note that if p,q e P , 3 < a, support(p) u
support (q) c_ 3 and p|$, q|3 are compatible in P R / then p and q are compatible.
(If r < p|$, q|3/ define r e P
r(y) = 1 for 3 ^ Y < a.
Then r < p,q.)
Suppose A is an antichain in P by (a) , if p £ P
by r|3 = r,
of cardinality K .
then 33 < a support (p) c_ 3.
Note that
If cf a < K then
93 < a 3B £ A |B| =*K and Vp e B support(p) £ 3.
But then by our
remark above {p|3: P € B} would be an antichain of cardinality K in P , contradicting (b). A similar argument works if cf a > K . Suppose cf a = K .
Let A = {p : £ < K } .
Let
a continuous sequence of ordinals cofinal in a.
Define f: K -> K
by f(£) = the least ri such that support (p.. | a..) £ a .
Since
(3: P Q is the direct limit of
is the direct limit of
Fodor's Theorem, f is constant on a stationary set S.
By
Say
f"S = (n). By thinning out S if necessary we may assume that if E , 5 O £ S and £
< £ 0 then support (p
the K-chain condition, -3£T , 5 O e 1 2 compatible. Let r < p |an ,p r K ^1 2 and
f
) £a . 1 2 s 5n < ?o a n ^ Pt 1 2 |a , and define n
p
(3)
if
p
(3)
if a
Then q ^ p r /P>. r contradiction.
Since P has 1 V I0* 'Pr I a r ^l n ^2 T| q £ P by q|a
a^ < 3 < a ?
< 3 < a.
D
11 Corollary 2.3.
Assume that for every 3 < a, II- Qo has the p
countable chain condition. then P
p
If direct limits are taken everywhere
has the countable chain condition. a
Proof.
By induction on a.
Theorem 2.1 handles the successor
case and Theorem 2.2 handles the limit case.
D
Corollary 2.3 is the heart of the consistency proof for Martin's Axiom.
See Section 3.
Another corollary, useful in
reverse Easton extensions, is the following: Corollary 2.4. If K is a Mahlo cardinal, |p | < K for all 3 < K , and P
is the direct limit of
is strongly inaccessible, then P
has the K-chain condition.
Now we turn to closure conditions. Theorem 2.5.
Suppose that for all 3 < a, ||- Qfi is K-closed.
Suppose also that all limits are inverse or direct and that if 3 ^ a, 3 is a limit ordinal and cf 3 < K , then P
is the inverse P
limit of
i
?
support (p|3) =
\^J {support (p | 3) : ? < £}. Given p|$/ we obtain p(3) (hence
p| 3+D
as follows.
Since p| 3 ^ P r | 3 for all 5 < £ it is clear
that P| 3 ||-
If support(p|3) is not cofinal in 3 then this is easy,
12 so suppose otherwise.
Since support(p|3) = u{support
(p |3): 5 < C) we must have either sup(support(p.|3)) = 3 for some ^ < i; or else cf 3 ^ |?| < K.
In either case P
is the
inverse limit of
for all y < 3.
•
If P is K-closed and ||- Q is K-closed, then
P 0 Q is K-closed. A similar result holds for K-directed closed forcing. Theorem 2.7.
Suppose that for all $ < a, II—_ Q o is K-directed p p closed. Suppose also that all limits are inverse or direct and that if 3 ^ a, 3 is a limit ordinal and cf 3 < K , then P is the P
inverse limit of
13 3.
Martin's Axiom. Here we show the relative consistency of Martin's Axiom with
2
> }$•,• This was historically the first application of
iterated forcing, due to Solovay and Tennenbaum [21]. Martin's Axiom (MA) is the following assertion:
Let P be an
arbitrary partial ordering with the countable chain condition, and let
G n D
i 0. a '
We sometimes express (i) and (ii) by saying G is generic with respect to the D . £_
a
Note that MA is implied by CH. relative consistency of MA with
We will be interested in the
—i CH.
Let MA* be the same as MA except for the additional hypothesis that |P| < 2-**°. Lemma 3.1. Proof. ________
MA* implies MA.
Let P and
be given with |p| arbitrary.
It is not difficult to see that there is
P1 £ P
such that
P'| = X and (*) if p,q £ P' and -3r e P r < p,q then -3r e P' r < p,q and (**)
D
n P' is dense in P'.
(The logicians will see that an elementary substructure of (P, <, D ) will suffice.) By (*) compatibility in P' is the a cx
Now apply MA* to P' and
n P': a < X>
to obtain G c p' satisfying (i) and (ii). But then G also works for P.
14 Thus it will suffice to prove the relative consistency of MA* with —| CH.
Before giving the proof, however, it is nec-
essary to have some simple facts about forcing. Lemma 3.2. K
y
Suppose P has the X-chain condition, |p| < K , and
= K for all y < X.
Proof.
If ||-
|o| < K , then |p ® Q | < K .
Without loss of generality we may assume ||- Q <=_ K .
Fix q such that I-
q e Q.
We associate with q a function f on
K such that Va f(a) is a maximal pairwise incompatible subset of {p £ P: p ||- q = a} . Note that since P has the X-chain condition, | {a: f (a) =j= 0} | < X.
It follows that the maximum number of such
functions f is K, since K = £ { K : y < X} . But if the same function f is associated with both q_ and q,-, then clearly ||- q
= q^.
Thus {q: ||- q 6 0} has cardinality < K
conclusion of the lemma follows. Lemma 3.3.
D
Suppose P has the X-chain condition, |p| < K, K
for all y < X, and K Proof. —————
and the
= K.
Then ||- 2
Fix X such that ||- X c v. P —
= K
< K.
Again we associate a function
f on K such that f(a) = (A ,B ) , where A
is a maximal incompa-
tible subset of {p € P: p ||- a £ X} and B
is a maximal incom-
patible subset of {p £ P: p ||- a \ x} . The number of such pairs (A ,B ) is at most Z { K : y < X} = K , S O the number of such functions f is at most K X 2 then clearly ||- X
= K.
If f is associated with both X
= X2«
Thus ||- 2° < K .
and
D
Now we are ready to attack the consistency proof. Theorem 3.4.
Assume GCH and let K be a regular cardinal, K >J^>, .
Then there is a partial ordering P with the countable chain condition such that ||- MA ans 2 Proof.
= K.
We will obtain P as a K-stage iteration P . By induction K on a we determine P (a < K) and Q (a < K) S O that ||- Q has the
15 countable chain condition and
|Q | < K .
Direct limits are always taken at limit ordinals. Corollary 2.3 each P
Hence by
has the countable chain condition.
Lemma 3.2 we note by induction on a that |p | < K . then follows that for all a < K , ||- (VX < K ) 2 Given P , it remains to find Q . a a
Using
Using GCH it
< K by Lemma 3.3.
This will be done in such
a way that for any Q e V[G ] , if | Q | < K and Q has the countable chain condition in V[G ] , then Q "appears" as Q
for arbitrarily
large a < K . Fix a mapping TT: K -* K X K such that for every
($,Y) € K X K,
there are arbitrarily large a < K such that 7r(a) = ($,y) , and whenever TT (a) = ( $ , y ) , then $ < a. Now suppose P
has been determined.
As remarked above,
||- (VX- < K ) 2 < K , so there is a term such that II *Y ||- "
enumerates all partial orderings with universe some element of K " . Let TT (a) =
(3/Y) •
L e t
Q
b e the term which denotes the same
partial ordering Q in V[G ] a
that QI denotes in V [ G O ] p P
(recall
V [ G _ ] C V [ G ]) provided Q still has the countable chain condition in V[G ] , and which denotes the trivial partial ordering otherwise. Note that the term 0 since Q
cannot be the same as C K unless 6 = a,
is a term of the language of forcing over P
and Q
is a term of the language of forcing over P o . P
Clearly
||-
Q
has the countable condition and
|Q
This completes the construction of the P , a < K .
| < K. For the
rest of the proof we need the following lemma.
Lemma 3.5. Proof. forcing.
If a < K , X £ a and X e V[G ] , then
(93 < K ) X e V[G ].
Suppose X is denoted by a term X of the language of For each Y <
a
/ let A
be a maximal incompatible subset
16 of {p € P^: p||- Y e X } . is countable, and since P
Then X = fy: G ^ n A ^ O } ,
Now each A^
is a direct limit support(p) is
bounded below K for every p e A . Since K is regular there is 3 < K such that support(p) c 3 for all p e A But now if A
and all y < a.
= {p| 3: P e A }, then we have X = {y: G
and X e V [ G O ] .
n A
4= 0},
D
P
Now we check MA* in V[G ]. Suppose V[G ] |="|Q| < K, Q K
K
has the countable chain condition, and
By Lemma 3.5 there
is 3 < K such that Q,
Tr(a') = (3/Y) .
p
Since Q has the countable chain condition in
V[G ] it must certainly have it in V[G , ] . K
Hence Q is the
Ot
denotation of Q ,.
But then V[G , .] contains a set G which is
Q-generic over v[G , ] ; hence G n D
^=0 for all a < A and G is
directed. It remains only to check that ||- 2 we have ||-
2
< K.
= K.
By Lemma 3.3
By the argument in the preceding para-
graph, if Q is the ordering for adding one Cohen real, then Q is the denotation of more Q for arbitrarily a <||-K;2each >time force with Q one real is added. large Hence K. we This force with Q
one more real is added.
He
completes the proof. Q Remark. GCH was needed only in order to find a regular cardinal K > &
such that 2
< K for all A < K.
Any such K would work.
17 4.
Generalized Martin's Axiom. The search for extensions of MA for larger cardinals has
proved to be rather difficult.
One natural candidate for such
an extension is the following:
If P is countably closed and has
the
wfVo-chain v
z
condition and
dense subsets of P, then there is G £ P generic with respect to the D .
Unfortunately, it is not yet known whether this assertion
is relatively consistent (as one would hope) with CH + 2 *
> M .
On the other hand, several weaker versions are known to be consistent.
The one described here is perhaps the simplest (and
w e a k e s t ) , due to the author.
The first generalization of MA is
an unpublished result of Laver.
The strongest version is due to
Shelah [ 1 5 ] . Herink [8] has strengthed the version which appears here. A partial ordering P is
jy -linked iff there exists
f: P -> OK such that (Va e GO ) f
{a} is pairwise compatible.
refer to f as a linking function for P. $*,-linked then P has the
yf^-chain
We
Note that if P is
condition.
Let us call P countably compact iff for any countable set A ^ P, if for every finite F c_A there is p e P such that (Vq € F) p < q, then there is p e P such that (Vq £ A) p < q. Note that if P is countably compact, then P is f£ -directed closed and hence
$
-closed.
The generalized version of MA which we will consider is the following: GMA: If P is countably compact and }£ -linked and
There
First, the obvious version of GMA* does not seem to
imply GMA in every possible case, and second, the fact that the
18 iteration P
has the
J^* -chain ^ s
no
longer a trivial conse-
quence of Theorems 2.1 and 2.2. Let GMA* denote GMA restricted to orderings P such that |p| < 2 A . Lemma 4.1.
The analogue of Lemma 3.1 is Assume that (VX < 2T* ) X ° < 2 ^
. Then GMA* implies
GMA. Proof.
Suppose P is countably compact and
f£ -linked, and
Construct by
induction on £ < u)_ an increasing sequence
(i) \QK\
< ***•
(ii) (Vet < X) (Vp £ Q ) (3q e D^ n Q £ + 1 ) q ^ P (iii) if £ is limit and A is a countable (or finite) subset of u{Q : n < O
such that (9p € P)(Vq € A) p < q, then
(^P € Q ) (Vq € A) p < q. Then Q = u{Q r : E,
n Q. is dense in Q.
jy -linked and
Now apply GMA* to Q.
•
Assume GCH, and let K > f K be a regular cardinal.
Then there is a countably closed partial ordering P with the ft -chain condition such that ||- GMA and 2
Proof.
Q
and 2 ** = K.
As before, P will be a K-stage iteration P . For all
a < K we will have ||and
° = $>
< K".
"Q
is countably compact and }^*,-linked
If a is a limit ordinal and cf a = w then P
the inverse limit of the Po,3 < a; if cfa > co then P p
direct limit of the P ,$ < a.
is
is the
ot
By Theorem 2.5 P
is /^ -closed.
Note that support(p) is countable or finite for each p e P . We assert that P
has the
fvo-chain condition.
This almost
follows from Theorems 2.1 and 2.2 by induction on a ^ K; only the case cfa = a) is not covered.
It turns out, however, that the
19 proof for this case is no easier than the proof for the general case, so we give a direct proof for P . For each a < K let f II-
f
a
be a term such that
is a linking function for Q .
Now fix p e P .
If a < K then there must be r e P a %
K
that r < p|a and for some £ < co,, r ||q € P
f (p(a)) = £.
so that q|a = r and q(3) = p(3) for a < 3 < K .
q < p and q|a
||-
f
such
Define Then
(p(a)) = £.
Using this remark repeatedly it is easy to find
: n £ 0)> such that
(2) V n p n + 1 | a n II-
f (p (an)) = ^ n n (3) Va e u{support(p ) : n e OJ}, {n: a
= a} is infinite.
Let a (p) =
(n) =
(a ,£ ) .
Suppose I is a pairwise incompatible subset of P |l| = JVp. I
1
A standard A-system argument
<=_ I, |i'| = £>
kernel D
on
and
(using CH) yields 1
such that {A(p) : p e I } forms a A-system with
(i.e., A(p) n A(q) = D whenever p,q e I 1 and p =f q) .
Moreover, we may assume that there is a function h such that for all p £ I 1 , g |{n: a
£ D} = h
]>, such functions g |{n: a
(Since in any case there are only
£ D } , assuming C H ) .
W e assert that any two elements of I 1 are compatible; this contradiction will establish the
,fv 2 -chain condition.
1
p,q £ I , a (p) = . r < p,q. Vn r|a < p
Let
We must find
We will determine r|a by induction on a < K S O that |a,q |a and if Vn p
The limit case is trivial. let r(3) = 1.
(3) = q
(3) = 1/ then r(3) = 1-
Supppose a = 3+1.
If not, then 3 e A(p) u A ( q ) .
3 e A(p) - A ( q ) .
If P
(3) = q (3) = 1/
Suppose
(The case 3 £ A(q) - A(p) is symmetric.)
Since
20 3 ^ p | 3 for all n, we clearly have r
l$ Il""g KP
(^) :
n
e
°L)> i s
a
decreasing sequence in Q .
Since ||-o Qo is countably closed, there is r($) such that p
p
r|B ||-p Vn(r(3) ^ P n (3)). Then r|e+l < pn |e+l, qnj 3+1 for all n. Finally, suppose 3 e A(p) n A(q) = D.
As above,
(*) r|3 ||-o are decreasing, p n n We claim that for each n, (**)
r|3 ||-o p (3),q (3) are compatible, p n n
Fix m > n such that h(m) = (3,£) • Then
=C
' S°
compatible, and (**)
follows by (*).
with the assumption that ||-
"Q
But (*) and (**)
together
is countably compact" imply
that for some r(3)/
|hp Vnr(3) < Pn(3)/qn(B)• This completes the proof of the
^j> -chain condition.
If now (VX < K) "X"°< K then we may finish the proof using Lemma 3.1 just as in the case of ordinary MA.
If
(•3X < K) "A ° = K then we must be circumspect. Let O be a partial ordering.
A finite or countable set
A c^ Q is consistent iff for any finite F c A there is p e Q such that for all q e F, p < q.
Let Q be the set of all consistent
subsets of Q, partially ordered by reverse inclusion (i.e., A < B iff B c^ A) .
It is easy to see that Q is countably compact.
We claim that if Q is
£y -linked, then so is Q.
f: Q -> a) be a linking function.
Let
Let us say that A,B e Q are
isomorphic if there is a bijection cj>: A ~> B such that (i) for p,q e A, p < q iff $ (p) < c|> (q) , and (ii) for all p e A, f (p) = f (<J> (p) ) .
21 It is clear by CH that there are only
fy^ isomorphism types.
If g(A) = the isomorphism type of A, then we assert that g is (essentially) a linking function for Q.
This requires showing
that if A and B are isomorphic (via (J>, say) then A and B are compatible, i.e., A u B is consistent. finite.
Suppose F ^ A u B is
Find finite F <^_ A and F 2 c_ B such that <J> carries F..
onto F
and F <=_ F.. U F^.
Fix p such that p < q for all q e F .
Then f (p) = f (<j)(p)) so there is r e Q such that r < p, cf) (p) . But then r < q for all q e F
u F^.
Now at stage a in the iteration we determine Q
as follows.
Let ir(a) = ($,y) , and let Q be the denotation of QI in V[G_]. _
Let Q
p
be the term which denotes Q if Q is
$
p
-linked (in
V[G ] ) , and which denotes the trivial ordering otherwise. We must verify that this works. countably compact and
j^y -linked, and
sequence of dense subsets of Q. compatible. Q
In V[G ] , suppose Q is
Then |l | < f>2 •
Let I c D be maximal ina — a
As
in tne
c Q so that |Q I = X, (Va < X)I
previous section, find
£ Q. and if p,q e Q. and
(3r e Q)r < p,q, then (-3r e Q ) r < p,q.
Then Q.. is still
F)-, -linked, so there is a linking function f: Q -> oo-^. As in the previous section, there is 3 > K such that Q ,f,I 6 V[G O ] for all a < X. Ja p tion of QI in V [ G D ] . tion of Q
If TT (a) = (&,y) , then clearly the denota-
is Q .
Let E Q
We may assume Q, is the denotai.
a
= {A e xQn : I n A =t 0} . We claim E is dense in l a ' a
(and clearly belongs to V[G ]) .
If A e Q , then since Q is
countably compact there is p e Q with p < q for all q e A. Since I
is maximal in Q there is p 1 e l
are compatible. Now V[G
such that p and p 1
But then {p 1 } u A < A and {p 1 } u A e E . ] contains a set G which is Q -generic over
V[G ]. Let H = UG.
Since G n E
A 0, H n I
4= 0.
Also every
22 finite subset of H is compatible. directed.
We need only check that H is
But without loss of generality we may assume that for
all a, 3 < X
there is y < X such that
D
= {p e Q: G q e I ) H e I j p < q,r} . Then H n u{l : a < X} Y o i p a is directed and is generic with respect to the D . •
Remark.
As in the last section, GCH was not necessary.
we needed to assume was CH, K is regular, and (VX < K ) 2
All < K.
23 5.
Intermediate Stages. It seems clear intuitively that if P
ation and $ < a, then forcing with P forcing with P
is an a-stage iter-
should be the same as
followed by an (a-3) -stage iteration in the sense P
of V[G ]»
Here we make this intuition precise.
P o
If 3 < a and p e P , let p i
p = (p 3) u p
8
Thus
= {p : p £ P .} Given a P -generic set pa a p GQ, define an ordering on P by setting f < g iff P pot (3p € G )p u f < p u g. (The last < is in P . It is an exercise p a to verify transitivity.) Let P o be a term of the language of pa forcing with P which denotes P with the ordering above. p pa Theorem 5.1. P is isomorphic to a dense subset of P_ ® P o . a * 3 3a Proof.
•
= p|(y: 3 ^ y < a} .
R
Let P
Let (j)(p) = (p|3,p ) . If p < q then p | 3 ^ q| 3 and it
follows that p < (p| B) u q . <J> (p) < (j) (q) .
Hence p| B ||-g P
It is also easy to show that if $ (p) < <|> (q) then
p < q.
Suppose (p,f) e Pfi ® P R .
f e P
such that q ||- f = f.
Then there is q < p and
But then (j) (quf) = (q,f) < (p/f) ,
and the range of (j) is dense in P Theorem 5.2.
^ q # and
® P
Let a and 3 be as above.
.
Q
Then
II- P. is isomorphic to an (a-3)-stage iteration. p pa Proof.
We work in V
= V[GO].
By induction on y < a - 3 we
p
will construct a y-stage iteration P
(with associated terms
O ) and an isomorphism h : P n n -> P such that —y y 3f3+y -y (*) If 6 < y and f e P g B + then (h (f))|5 = h 6 ( f | e + 6 ) . Note that (*) allows us to handle limit ordinals easily; simply define P f
to be all y-sequences s such that for some
e P
Q o^ 9VS
Details in this case are left
has been obtained.
We must find Q
and
24 h
.
The natural choice for Q
incorrect because Q V with P over V Q
is Q
is a term in the language of forcing over
, whereas Q
must be a term of the language of forcing
with P . Nevertheless it can be arranged that Q
both denote the same object.
generic over V . Po ® Po
and
This is done as follows.
Suppose H is P -generic over V .
p
, but this is technically
Then h
As in Theorem 1.1 (a) , G
(H) is P
® h
-
(H) is
-generic over V, and if $ is as in Theorem 5.1, then
p,p+Y
<J) (G
® h
(H) ) is P 1
V'[H] = V[())~ (G
-generic over V.
X
® h"" (H))].
Now let Q
Moreover,
be the canonical term
representing the same object in V'[H] that is represented by in "1 Since ||-R+
^ Qft+
is a partial ordering, we must have also
Q is a partial ordering; this determines P _ . P ^v —v+1 —v If H-QJ^T £ Q , then just as before there is a canonical 1 term T which represents in V'[H] the same object that T represents in V ^ " h
+ 1 (f)
1
^
® f ^ (H) ) ].
Define h^±:
P ^ ^ 1
= p iff P|Y = h (f | B+Y) and P ( Y ) = ftB+y) .
to see that h
is an isomorphism of P
We must show h
_ is onto. Y+l
into P
•> P y + 1 by It is easy .
It will suffice to show that
if a is a term such that ||- o e Q , then there is x such that lr B + Y T e Qg+y' a n d Ihp cr = T 1 . Now P^, 0^ and a are all elements of V = V [ G O ] . Since P and 6 are canonically determined 3 —Y 7T . from G o , there are canonical terms P and Q of the language of P
forcing over V with P
which represent
them.
(Note that Q is
P
a term denoting a term.) It is not difficult to find a term a representing a in V[G ] such that P
||If G
"a and Q are terms and ||-. a e Q." is P
-generic over V, then cj) (GR
) determines a
25 generic set G
® G
c_ p
P -generic over V[G ]„
® p
Now let T be the term of the language
of forcing over V with P compute a
. If H = h " (G ) , then H is
obtained as follows.
Given G
,
and then compute the denotation s of this term in
V[G O ][H].
Let T denote s.
P
Naturally T makes use of canonical terms denoting G_ and H, p
but such terms are easily obtained from the term G R G
.
denoting
It is easy to see that x works, and this completes the
proof.
D
Remarks 1.
In a sense Theorem 5.2 is completely trivial, and it
should always be thought of as such.
The subtleties about terms
of different languages may be confusing at first, but they must be brought to the reader's attention.
Since the correspondences
are invariably canonical, however, henceforth we shall treat, say, Q
and 0
as being virtually the same thing.
2.
In view of Theorem 5.2, we may speak of P
inverse (or direct) limit of P
as being the
,y < a, even though technically
Po is not an iteration. $a One question which will come up later on is the following: If P
is the inverse limit of the P ,y < a, does it follow that a Y in V[G_], P. is the inverse limit of the P o ,y < a? The answer P pot py in general is no, although in practice the answer is usually
yes.
We formulate a condition in Theorem 5.4 below which will
make the affirmative answer easy to verify in most cases.
First,
however, we treat the case of direct limits. Theorem 5.3.
If P
in V[G ] , P
is the direct limit
Proof.
is the direct limit of the P ,y < a, then of the P
,y < a.
Simply observe that every member of P
ends in a string pa
of i's.
a
26 As before, suppose P Let K = { 6 < a : $ < < $
is an a-stage iteration and 6 < a.
and P
P
is the direct limit of the o
P ,y < 6}. Call a set X c a - 3 Ko-thin iff (V6 £ KD)sup(Xn6)< 6. Y ~~ P p Note that if p e P then support(p) is K -thin, pa p Theorem 5.4.
Suppose that for every limit ordinal y < a, P
is
either the direct or inverse limit of the Pr,6 < y. Fix |3 < a, o and assume that V[G D ] |= For every K -thin set X there is p P K -thin Y € V such that X c Y . P If P
is the inverse limit of the P ,y < a, then in V[G
is the inverse limit of the P Proof.
,y < a.
Suppose f e V [ G Q ] is such that dom p
and V y ( 3 ^ Y < a - > f | y £ P R ) . f < g < f.
],P
f = { y : $ ^ Y
We must find g £ P
< a
}
so that
(This means that for all y such that
3 < Y < OL, f |y < g|y ^ f | Y-) Let f be a term denoting f in V [ G D ] , and such that P
||-6 f £ P ^ .
Suppose Y € V is Kg-thin, p j l ^ support(f) £ Y,
and p
Now for 3 ^ Y < c t /
o
£ G_. p
of the language of forcing with P obtained from G in V [ G Q ] . VCG
as follows:
if Y 6 Y let g(y) be the term which denotes the object x
First compute the denotation of f
This is possible since y > 3.
Then
]
• 6 f (y) is a term of the language of forcing with P which in V[G ] denotes a set x. Then g(y) denotes x. If 3 ^ Y < OL and Y • # Y ^ Y let g(y) = 1. In either case ||- g (y) e Q . We claim that p u g £ P , where p is a 3-sequence of l's. It is clear that if q = p u g, then Vy ||- q(y) £ Q
so the only
difficulty occurs at limits.
If P. is the direct limit of o P ,y < 6, then support(q) n 6 is bounded in 6 since support(q) = Y, which is K -thin. Since the only other alterP
native is an inverse limit we must have p u g £ P , and thus g £ P^.
But clearly po||-g f < g < f.
•
27 A good example of the use of Theorem 5.4 is given by the iteration P
of Section 4. K
If $ < K then the K -thin sets are 3
precisely the countable (or finite) subsets of K - 3 - But since P
is countably closed, these sets are the same in V and in
V[G ] .
Thus the hypothesis of Theorem 5.4 is satisfied.
It
follows that from the viewpoint of any intermediate stage V [ G O ] , P
the rest of the extension (via P ) looks very much like the entire original extension (via P ) . This accords very well with intuition. Another corollary of Theorem 5.4 is the following, which will be used in Section 6. Theorem 5.5.
Suppose K is a regular uncountable cardinal,
3 < a, P o has the K-chain condition and if 3 - Y
< a
then
P
||- Q
is K-directed closed.
y is limit then P
Suppose also that if 3 < Y -
is the direct or inverse limit of the
P., 6 < y, and if cfy < K then P P., 6 < y- Then o IIProof.
P
is the inverse limit of the
is K-directed closed.
By Theorems 2.7, 5.3 and 5.4, it will suffice to show
that II-
"VX if X is K -thin then (3Y e V) X c y and Y is K -thin" P — p Suppose p||-DX is K -thin. Let Y = {£: (3q < p) q II- £ e x) • P . P P Obviously pll- X c y. We claim Y is K -thin. If 6 e KQ then P
P
cf6 ^ K.
~~
P
P
Now p | | - D sup(X n 6) < 6, and s i n c e P
has t h e K-chain
P p condition there must be 6 ' < 6 such that p||-o sup(X n 6) < 6'. P
But then Y n 6 c 6' and we're done.
Q
28 6.
Reverse Easton Forcing. If K is a regular cardinal and X > K, then let P(K,A) be
the usual ordering for adding A subsets of K.
Elements of
P(K,A) are functions p such that |p| < K, dom(p) ^ X and range (p) £ 2.
Let p < q iff p ^ q.
Easton [6] showed that a model of 2
° = $* 2 A 2
= JT>3
results if one begins with a model of GCH and forces with the product P ( j ^ ,}vJ x P ($>.,, $*») . This amounts to adding the subsets of 0) before the subsets of 0) are added.
This observa-
JL
tion remains true even when subsets of many cardinals are added at once as in [6]. One can always regard the subsets of larger cardinals as having been added before the subsets of smaller cardinals. What happens if, as may seem more natural, this process is reversed?
In the example above, if we force first with P (yyQr ,HV
and then force over that model with P ( jy , fv~) as_ defined in that model, then
jy 2 will be collapsed to
a model of CHI
jy, and we will arrive at
Nevertheless it still turns out that this reverse
Easton forcing has many applications.
As Silver [20] has shown,
it is extremely useful in a number of problems involving large cardinals.
One of the simplest of these problems is treated in
this section. A treatment of reverse Easton forcing from a Booleanalgebraic point of view may be found in [13]. For the rest of this section we will assume that the reader is familiar with the elementary theory of measurable cardinals. See [9]. A cardinal K is A-supercompact iff there is a transitive class M and an elementary embedding j : V ->• M such that (a) j is the identity on V , the set of sets of rank < K (b) if x £ M and (c) j (K) > K.
|x| < X then x € M.
29 The following is a slightly weakened version of a result due to Silver. Theorem 6.1.
See the remarks at the end of the section.
Suppose K is K
-supercompact and 2
= K .
Then
there is a partial ordering P such that ||- K is measurable and 2 P Proof.
= K
We will obtain P as a (K+l)stage iteration.
obtain Q
as follows.
||- y CX
Let y
Given P , ex
be a term such that
is the least regular cardinal > $ a CX
for all 3 < a. p-*~.j
Then let Q^ be P ( y a , y a + + ) , i.e.,
If a is a limit ordinal but a is not strongly inaccessible, let P
be the inverse limit of the P O ,B < a; if a is strongly inot p accessible let P be the direct limit of the P o ,3 < Ot. a p Let P = P
hC i 1
.
If G is P-generic, then we assert that in
V[G], K is measurable and 2 Lemria 6.2.
Proof.
P
= K
. W e need several lemmas.
has the K-chain condition, |P I = K and
Since K is K
-supercompact, K is measurable, hence
Mahlo, hence strongly inaccessible.
In view of this fact, it
is easy to check by induction that if a < K then |p | < K |- y
< K.
(Use Lemma 3.2 for the successor case.)
and
Since P
is
the direct limit of the P , a < K, it follows immediately that jp | = K. that P
Also, since K is Mahlo Corollary 2.4 applies to show
has the K-chain condition.
implies immediately ||- y
= K.
Thus ||- K is regular. •
Since |p | < K for all a < K we shall assume that
This
30 (Va < K ) P
£ V , and hence j(P ) = P 0t
P .
K.
Ql.
(Recall j: V -> M by K
and j is the identity on
CL
-supercompactness). Of course this
assumption depends on the way the forcing apparatus has been set up; if it should turn out not to be the case that P
e V
still true that j yields a canonical isomorphism of P
it is
and
(
* V• Now in M, j(P) is an iteration of length j(K+l) = j(K) + 1. M We shall denote the stages of this iteration by P . The assumption above then says P
= P
for all a < K .
mains strongly inaccessible in M, we have P
Proof. ——————
Also, since K re= P .
Suppose ||- q e Q . Since P has the K-chain condition, HC K K
K
is the same whether computed in V or V[G ] . Thus
||-
"domain (q) £_ K
and |domain(q) | < K".
K-chain condition, there must be D c K ||- domain (q) <=_ D.
Again by the
, |D| < K, such that
For each a e D let A
be a maximal in-
compatible subset of {p £ P : p||- q(ot) = o } , and let B
be a
maximal incompatible subset of {p e P : p||- q(a) = l}. Note that if q
and q 2 both give rise to the same sequences
and then ||- q
= q0.
and completely describe q. (b) of the definition of K
Thus
Moreover, by condition
-supercompactness, M and V both have
the same sequences and . 06
Thus P
Ot
will K T J.
come out the same whether it is defined in M or in V.
0
Now G is P -generic over V; by Lemma 6.3 G is also M P .-generic over M. Lemma 6.4. x £ M[G].
If x c M [ G ] , X e V [ G ] and M [ G ] f= |x| < K + + , then
31 Proof.
Since M[G] is a model of AC, it will suffice to assume
that x is a set of ordinals. V[G].
Let x be a term denoting x in
By the K-chain condition for P , there is a set D e V
such that |D| < K
and ||- x £ D.
incompatible subset of {p e P : ++ K definition of K -supercompact, x £ M[G] since x = {a 6 D: A nG a
For a e D let A
be a maximal
p||- a e x} . By (b) in the K e M. But then i o}. D '
For notational convenience let us denote K + 1 henceforth M by £. The ordering on P r . ,r. (recall that in M MM M,P r = P ® P r r ) is definable in M[G], hence in V[G]. 3 \t>) s s / D \t>) Lemma 6.5.
Pr
, .r
M If A c p
Proof.
is K
, every finite subset of A has a lower
bound in A, and |A| < K M [ G ] , Pr
. .r. is K
-directed closed in V[G].
, then A £ M [ G ] by Lemma 6.4.
Now in
-directed closed by Theorem 5.5 so A has
a lower bound.
D
Now let A = {q : (5p £ G)q = j(p)}. Lemma 6.6. Proof.
Every finite subset of A has a lower bound in A.
Suppose p. e G and q. = j(p.), i = l,...,n.
is p £ G such that p < p., i = l,...,n. P
Then there
Let q = j(p). Since
is a direct limit, there is some C c K , sup C < K, such that
support(p) c_ c u { K } .
Since j is elementary,
support (q) c_ j (c) u {j (K) } = C u {j (K) } . Moreover, we have II £ M q|K = p|K. Since p e M we have p u q e P . (r\ • Also, since q(K) = 1, p u q^ < q. But q < q. so p u qs < p u q., i = l,...,n. 1 M E r1 And by the definition of P r . /r. , this means q ^q^, i = l,...,n. It follows now from Lemma 6.5 that A has a lower bound q
£ Pr
Let V
.,r* . Let H be P. . r.-generic over V[G] with q = V[G] and M
= M[G][H].
= H.
32 Lemma 6.7.
In V [H] the map j can be extended to an elementary
embedding j : V Proof.
-*• ML.
Let G ' = G ® H .
p < q|hC + 1, and q 1
If p £ G and q = j (p) then
e H since a
e H and q
< q . Therefore,
1
q e G .
So p € G implies j(p) e G .
Define j
as follows. If x 6 V. then there is some "term x •VTGI * . MTG'1 * such that x = x . Let j (x) = j(x) . We claim j is elementary. x-,...,x
Suppose V.. \= (f) (x , . .. ,x ) . Choosing terms
as above, there must be p e G such that
p||--.
so
M |= (J) (j* (x ) , . . ., j (x ) ) . Note that this shows j defined, since if p||-> ^
is well-
= x 2 then j (p) lh-(r} j (x^ = j (^2) .
Now we are nearly done.
Let U = {x c K: X e V
D
and
K € j (X)}. By Lemma 6.5, every subset of K in V [H] is already in V , so U is a measure ultrafilter on K in V [H], It is easy 2K = K
to see that ||V [H] |= |u| < K K
||-
++
(||- 2 K = K
and 0
is P(K,K
, and by Lemma 6.5 again, U e V1.
)) so
Therefore
t
"2 = K and K is measurabl "2 = K and K is measurable", and the proof of Theorem 6.1 is complete. • Remarks 1.
In order to obtain the consistency of the assertion,
"There is a measurable cardinal K such that 2
= K
" it is not
sufficient to assume simply the consistency of the existence of a measurable cardinal.
Kunen [10] has shown that if K is
measurable and 2
then there are inner models with many
> K
measurable cardinals. 2.
The argument
we have given is not quite optimal, since
by means of a trick one can show that in the situation we considered K is actually K -supercompact in V [ G ] .
The same trick
would yield our conclusion that K is measurable in V [ G ] from the weaker assumption that K is K -supercompact in V.
Since
33 the trick tends to obscure the important ideas, however, we have omitted it.
34 7.
Axiom A forcing. In this section we consider a type of iteration which covers
a great number of cases arising in practice. A partial ordering
(P,^) satisfies Axiom A iff there exist
partial orderings < < : n e 0)> such that (1) p
<
Q
q iff p < q
(2) if p <
q then p < q n+1 n (3) if
(-3q e P) (Vn) q <
p .
(4) if I is a pairwise incompatible subset of P, then (Vp € P) (Vn) (9q £ P) q <
Condition (4 1 )
p and {r e I: q is compatible
with r} is countable. (4) may be rephrased in terms of forcing as follows: (Vp £ P) (Vn) if p set x £ V and q <
To see that
(4) implies
||- a £ V, then there is a countable p such that q ||- a £ x. (4 1 ) suppose p ||- a £ V.
Let I be
a maximal incompatible subset of {r e P: -3a £ V q II- a = a }. r r If q < q
p and {r £ I: q and r are compatible} is countable, then
||- a. £ x, where x = {a : r £ I and q and r are compatible} . For the other direction, let I be a pairwise incompatible
set.
Without loss of generality we may assume I is maximal.
Let a denote the unique element of I belonging to the P-generic set G. q <
Given p e P we must have p ||- a e V.
p and countable x such that q
By
||- a e x.
(4 1 ) there is
But then
{r £ I: q,r are compatible} c_ x. Following are some examples.
In each case details are left
to the reader. Example 1. Countable chain condition forcing.
If P has the coun-
table chain condition, then for n > 1, let p < 2.
Countably closed forcing.
Let p <
q iff p = q.
q iff p < q.
35 3. Perfect-set forcing
(Sacks f o r c i n g ) .
Conditions are
sets p c_ u{ 2: n £ U)} satisfying (a) Vs £ p Vn £ domain(s) s|n £ p . (b) Vs £ p
-3t,u £ p
s<^_t,u and t and u are
incomparable with respect to inclusion. Let p < q iff p £ q,
Conditions may be thought of as trees of
height co in which forking occurs above each element. are called perfect since {f e subset of
Such trees
2: (Vn)f|n e p} is then a perfect
2 (with the product topology).
If p is perfect and
s £ p then the degree of s in p is | {n: -3t £ p s | n = t n and s(n) =f t ( n ) } | . tree p .
This is just the number of f
Let p <
q iff p < q and
in q is < , then s £ p .
ks below s in the
(Vs e p) if the degree of s
See [1] for a treatment of iterated
perfect-set forcing. 4. Prikry-Silver forcing.
Conditions are functions p such
that domain(p) ^_ 03, range (p) £ 2 and O)-domain(p) is infinite. Let p < q iff p ^_ q.
Let p ^
q iff p
D
q and the first n
members of GO-domain (p) and co-domain (q) are the same. 5. Laver forcing.
Conditions are sets p ^ u{ (JO: n £ 0)}
such that (a) (Vs £ p)(Vn £ domain(s)) s|n £ p . (b)
(-3s £ p) (Vt £ p) t £ s or s £ t, and if s £ t then |u £ p : u is immediate ^-successor of tj is infinite.
Let p < q iff p <=_ q. for the reader.
The definition of <
is left as an exercise
See [ 1 1 ] .
6. Mathias forcing.
Conditions are pairs
(s,A), where s is
a finite subset of co and A is an infinite subset of co. (s,A) <
(t,B) iff s £_ t, A ^ B and s - t £ B.
Let
This ordering will
be discussed at length in Section 9. 7. Adding a closed unbounded set with finite conditions. This is perhaps the simplest cardinal-preserving ordering which does not satisfy Axiom A.
Conditions are finite functions p
36 from u). into OJ, such that for some closed unbounded set C c 00 , if f: co ->• oj enumerates C in increasing order, then p £ f. Let p < q iff p 3_ q. In work at present unpublished, Shelah [17] has found a class of partial orderings, the proper partial orderings, which is more extensive than the class of Axiom A orderings, and for which the analogue of Theorem 7.1 below is still true. For example, the ordering in Example 7 is proper but fails to satisfy Axiom A. Nevertheless we have decided to treat only the Axiom A case, for two reasons. First, the Axiom A orderings are nearly as extensive as the proper orderings, and iterated Axiom A forcing seems to be a little easier to handle than iterated proper forcing. Second, in the author's opinion the proof of Theorem 7.1 is conceptually simpler than the proof for proper forcing. The reader who understands the proof of 7.1 should be able to follow Shelahfs argument without great difficulty. We shall be interested here in iteration with countable support, i.e. iterations in which inverse limits are taken at ordinals of cofinality GO and direct limits are taken elsewhere. For example, the iteration in Section 4 has countable support. One checks easily that for any element p of such an iteration, support(p) is countable or finite. Theorem 7.1.
Suppose "P
is an a-stage iteration with countable
support, and (v$ < a) \\- Q R satisfies Axiom A. collapse oj .
In addition, if a < OJ , CH holds, and (V$ < a) ||-
| Q J ^ ^ x , then ? Proof.
Then "P does not
has the 5^ 2 " c h a i n condition.
The fundamental notion in treating Axiom A forcing is the
following.
Suppose F c_ y < a,F is finite, and n e oo. If
p,q e P
then let p < q iff p < q and (V3 e F) p|& ||- p(g) y r ,n p < q ( 3 ) . (Here, of course, < refers to the ordering on Qo.) n n p A set D £ P is (F,n)-dense iff (Vp e P ) (3q e D) q < p. Note that < is a partial ordering. F, n A sequence < (p ,F ) : n e co> will be called a fusion n n sequence iff
37 (a) Vn p < p *n+l F ,n n n
(c) L K F : n e 00} = U ( s u p p o r t ( p ) : n € GO}. Lemma 7 . 2 .
If < (p ,F ) : n e OJ> i s a fusion sequence in P , then
t h e r e i s a p e P such t h a t Vn p < p . ^ a ^ F ,n ^n n Proof.
By induction on 3 < a we determine p|3 e P Q so that p
Vn pi 6 < p | 8 , and if p (g) = i for all n, then p(3) = 1. F no p / n n n n The latter condition guarantees that support (p|S) is at most countable, and it makes the case 3 limit a triviality. B = y+1 .
If y | U(support p : n e a)} then let p(y)
wise fix n minimal such that y e F .
Let q
n
and let a p|y
= p (y) if m < n.
||- (¥m) q
<
q .
Til
=
Suppose
1-
Other-
= p (y) if m > n m
Now p|y < p |y for all n, so
Hence there is p(y) such that p|y \\-
(Vm)p( Y ) < m q m . Lemma 7.3.
Suppose $ < a.
Then:
(a) If |U
a e V, then {q e P : 9 countable x e V, p q II- a € x} is (F,n)-dense for all finite F c 3 P
p
—
and n € a). (b) If ||-
"x is a countable subset of V", then {q e P :
3 countable x e V, q |[- x £_ x} is (F,n)-dense for all finite F c 3 and n € a). (c) If 3 < Y ^ a and ||- f € P Gf
e P
, then {q e P :
) q ||- f = f} is (F,n)-dense for all finite
F c 3 and all n e a).
38 Proof.
The proof is by induction on 3 (a) First suppose 3 = Y + 1.
P|Y Ih
If p ||- a e V, then
"p(Y) ||- * a £ V" .(Note that a is here being used in
two different senses.
See the remarks in Section 5.)
Hence by
1
condition (4 ) applied to Q , there must be terms x and q of the language of forcing with P p|y ||-
such that
"x is a countable subset of V,
q <
p(y) , and q |ky y
a e x"
Now, using (b) of the inductive hypothesis, there is countable x e V and q1 <
p|y such that q' ||- x £ x.
by q|y = q' and q(y) = q.
Clearly q ||-ft a e x, and q <
Now suppose 3 is limit. that F c y < g. —
Then p|y
||-
p.
Fix F and n, and suppose y is such
If p £ P D , let p Y denote p|{6: y < 6 < 3 h p
" p ^ ||- a e V", where the last "||-" refers to
forcing with P p|Y
Define q e P
R.
Hence there must be f and b such that
l|-y "f € P y B , f < p Y , b £ V and f ||- a = b " .
(Here we are using a both as a term of the language of forcing with P R and as the corresponding term of the language of forcing over V[G ] with P ^.) applied to P
F, n
(a) and
there must be q <
x £ V such that q ||q u f <
By
p | y , f £ P
f = f and b £ x.
p , and q u f ||-Q a e X. p
(b) Suppose p
(c) of the inductive hypothesis o
But then q u f e P R ,
This proves
||-oX = .
and countable
(a) .
Fix F c^ 3 and n.
Using
(a) repeatedly, it is easy to construct a fusion sequence < (p ,F ) : m £ co> such that p
= p for m < n, F
mm m n m = n + i there is countable x. £ V such that p
= F, and for
||-o a. £ x. .
Let x = u{x. : i e b)} . If q £ P is such that q < p i ^ F ,nrm m, then q ^ F R P and q \\-^ x £ x.
for all
39 (c) Suppose p e P
and II- f e P P
q <
P
.
By (b) there is
PY
p and countable x e V such that q ||- support (f) £_ x.
Now define f e P
as follows.
If 6 f x then f (5) = 1.
If
6 e x then f (6) is the term denoting the same object denoted by g(6), where g is the interpretation of the term f.
It should be
clear that f e P o .
D
PY
But also q ||-o f = f. P
Lemma 2.3 (b) shows that oo1 cannot be collapsed by forcing with P .
There is also a version of Lemma 2.3(a) which is related to
2.3(a) as (4) is related to (4'), namely: (a1) If I ^_ P
is pairwise incompatible, then
{q € P : { p e l : p,q are compatible} is countable} is (F,n) -dense for all finite F c a and all n e oo. Now we prove the final sentence of Theorem 7.1. to prove that P
has the
It suffices
H^-chain condition for all a < a^; the
case for a = 0)2 will* then follow by Theorem 2.2.
The following
lemma will therefore complete the proof. Lemma 7.4.
Assume CH.
If a < w
and (V$ < a) ||-ft | Q J < rS , p
z
p
p
i
c p such that |D I < H H, , and and D is (F,n)-dense a — a ' a' 1 a ' 1 for all finite F c a and all n e w .
then there is D
Proof.
The proof is by induction on a.
Suppose a = 3 + 1.
Let D
c p be as in the Lemma. Fix — P (d : g < a)_> such that ||~o
s
1
p
t,
1
p
countable pairwise incompatible set I c D and each function — p f: I -> a) , let q = q(f) be the term determined as follows: if I n G
= {p} then q denotes the same thing as d
I n G
= 0 then q denotes the same thing as 1.
; if Now by CH the
P
total number of such functions f is at most j ^ so the set T of all such terms q(f) has cardinality < yy . Let D = {p e P : pi 3 e D o and p($) e T } . Then |D | ^ }y. Ot
Ot
and we claim D works. a
p
Fix p e P , F and n. a
Ot
1
Let I be a maximal
40 incompatible subset of {d e D : d < p and for some P
£, d II- p(3) = d r } . By (a1) above we may find d e D P s
p
such that
d < p and J = {q e I: d,q are compatible} is countable. F,n Define f: J -> u> by f (q) = £ iff q ||- p(3) = d . Now it is easy to see that d ||- p(3) = q(f). Hence if r e D
is such
that r | 3 = d and r(3) = q(f) , we have r < p. Finally, suppose a is a limit ordinal. then let q e P
If 3 < a and q e D
be such that q| 3 = q and q(y) = 1 for 3 ^ Y < ex.
Using CH, it is easy to find a set D c p such that D | < Jy_ a — a a' 1 and (i) if 3 < a and q e D then q e D p a (ii) if < (p ,F) : n £ a)> is a fusion sequence and Vn p e D then there is p e D such that Vn p < n p . ^ a * F ,TV n n* We claim D works. Fix p e P , F and n. a * a Case 1.
cf (a) > w.
Then there is 3 < a such that support (p)
c_ 3 and F <^ 3. By inductive hypothesis there is q € D p I 3. But then q <_ p. that q < ^ F,n ^' ^ F,n ^
Case 2.
cf a = w.
such
Let be an i n c r e a s i n g
———__
Yd
sequence with supremum a, and suppose F c a . For each m > n,
o let f 6 P be defined by f (6) = rp(5) for a < 8 < a ,.. m a ,a ,. m m m+1 m m+1 Let q € D be p a . Now it is easy to find a be such such that that q q < ^ Fn * a ^ F,n n
n n fusion sequence < (p ,F ) : m e ca> such that p
= q for all
m < n, F = F, and for m > n, p _ = q__ . , where q_ _ is such m+1 TH+1 Tn+1 n that q^ _ € D and q . <,_, q u f . ^m+1 a Tn+1 F ,m ^m m m+1 m By (ii) there is q e D
such that q ^
But it is easy to check that q < Jb / m
p.
p
for all m. D D
41 8.
An application to trees.
The rest of this paper is devoted to applications of the ideas in Section 7. In this section we will show how, given an inaccessible cardinal, to construct a partial ordering P such that ||- MA + "Every
jy, -tree has at most
n
branches" + 2 ° = tf
2
py
uncountable
.
A partial ordering (T,<) is a tree iff (Vt e T){s e T: s < t} is well-ordered by ^. The level of t e T is the order type of {s e T: s < t } . The ath level of T is {t e T: the level of t is a } . The height of T is the least a such that the ath level of T is empty. A branch through T is a maximal linearly ordered subset. An
Aronszajn tree is a tree T of height 00 with no uncountable branches such that for all a < OJ^ the ath level of T is countable. A Souslin tree is an Aronszajn tree in which every pairwise incomparable set is countable. An ft.-tree is a tree of height U)]_ and cardinality -^]_» A Canadian tree is an ft -.-tree with at least _K2 uncountable branches. A Kurepa tree is a Canadian tree in which the ath level is countable for all a e U), . An n -tree T is special iff there is a function f: T •* OJ such that Vs,t,u e T if s < t, u and f($) = f (t) = f (u) , then t and u are comparable. [Note that this is different from the usual definition, which says T is special iff (if: T ->• OJ) (Vs,t e T) if f(s) = f (t) then s and t are incomparable. It is easy to see that a tree is special in the usual sense iff it is special in our sense and it has no uncountable branches. Thus our definition extends the usual one to trees with branches.] Theorem 8.1. If T is special then there are at most fos uncountable branches through T. Proof. Let f: T -* 0) be as in the definition, and let B be a branch. There must be s e B such that {t e B: f(t) = f (s) } is uncountable. But then B is determined by s since B = {t € T: t < s or else -]u s < u and f(u) = f(s)}. Since there are only /^i choices for s, there are only j^ i choices for B. Theorem 8.2. Suppose (T,<) is an H -^-tree with at most H]_ uncountatLe branches. Then there is a partial ordering P with the countable chain condition such that - is special.
42 Proof.
Without loss of generality, we may assume T has exactly
££. uncountable branches (otherwise simply enlarge T) . B
For each a < OJ , let s
- u{B :3 < a } . Ot
Let T
1
be the <-minimal element of
= {t: 3a s
p
< t e B }, and let
CX
Ot
S = T - T 1 . Now S, considered as a substructure of T, is still a tree. Moreover S has no uncountable branches: if B were such a branch then for some a, B ^ B , and then clearly B n T 1 =1=0, a contradiction. Suppose there were f: S -> OJ such that if f (s) = f (t) then s and t are incomparable.
Then f could be extended to all of T by
defining it on T" so that f(t) = f(s ) if s < t e B . ^ a a a
But now
it is easy to see that f shows T is special. Thus we need only find P such that for some f, ||- f: S -> a) is as above. Let P consist of all finite functions p mapping S into co such that if p(s) = p(t) then s and t are incomparable. iff p =>_ q.
Let p ^ q
A proof that P has the countable chain condition may
be constructed along the lines of [2], but for the reader's convenience we outline a (slightly different) proof here. Suppose I = {p : a e OJ } is a pairwise incompatible set. Without loss of generality we may assume |p | = n for all a € OJ.. , and that n is minimal; also that domain (p ) n domain(pD) = 0 a p when a =j= 3 • Also, by thinning out I if necessary, we may assume that whenever a < 3> p (s) = p (t) , s =|= t and s and t are compact p rable (which must happen for some s and t if p and p are ina p compatible) then s < t.
Let U be a uniform ultrafilter on OJ. , and
for each a e OJ, let domain (p ) = {s , ...,s _-, } • Now for each a there must be i (a) , j (a) < n such that { 3 e OJ, : s . , . < t. . .} e U by the assumption above. Furthermore there must be i and j such that A = {a: i(a) = i and j(a) = j} e U. But now if a ,a 2 e A then there must be 3 > a T ' a 2
suc
^
tnat
s
-
<
t«
for a = a
i' a 2
43 a l , a2 But since S is a tree, that means s, and s. are comparable.
Hence {s.: a e A} may be extended to an uncountable branch through S, contradiction.
D
From the proof may be extracted the following: Corollary 8.3. If every ££ -tree with no uncountable branches is special, then every
jr.. -tree with at most
y$_ uncountable
branches is special. Corollary 8.4. at most
MA + 2
> }£
implies that every
fy-, "tree with
yV. uncountable branches is special.
Two more technical theorems will prepare us for the main result. A partial ordering P has property K iff every uncountable subset of P contains an uncountable pairwise compatible set. Property K clearly implies the countable chain condition.
As is
well-known, the usual ordering P(J£ , A) for adding X Cohen subsets of a) (see Section 6) has property K. Theorem 8.5.
Suppose T is an S> -tree and P has property K.
Then forcing with P adds no new uncountable branches through T. Proof.
By way of contradiction, suppose p \\- B is a new branch.
Let S = {s 6 T: (9q ^ p) q ||- s e B} . each a £ con there is s s
has level a in T.
e S and p
It is obvious that for
^ p such that p
—
1
e B and
Since P has property K there is uncountable
A c to such that {p : a e A} is pairwise compatible. 1
11— s
But then
Ot
{s : a £ A} is linearly ordered, so there is an uncountable ii * i branch B through S. Since p ||- B f V, it follows that (Vs £ S) (-3t, u £ S) s < t,u and t and u are incomparable. S'={s£S:sis<
Hence
-minimal such that s \. B} is an uncountable
pairwise incomparable set.
But if s and t are incomparable,
q.. |(- s £ B, and q^ \\- t £ B, then q_ and q^ are incompatible.
44 Since P has the countable chain condition S can therefore have no uncountable pairwise incomparable set, contradiction.
H Theorem 8.6.
Suppose T is an
countably closed.
$
-tree, 2
> jj
and P is
Then forcing with P adds no new branches
through T. Proof.
As before, suppose p ||- B is a new branch.
a € u{ 2: n € a)} we will find p p a
II- s e B. o
Also, if a
u
c
< p and s
x then s
—
o
arbitrary such that p
||- s
such that
< s and if a and x are x
incomparable (with respect to c) then s We proceed by induction on |a|.
and s
are incomparable.
If a = 0 let p
e B.
For each
Given p
< p and s
be
and s , let x, and
T 9 be such that | x | = \TO\ = \o\ + 1 and a = x - n x0 (i.e., x £
£
JL
JL
*L
x.
and Xp are the two immediate successors of a) . Since p ||- B is a new branch, there exist incomparable s and sT > s and there Tl 2 ° exist p / p < p so that p \\- s £ B, i = 1,2. This comT T l 2 cr x± T± pletes the construction. For f e
2 let p
< p I
since P is countably closed.
for all n e w .
This is possible
We may also assume that p_ is
chosen so that for some s £ > s^i , n e co, p^ ||- s^ e B. r f|n ^f " cxf
But if
f =j= g then s
this
and s
are incomparable.
Since 2
° > ^
means | T | > j^ , contradiction, Corollary 8.7.
If T is an
/*£ -tree then there is a partial
ordering P(T) satisfying Axiom A such that
'hp(T) T is Proof.
Let P
= P(}£ , $ ) , the ordering for adding
reals with finite conditions.
Let P p be such that ||-
the standard partial ordering for collapsing 2 countable conditions." onto
rv
jy 2 Cohen "P 2 is
onto P* with
(The standard ordering for collapsing X
consists of countable functions mapping o> into X.)
By Theorems 8.5 and 8.6, forcing with P
® P 2 adds no new
45 branches through T, while at the same time, ||Let P
T has at most $
be such that ||-
*
uncountable branches.
"P~ is the partial ordering with
the countable chain condition which makes T special". Let P (T) = (P
® P 2 ) $ P . Clearly ||-
T is special.
To
see that P(T) satisfies Axiom A it will suffice to observe that if P satisfies Axiom A and ||- Q satisfies Axiom A, then P ® Q satisfies Axiom A. and p_ ||- q
<
Simply let (p ,q ) <
(p^/^o) ^^ f Pi ~
q . Details are left to the reader.
Po D
Now we are ready to prove the main result. Theorem 8.8.
If there is a strongly inaccessible cardinal, then
there is a partial ordering P such that ||- MA + 2 Proof.
° = h S 2 + "Every
The proof is very similar to Theorem 3.4.
strongly inaccessible. P
-ftx -tree is special". Let K be
We will construct P as a K-stage iteration
with countable support (recall this means inverse limits at
ordinals with cofinality GO; direct limits elsewhere) . For each a < K we will have I1I1
a
Q
satisfies Axiom A and IQ 1I < K . 'a
a
Using the inaccessibility of K it is easy to check by induction on a < K that |p | < K (use Lemma 3.2 for successor ordinals). Hence each P
has the K-chain condition so by Theorem 2.2, P
has
the K-chain condition. It remains to determine Q . As in Section 3, for each 3 < K, let
11—
*Y u<
Qol
y
< K>
enumerates all partial orderings with universe an element of K".
46 Also, let
Here L
is a term denoting 2
in V [ G j . p Let TT: K -> K X K b e a pairing function as in Section 3. P
Now if a is even, say a = 2-a 1 , let TT(a') = (3/Y) • be a term such that Q
Let
Q
denotes the same ordering Q in V[G ] that
QI denotes in V [ G O ] , provided Q has the countable chain condition P p in V[G ] ; otherwise Q denotes the trivial ordering. If a is odd, say a = 2-a1 + 1, let IT (a1) = ($,y) • Let Q^ be a term such that if T is the tree denoted by T^ in V [ G O ] then •Y 3 3 Q a denotes P(T); if T' is undetermined (i.e., if Y - t n e denoP
tation of X in V[G ]) then Q denotes the trivial ordering, p P & Since |p | < K and K is inaccessible, it is straightforward to check that II- IQ I < K . 11 a ' a' Hence by Theorem 7.1, P
Clearly II- Q satisfies Axiom A. "a a
does not collapse a) . One checks that
MA holds in V[G ] for orderings of cardinality < K just as in Section 3. To see that ||- "Every n , -tree is special", argue first as in Lemma 3.5 that if T £ V[G ] and T is an K - t r e e then (•33 < K ) T £ V [ G O ] . (This requires the K-chain condition.) -Y Hence T is the denotation of some T^. But if IT (a1) = (3/Y)/ P 1
then at stage a = 2-a + 1, T was made special.
And, of course,
once T was made special it remained special in all further extensions. Since r>, and K both remain cardinals in V[G ] , we have K V[G ] X [G ] VCG ] K K > i^2 . We claim K = & 2 K . If not, let X = ry2 Since X < K, one sees as in Lemma 3.5 that for some 3 < K , V [ G O ] P
contains onto mappings f: co -»• p for all y < X. But then X = f^o Z
. If T is an
/^n-tree in V [ G O ] , then at some later 1
(3
point T is made special by forcing with P ( T ) . But among other
47 things P(T) collapses 2 j ^ ,
contradiction.
onto S> . Hence X is collapsed onto
Thus K =
The proof that 2
H*2V[GK] .
= K in V[G ] is left to the reader.
This completes the proof. Remarks. 1.
D
Silver [19] showed that in the model obtained by
collapsing all cardinals below an inaccessible onto f>., CH holds and there are no Kurepa trees.
However, CH implies that Canadian
trees exist; just consider the complete binary tree of height QJ, . Silver also showed that in order to obtain a model with no Kurepa trees it must be consistent that an inaccessible cardinal exists. Hence the inaccessible is needed in Theorem 8.8. Mitchell [14] showed the consistency (assuming an inaccessible) of the assertion that there are no Canadian trees.
Our
argument, particularly Theorems 8.5 and 8.6, bears the imprint of Mitchell's approach. Finally, K. Devlin [4] showed with a long and difficult argument that by forcing over Silver's model with an ordering as in Section 3, a model of MA + 2 trees" results.
= ji 2 + "There are no Kurepa
Unfortunately, there are Canadian trees in
Devlin's model since any Canadian tree in Silver's model remains Canadian in Devlin's model. 2.
In recent work, Shelah and the author have found several
strengthened versions of MA for proper and Axiom A forcing. stronger axioms imply that every
These
yV -tree is special, as well as
other results that do not follow from ordinary MA, but to prove the axioms consistent some fairly large cardinals seem to be needed.
These axioms will therefore be discussed elsewhere.
48 9.
Iterated Mathias forcing and the Borel Conjecture.
A set X of real numbers is said to have strong measure zero iff for any sequence <£n' n e co> of positive real numbers there is a sequence of intervals such that the diameter of I n is < e n and X c_ u{l n : n e co} . The Borel Conjecture asserts that every set of strong measure zero is countable. In [11], Laver proved the consistency of the Borel Conjecture using iterated Laver forcing with countable support. He remarked that the same could be done with iterated Mathias forcing; that is the goal of this section. For a more complete discussion of the Borel Conjecture, see [11]. Mathias ordering P consists of all pairs (s,A), where s is a finite subset of w, A is an infinite subset of co and (Vm 6 s) (Vn e A)m < n. Let (s,A) < (t,B) iff s =>_ t, A c_ B and s - t c^ B. For a thorough discussion of Mathias forcing, see [12]. First we argue that P satisfies Axiom A. For n > 1, let (s,A)
If (s,A) ||- a. e V, then there is B c_ A and countable
x € V such that (s,B) ||- a £ x.
Moreover, B may be chosen so that
if (t,C) < (s,B), a e V and (t,C) ||- a = a, then (t,B-(max(t)+l)) ||- a = a. Proof. We will construct inductively a sequence b^ < b, < b o <... ————— o i. £ of elements of A and a sequence B ^ B i i ^ 2 2- • * • °^ subsets of A such that (Vb e B
,)b < b. Let B^ = A. Given B , let n+1 n 0 n s,,...,s, enumerate all the subsets of {b. : i < n} . Now construct a sequence B0 —D B. B . Given *i i —D ... —D B. k as follows. Let B^ = On B., if there exists C c_ B , such that for some a e V, (s u s. ,C) ||- a = a, then let B ^
= B".
be such a set C; if not let
Finally let b n be the least element of B £ and let
B _ = Bf - {b }. n+1 k n
Let B - {b : n £ 0)} . n
49 Let x = { a £ V : ( 3 t £ B ) ( s u
t,B-(max (t)+l) ) ||- a = a} .
Then x is countable, and we claim (s,B) ||- a e x. (t,C) < (s,B) and (t,C) ||- a = a.
Suppose
Let t - s £ { b . : i < n} . Using
the notation for step n in the induction above, assume Since (t,C) ||- a = a, we must have chosen B.+-,
t - s = s. .
that for some a 1 , (su s.,B.
) 11- a = a' .
But 1
(t,C) < (s u sj,/B^+1) = (t/B*?+1), so a = a . (t,B-(max (t)+l) ) < (t,B.
) so a e x and we are done.
Suppose (s,A) ||- a e V.
of the first n elements of A. sets of t.
But also D
P satisfies (41) in the definition of Axiom A.
Corollary 9.2. Proof.
so
Determine B
Let n be fixed and let t consist Let t , ...,t
B
B
=>_ -i zL • • • £ v
Given B., apply Theorem 9.1 to find B.
c
enumerate the sub-
as follows>
Let A =
B
o-
B. and countable
£ V such that ( s u t . , B . )||- a £ x. . Let B = t u B . i+l i i+1 i+1 K If x = x u x 0 u ... u x , then it is routine to see that
x
(s,B) ||- a £ x.
D
Thus the Mathias ordering satisfies Axiom A. There is one additional property, however, which is crucial for the Borel Conjecture. Theorem 9.3.
Let (f be a sentence of the language of forcing.
For any (s,A) e P there is B £ A such that either (s,B) ||- $
or (s, B) 11- —! 4>. Proof.
First note that there exists B' ^_ A such that if
(t,C) < (s,B') and (t,C) ||- <|> (resp. ~i <|>) , then (t,B'~ (max (t) +1) ) ||- $ (resp. —i <()) . This may be deduced from Theorem 9.1 using the following trick.
Let a be a term which
denotes 0 if <J> holds in V [ G ] and which denotes 1 if —i <j) holds in V [ G ] .
Then ||- a e {0,1} and (Vp e P)p||- a = 0 iff p||- <()
(and similarly for —i $) . For the purposes of this proof, if t c B 1 , let t||- cf) abbreviate (s u t,B '- (max (t) +1) ) ||- (j), and similarly for —I $ .
50 < b. < ... of B 1 and subsets
Now we construct elements b
=> B.. 3 ... of B 1 by induction as follows. Let B = B 1 . o — J- — o Given B find B' <= B S O that for all s f c {b.: i < n} one of n n+1 — n — 1 B
the following alternatives holds: (1) (Vb e (2) (Vb € B^ +1 ) s'u{b}||(3) (Vb 6 B 1 ) neither s'u{b} - <j> or s'u{b} - —\ $. n+1 " " be the least element of B 1 , and let B . = B 1 - {b }. n n+1 n+1 n+1 n Let B = {b : n £ u)} . n
Let b
Now suppose (t,C) < (s,B) and (t,C) ||- <|>. —i <|) is exactly similar.)
Let |t| be minimal.
t = s and (s,B) ||- cf> by the assumption on B' . max(t) = b
for some n.
above for s (s u s',B
1
(The case for If |t| = | s | then If |t| > |s| then
Then at stage n we must have had (1)
= t - (su{b}). n
But then we would have had
) 11—
dicting minimality of |t|. Corollary 9.4.
D
If x e V is finite and (s,a) ||- a e x, then for
any n e u) there is (t,B) < (s,A) and y c x such that |y| ^ 2 n — and (t,B) ||- a e y.
There is also B<^_A and a £ x such that
(s,B) ||- a = a. Proof.
For n = O this is trivial.
Repeated use of Theorem 9.3
shows that if (s,A)||- a e x then there is B c A and a e x such that (s,B) ||- a = a. that a = a.
(Choose a £ x and let (|> be the assertion
If B £ A and (s,B) ||- cj> we are done; if (s,B) ||- — | $
then (s,B) ||- a e x - {a}, and we proceed inductively.) n ^ 1.
Let u consist of the first n elements of A.
s,,...,s, 1
(k = 2 ) enumerate the subsets of u.
Now fix
Let
Determine
JC
B £_ B_ ^ ... 3 B as follows. Let B = A - u. Given B., use O i K 0 i the observation above to find B, c_ B. and a £ x such that ( s u s . -,B. )||-a = a. _. Let B = B, u u and let 1+1 1+1 1+1 X
51 y = {a ,ao,...,a }.
Then it is easy to see that (s,B) ||- a £ y,
and of course (s,B) < (s,A) .
D
From now on, let P denote an a-stage iteration with a countable support (inverse limits at ordinals of cof inality (indirect limits elsewhere) such that for all 3 < a, ||- Q is the Mathias ordering. We need some technical lemmas before we can prove the theorem. The notation of Section 7 will be used. Lemma 9.5.
Suppose x e V is countable and p||- a e x.
finite F ^ a and any n, there is q < n
For any
p and y c x such that
F
|y| < 2 ' ' and q||-a a £ y. Proof. The proof is by induction on a. First suppose a is limit.
Fix 3 such that F c_ 3 < a.
There
must be terms b and f of the language of forcing with P o so that P P|B||-D
"b
£ x,
f £ Po
P
, f < p
3
and
f ||- a =
b."
pOt
(The last forcing symbol refers to forcing with P
.)
Now by
pCX
Lemma 7.3 (c) and the inductive hypothesis,
there exist
q'<
p|3, f e P ^ and y c x such that lyl < 2 ' ' and F, n ' 3a — ' ' q'||b £ y and i. = f. But then q 1 u f < p and q 1 u f||p F, n d
Now suppose a = 3 + 1above.
So assume 3 € F.
a e y.
If 3 f F then proceed as in the case
By Corollary 9.4 there exist q and
y ,...,y., where k = 2 , such that B"y1'--wyk
e x, 4
(Here, of course, the last forcing symbol refers to forcing with Q o .)
By inductive hypothesis there exist q1 <^ /1 -ci I T \
y n ,...,y
c x such that |y.| < 2
n
'
'
p(3) and
Ffip,n
,i=l,2,...,k,
and
52 u ... u y, then lyl < 2 ' ' and if q|$ = q 1 1 K and q(3) = q then q ^ p and q||- a e y. D
But now if y = y
Lemma 9.6.
Suppose <x : n e u)> e V is a sequence of finite sets.
If p||- Vnf (n) € x , then there exist q < p and a sequence
^ x
and |y | < 2
, and
q||-a Vnf (n) 6 y n . Proof.
Use Lemma 9.5 repeatedly to construct
______^_
j^
fusion sequence <(p ,F ) : n e o)> such that p € V q||-a Vnf(n) e y n>
If q
-F n ,n P n
for a11 n
' D
= P,|F | = n, and
then
Now we are ready for the main theorem. Theorem 9.7. Proof.
Assume CH.
Then ||-
The Borel Conjecture.
By Theorem 7.1 we know that P
has the
J^-chain condition.
verification that
^2
preserves cardinals and
We leave as an exercise the o 2
In order to treat the Borel Conjecture, it will be convenient to reduce it to a proposition about the space
2 with
the product measure generated by the measure on 2 giving {o} and {1} both measure y.
It is easy to see that the union of
countably many sets of reals of strong measure zero still has strong measure zero, so it suffices to prove the Borel Conjecture for subsets of the unit interval [0,1].
Moreover, the
mapping (j) which associates to each real in [0,1] the function in 2 obtained from its binary representation has the property that <j) is one-to-one, its range is all but a countable subset of and <)> is measure-preserving.
Let us say that X £
2 has strong
measure zero iff
2,
2 has a basis consisting of Baire intervals of the
form [s] = {f £ W 2 : s c f}, where s e u{ n 2: n e a)} .
If s e n 2
53 then the measure of [s] is — .
With this in mind it should be
easy to see that a set X ^_ 2 has strong measure zero iff (Vf e ^CJ) (3g) (Vn)g(n) e
f(n)
2 and (Vh € x)(3n)g(n) c_h. The
latter statement is sufficiently set theoretical for our purposes . Suppose X e vCG
]
and X is an uncountable subset of s
Arguing as in Lemma 3.5 and using the j f2~
- condition, we
see that there must be $ < 03o such that X e v [ G o ] . z
2.
cna:i n
Defining K o
p
p
as in Theorem 5.4, we see that a set is K -thin iff it is P
countable. Hence by Lemma 7.3(b) the hypothesis of Theorem 5.4 is satisfied, and it follows that in V [ G O ] , V is (isomorphic P
P(A)2
to) exactly the same kind of iteration as P . To be precise, Po is isomorphic to P as defined in V [ G O ] . Thus without 3a)2 032 & loss of generality we may assume X e V. Now since P. is canonically isomorphic to the Mathias ordering P, the P..-generic set G, determines a P-generic set G. Let Me
0) enumerate u{s: (-3A) (s,A) e G} in increasing order.
For each n e o), define M
€
w by letting M (m) = M(m+n) .
assert that there exists.n such that in v[G such that Vm g(m) e
We
] there is no g
2 and Vh £ X in g(m) c_ h.
Now by
Lemma 9.6 applied in V[G,] (and using the fact remarked above that over v[G,], P, is the same kind of iteration as PW ) , 1 1,032 2 for any such function g there is y e v [ G , ] such that Vm g(m) e y(m) and |y(m)| < 2 Let Z
= {y e v C G , ] : Vm y(m) C_
M (m) m2 2 and |y(m)| < 2 }.
Hence it will suffice to prove in VEG.. ] that for some n, (*)
(Vy £ Z n )(5h € X)(Vm)(Vs e y(m)) s ^ h.
This, of course, will involve only forcing with the Mathias ordering P.
54 Since X is uncountable it is easy to see that for each m e 0) there is k (m) e u) such that |{s £
k(0)
2 : [s] n X is uncountable] > 2 and
for all m, if s € k(m+1)
|{t €
2:
2 and [s] n X is uncountable then
s c t and [t] n X is uncountable}| > 2 ( m + 1 ) .
Note that if y e Z
and j e u) then there are uncountably many
h € X such that (Vi < j)(Vs e y(i))s £ h. Now let (s,A) e P be arbitrary. be such that if N e
Let n = |s| and let B <=_A
u) enumerates B in increasing order, then
N(m) > k (m) for all m.
We have chosen n minimal so that (s,A)
decides no value of M . We assert that (sfB) ||- (*) , and this will complete the proof. Suppose on the contrary that (s,B) ||/ (*) . Then there exist (t,C) < (s,B) and y such that (t,C) ||- y e Z n and (Vh € X) (3m) (3s1 e y(m))s' c_ h. Here Z
n
is the name for Z . n
Lemma 9.8.
There is D £ C such that for every finite u £ D and
every m such that m + n < |t u u|, there is z = z(m,u) such that (t U u,D-(max(u)+l)) ||- y(m) = z. Proof.
Note that the condition on m and u guarantees that
(t u u,C-(max(u)+l)) decides M (m) and hence restricts the set of n possibilities for y(m) to a finite set. We construct elements d
< d. < ... of C and subsets D
Let D
= C.
D_D, ^ ... of C as follows.
Given D., let (u ,m ) ,
,(u ,m ) enumerate all
pairs (u,m) such that m + n < |t u u| and u c _ { d . : i < j } . We obtain D^ => D"? => ... 3 D^ as follows. Let D^ = D. . Given D"?, O — i — — K . OD i note that for some r e CD, (t u u,,D.) II- M (m.) = r and I I " n l . y(m.) c r 2 . Hence by Corollary 9.4 there is D"?,. c D"? and z l — l+i — l such that (t u u.,D. ) ||- y(m.) = z. Let d. be the least element of D w and let D. _ = D.3 - {d.}. Finally, let D = {d. : i e co} .
55 It is easy to check that D works.
D
Now if h e X and u is a finite subset of D, let us call h u-good iff (Vu1 c_ u ) (Vm) if m + n < | t u u ' | then Vv e zOi^u1) v £ h. then 3u' c^ D u s u
1
Call h avoidable iff Vu c D if h is u-good and h is u'-good.
If there is h e X which is avoidable and O-good, then we may construct a sequence 0 = u^ 5 u. 5 ... such that h is O f 1 f u.-good for all i.
But if E = u{u.
: i e u>} then
(t,E) ||- (Vm) (Vv £ y(m))v j^h, and this contradicts our original assumption about (t,C). Now if m is such that m + n < |t| then by Lemma 9.8, (t,D) decides the value of y(m). Hence by the remark following the definition of k(m) there must be uncountably many h e X which are O-good.
Since none of these O-good h can be avoidable,
there must be u c^ D such that if K = {h e X: h is u-good but not u u {k}-good for any k e D - u } , then K is uncountable. Note that if h e K, k e D and u c_ k, then we must have h|k e z(u'u {k}, m(u')) for some u 1 f_ u, where m(u') is chosen so that m(u') + n = |t u u' | . Let % > 2 K.
, and let h- f ...,h 0 be distinct members of
Choose k e D so large that u ^_ k and h1|k,...,h |k are
distinct.
J
Now h. Ik, .. . ,h J k e u{z(u'u {k} ,m(u ')) :u ' c u}
the other hand, |z(u'u {k},m(u'))| < 2
/ i\9
m(U }
|u{Z(u'u {k},m(u')): u 1 c u } | < 2 l u l 2 m ( u ) contradiction.
mW
< 2
< 22m(u)
This completes the proof.
On
( \ 2—
. Hence
Remarks 1. It is clear from the proof of Theorem 9.7 that it is not necessary to iterate Mathias forcing at every stage. It will suffice to have a cofinal set of a such that II11
a
Q
a
is the Mathias ordering ^
provided that for every a < au we have
56
If- " | Q | = H,
a
nd Q
satisfies the first sentence of Corollary 9.4."
Examples of orderings satisfying Corollary 9.4 are Laver forcing, Sacks forcing, and Prikry-Silver forcing. (See Section 7 for the definitions.) 2.
One naturally asks whether it is possible to obtain the
consistency of the Borel Conjecture with 2
.Ho = JT3• c v. This problem
is still open, and it is instructive to see where the obvious proof (an iteration of length w^ with countable support) breaks down.
Roitman was the first to observe that in many iterations
cardinals are collapsed for reasons having nothing to do with the combinatorial properties of the component orderings; this is a minor variation on her observation. Suppose 2
> )^_ and P JL
is an 0),-stage iteration with 1
OJ -1
countable support such that for all a, (**)
||_ Q ftas
at
least two incompatible elements.
Then forcing with P as follows.
collapses 2
onto
For each a, choose p , q
||- p ,q
-M-, • The proof goes
so that
are incompatible elements of 0 .
Now, in v[G
] define for each a < wn a set S = {n e 00: the 0)-^ l a interpretation of p in V[G ] lies in the 6 -generic set co-a+n ooa+n a
G obtained from G
+!•''•
the countable support of P
An
elementary denseness argument using shows that every subset of a) in V
appears as some S . In the case of iterated Mathias forcing, 2
= _}& in
v[G
] S O the argument above shows that j ^ o is collapsed onto r a>2 2 .H, in V C G ]. Hence an oo^-stage iteration still yields a 2
model of 2 °
= ,H> I This problem occurs with many other itera-
tions as well. Note, by the way, that our iterations in Sections 3, 4, and
57 8 do not satisfy (**), since there may be p e P p||-
Q
Nevertheless, (**) as well.
is the trivial ordering. can be weakened to include these iterations
This yields another proof that 2
of Section 8.
such that
= ft „ in the model
58 References 1.
J. Baumgartner and R. Laver, Iterated perfect-set forcing, Annals Math. Logic 17 (1979), 271-288.
2.
J. Baumgartner, J. Malitz and W. Reinhardt, Embedding trees in the rationals, Proc. Nat. Acad. Sci. U.S.A. 67_ (1970), 1748-1753.
3.
P.J. Cohen, Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.
4.
K.J. Devlin, tf -Trees, Annals Math. Logic 13 (1978), 267-330.
5.
K.J. Devlin and H. Johnsbraten, The Souslin problem, Lecture Notes in Mathematics 405, Springer-Verlag, Berlin, 1974.
6.
W.B. Easton, Powers of regular cardinals, Annals Math. Logic 1^ (1970), 139-178.
7.
F. Galvin, Chain conditions and products, to appear.
8.
C D . Herink, Ph.D. Dissertation, University of Wisconsin, 1978.
9.
T. Jech, Set Theory, Academic Press, New York, 1978.
10.
K. Kunen, Some applications of iterated ultrapowers in set theory, Annals Math. Logic 1 (1970), 179-227.
11.
R. Laver, On the consistency of Borel's conjecture, Acta Math. 137 (1976), 151-169.
12.
A.R.D. Mathias, Happy families, Annals Math. Logic 12 (1977), 59-111.
13.
T.K. Menas, Consistency results concerning supercompactness, Trans. Amer. Math. Soc. 223 (1976), 61-91.
14.
W. Mitchell, Aronszajn trees and the independence of the transfer property, Annals Math. Logic 5 (1972), 21-46.
15.
S. Shelah, A weak generalization of MA to higher cardinals, Israel J. Math. 30 (1978), 297-306.
16.
S. Shelah, notes on the P-point problem.
17.
S. Shelah, handwritten notes on proper forcing, 1978.
59 18.
J.R. Shoenfield, Unram^fied forcing, in Axiomatic Set Theory, D. Scott, Editor, Proc. Symp. Pure Math. 13 (1), Amer. Math. Soc. Providence, R.I., 1971, 357-381.
19.
J. Silver, The independence of Kurepa's conjecture and two cardinal conjectures in model theory, in Axiomatic Set Theory, D. Scott, editor, Proc. Symp. Pure Math. 13 (1), Amer. Math. S o c , Providence, R.I., 1971, 383-390.
20.
J. Silver, unpublished notes on reverse Easton forcing, 1971.
21.
R.M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. Math. 94 (1971), 201-245.
Dartmouth College, Hanover, New Hampshire, U.S.A. 03755.
THE YORKSHIREMAN'S GUIDE TO PROPER FORCING
Keith J. Devlin University of Lancaster, U.K.
§0. INTRODUCTION Properness is a property of forcing posetswhich generalises the countable chain condition, is preserved under countable support iterations, and which, when applied to iterations of posets of cardinality
X. (the posets one usually wants to iterate)
guarantees the preservation of all cardinals.
The notion was
formulated by Saharon Shelah, and arose from his study of Ronald Jensen's technique of iterated Souslin forcing (see [DeJo]) and from his attempts (eventually successful) to show that the GCH does not resolve the Whitehead Problem (see [Ek], together with §4 of these notes)•
In order to motivate the notion, let us
review some basic forcing theory, which we will in any case need later. If IP is a poset (we always assume that IP has a maximum element, 1 , and that for each p e IP there are q,r e IP such that
61
q < p, r < p, and q , r are incomparable in IP) , there is a uniquely defined complete boolean algebra BAQP) which extends IP as a parti a l l y ordered set and has IP as a dense subset (rh the usual, poset QP) sense).
We d e n o t e by V
.
(BAGP))
t h e boolean universe V
[Je] for basic forcing theory).
(see
To simplify arguments, i t
is
common to speak of a "V-generic subset, G, of P" and thence of "the extension V[G] of V".
Though technically wrong, this pro-
cedure is easily eliminated by working entirely with countable models of set theory, but i t is more convenient to avoid this restriction,
so we shall make frequent use of the above mentioned
abuse.
In this connection, each element, x, of V[G3 is the OP) "collapse" of some (in fact many) boolean set of V^ . Any OP) member of V which collapses to x in V[G] is called a P-name of x.
(Shoenfield,
in [Sh], makes use of a canonical collection of
P-names in order to develop his theory of unramified We often use x to denote a P-name of x e V[G].
forcing.)
Elements of V,
which are a f o r t i o r i elements of V[G], have canonical names:
the
v canonical name of x e V c V[G] is denoted by x. The boolean truth-value of a sentence (f> of set theory when interpreted in V
is denoted by ||
.
The forcing relation
between members of IP and sentences < > j is defined by p tKp <)j iff p < ||*||P
(in BAOP)).
Since c a r d i n a l i t i e s are not in general preserved when we pass from one universe of set theory to another, we are interested in properties of posets which guarantee the preservation of cardinals.
62 The most common are as f o l l o w s . If P s a t i s f i e s the K chain condition ( i . e . IP has no pairwise incompatible subset of c a r d i n a l i t y K ) , where K i s a regular uncountable c a r d i n a l , then cardinals X > K.
||A i s a cardinal
If
||
=1
=1
for a l l
In p a r t i c u l a r , i f IP s a t i s f i e s the countable
chain condition ( = c . c . c . = cardinal
||
K, chain c o n d i t i o n ) , then
II V
||X i s a
for a l l cardinals X.
W i s K-closed ( i . e . every decreasing sequence of elements
of IP of length l e s s than K has an infimum in IP), where K i s a regular uncountable cardinal, then for a l l a < K, ||(Ka) = P(a) ||
=1
( i . e . V and V™' have the same subsets of a
for a l l a < K ) , so in particular || X i s a cardinal \\
= 1
for a l l
cardinals X < K. If IP i s tc-dense ( i . e . the i n t e r s e c t i o n of any family of fewer than K dense i n i t i a l s e c t i o n s of IP i s dense in IP), where K i s a regular uncountable cardinal, then for a l l a < K, | |(?(a) = (P(a)|| = 1 , so again || X i s a cardinal ||
=1
for a l l cardinals X < K.
Clearly, i f IP i s K-closed then i t i s automatically K-dense. The converse is false (very false!).
OP) Suppose IP is a poset, and that in V there is a boolean object Q such that ||Q is a poset || IP 0 Q as follows.
= I .
We define a poset
As domain we take the set of all pairs (p,q)
such that p e W and pU~"q £ Qn.
(To ensure that this collection
i s a set, we should identify pairs (p,q), (p,q ! ) such that p H"q =<j* "„
i n practice we usually ignore this point.) The partial
63 ordering i s defined by (p\q')
< (p,q)
iff
p'
There i s then a canonical s t r u c t u r a l TJP ® *** [ V ^ ] ^ .
isomorphism
Moreover, i f G i s V-generic on IP and H i s
V[G]-generic on Q, then G x H i s V-generic on IP 0 Q, and every V-generic subset of IP 8 Q has the form G x H for some such G,H. See, for example, [SoTe] for d e t a i l s . The above "product lemma" provides the key for performing (transfinite)
iterations of boolean extensions.
We say that a sequence ( P |v < 7i) of posets i s an iteration sequence i f there i s a sequence ^Q v < X, Q i s an element of V ^ v = 1 and IP _,, = IP ® Q . v+1 v v anything about IP
|v < X^ such that for each
such that
|
This d e f i n i t i o n does not t e l l us
in case v i s a limit ordinal, of course, and
there are various different requirements we can make to cover t h i s case. We say that <^IP
| v ^ X^ i s a full iteration sequence
for each limit ordinal v < X, W
if,
consists of a l l sequences
q =
p "qd+O qf
< q
su
e Q "> and the ordering of P^ i s
i f f q f (O)
P P o r t °f
a n
element q of IP
such that q(c) ^ 1 •
(We use I
maximum element of any p o s e t . that q(c) i s not the term :
i s the s e t of a l l those £ < v as a general symbol for the
Incase £ > 0 here, t h i s means the maximum element of tjr .)
64 The iteration sequence ^1P
| v < X } is said to have finite
support, if, for each limit ordinal v < X, P
consists of all
sequences q =
| v < X / is that
for any v < x < X there is a canonical element Q that IP
-P
® Q .
e V
v
such
Hence an iteration sequence corresponds to
a sequence of successive boolean extensions (of the same length). Now, if <1P
| v < A ) is an iteration sequence with finite
supports, and if IP
satisfies the c.c.c. and for each v < X, |JQ
satisfies the c.c.c, || v = 1 , then every poset IP , v ^ X, satisfies the c.c.c.
This is proved in [SoTe].
Thus finite
support iterations of c.c.c. posets preserve cardinals.
This
enabled Solovay and Tennenbaum to construct a model of Martin1s Axiom in [SoTe].
A natural question is whether or not there is
a more general property than c.c.c. which is preserved by iterations.
More specifically, suppose we want to perform an
iteration in which not only are cardinals preserved, but CH is preserved as well.
Finite support iterations are of no use
here, since new real numbers appear at each limit stage. countable support iterations destroy the c.c.c. posets being iterated are both
But
Well, if the
H-'dosed and of cardinality Ht,
then each poset in a countable support iteration is also
^.-closed
65 and satisfies the
S^-c.c, so such an iteration could preserve
both cardinals and the CH. one can do with observed!
But unfortunately there is not much
^-closed posets of cardinality H1S as is easily
Much more useful would be a countable support
iteration of posets of cardinality 8y which are fy-dense. Unfortunately,
^-density is in general not preserved by count-
able support iterations.
So what we require is some property
of posets which is more general than
S-closure, which is
preserved under countable support iterations, and which implies at the very least that (for posets of size are preserved.
H > say) cardinals
These notes discuss just such a property.
turns out to include c.c.c. posets as well:
It
an extra, perhaps
unexpected,bonus. The organisation of this paper is as follows.
In §1 we
give some basic preliminary results concerning a natural generalisation of the concept of a closed unbounded set of ordinals,
§2 introduces the notion of properness, gives some
examples of proper posets, and develops alternative characterisations of the notion for later use.
In
§3 we show that
countable support iterations of proper posets are proper (i.e. properness is preserved under countable support iteration.)
In
§4 we give an illustration of the use of proper forcing by establishing a combinatorial principle which leads to the solution of the Whitehead Problem in Group Theory.
In §5 we show how
proper forcing can lead to a powerful generalisation of Martin's
66 Axiom, known as the Proper Forcing Axiom (PFA).
Though
extremely interesting in its own right, and having many startling consequences (see [Ba]), it should perhaps be stressed that this does not really involve the full potential of proper forcing, which really comes into its own when used (often in more special forms) to perform delicate iterations which do not destroy CH. But that is another story. Prerequisites?
If you have got this far (and followed our
outlines) then you must have all of the preliminary knowledge required.
If you haven't got this far, then,
!
Finally, if you were present at the Cambridge Meeting of which this is the Proceedings, you may recall that I was not even present, let alone did I give any lectures on Proper Forcing, Indeed, though Shelah was present and did speak about such matters, the whole concept was still in its infancy, and certainly not in the polished (I hope) form presented here, which version I only wrote out in the Autumn of 1980.
But the Cambridge
Meeting did represent the first "world wide exposure" of proper forcing, and since in the interim no completely worked out account has been made available, the Editor of these Proceedings felt that it was appropriate to include the present paper. The notes were based first of all upon Shelah's original, rather sketchy notes written in Berkeley in 1978, and then improved considerably following some lectures given by Jim Baumgartner in Toronto in 1980.
Stevo Todorcevic pointed out
67 a significant error in what I thought was the final version, and suggested various other improvements.
The idea of publishing
the notes at all (which I originally wrote in order to sort out the ideas for myself with a view to trying to extend them - and no, I am not saying in what direction!) came independently from Jim Baumgartner, Rudi Gb*bel, and Stevo Todorcevic.
§1. PRELIMINARIES For any set A and cardinal K, we set [A]K
= {X c A | |x| =
K}
,
[ A ] < K = {X c A | |X| <
K}
. o
Let A be an i n f i n i t e s e t , and l e t C c [A] ° . unbounded in [A]' 0 i f for every X e [A] °
We say C i s
there i s a Y e C such
HA
that X 5 Y.
We say C is closed in [A]
if, whenever X
and X c x ,- for all n < oi, then \ ) X e C. n - n+1 ^f n
eC
As usual, "club"
abbreviates "closed and unbounded". Lemma 1.1 If C-,0^ are club subsets of [A] ° , then C- nC^ is also club in [A]H° . Proof:
Obvious, n
Lemma 1.2 If C is a club subset of [A] ° for each a e A, then the set a C = {X e [ A ] * 0 | (VaeX)(XeC )} a is club in [A] Proof:
Suppose that X € C and X c x ^ for all n < w. Let n n — n+l X = (JX . We show t h a t X e C. Let a £ X. For some lc < u>,
68 a e l , k
X = \J
For a l l n > k, a £ X : * n
X £ C .
k
n
soX
£ C . a
n
Thus
Hence a e X -> X e C , g i v i n g X £ C.
a
Thus C
a
is closed. u
To p r o v e unboundedness now, l e t X {a
|n
i n d u c t i v e l yJ , c h o o s e X n ' o
Choose X e C o a
e [A] °
be g i v e n .
t o extend X . o
Let Then, *
nC n...nC t o extend X n . an a . o o 1 n+1 ( T h i s u s e s lemma 1 . 1 , of c o u r s e . ) Let X, = V JX , t h e n a £ X 1 37 o o
implies X- e C . 1 a
eC
a
Repeating t h i s process inductively, pick
X ,- D X so that a £ X implies X .- £ C . r n+1 - n n n+1 a
Let X= wU
X . n
n
C l e a r l y , X £ X and X e C. n A s e t S c [A] ° i s said t o be s t a t i o n a r y i f i t meets every club set C c [A]*0. Lemma 1.3 A s e t S c [a; ]
i s s t a t i o n a r y i f f S n OJ- i s s t a t i o n a r y in
a) i n t h e u s u a l s e n s e . Proof: (-0 Let C c o) be c l u b . may assume t h a t C n a) = <>j • a = {3J3 < ct} e [w ] • .
We show t h a t S n C +
We
Thus i f a e C, then C l e a r l y , C i s club in [w ] ° .
Hence C n S f i|, as r e q u i r e d . (-<-) OJ-.
Let C c [OJ ] °
be c l u b .
Then c l e a r l y , C n a) i s club i n
Hence (CnaO n S + <)>, and again we a r e done, a
Lemma 1 . 4 . Let S c [A]
be s t a t i o n a r y , and l e t f: S -v A be such t h a t
f(X) £ X for a l l X £ S. such t h a t f fS i s c o n s t a n t .
Then t h e r e i s a s t a t i o n a r y s e t S c S
69 Proof:
For each a e A, set S
a
= { X e S | f(X) = a }.
We must show that S is stationary for some a € A. a
Suppose not.
Then for each a e A we can pick a club set C c [A] ° such that a *"* C n S =
Hence C n S + $.
Since X e S, f(X) is defined, say f(X) = a.
Let X e C n S. By the assumptions
on f, we have a e X.
So as X e C, we have X e C . a X k S , i . e . f(X) =f a, which is absurd, a a
Thus
Lemma 1.5 Let A be uncountable, and l e t C 5 [A] ° .
The following
are equivalent: (i)
C contains a club;
(ii)
there is a function f : [A]
* •> [A]
such that for any
set X e [A]H°, if f'CX]**0 c [ X ] h \ then X e C; (iii)
there is a function f : [ A]
°->• A such that for any
set Xe CA]H° , if f"[X]
(We shall
say that f concentrates on C in this case.) Proof:
Since the nature of the set A clearly plays no role in
this lemma, i t suffices
to prove i t for the case where A is an
uncountable cardinal X. (i) -*• (ii) •
By cutting down C if necessary, we may assume that
C is i t s e l f club. any X e [X]
,
By induction on n, define f f[A]
so that
for
70 X u [ \J f(Y)] c f(X) e C. YcX Since C i s unbounded, there i s no d i f f i c u l t y here. defined on [X]
°.
So f i s
Suppose that X e [X] ° is such that
f"[X]
° c [x] ° . Let {a I n
—
1
c f ({a. I i < n+1}) c X. —
—
I
H But ran(f) c C and C is closed in [X] ° .
Hence
X = {a i |i
(ii)+(iii).
y
Let f be as in ( i i ) .
For each X e [X]
(X) | n
h : [X]
and,
° , let Define
° -> X by s e t t i n g
h ( { a } ) - a + 1 (a < X) for n>0 and a < . . . < a < X, o n h({aQ,...,an>)
where n = 2 1 . ( 2 j + l ) - l . h"[X]
Suppose that X e [X] ° i s such that
We show that X e C.
f"[X] <>Jo c [X]W° . -
Let {a , . . . , a } o m
that f ( { a , . . . , a }) o m f, ( { a , . . . , a
= fi({aQ,...,a.}),
e [X]* °
c X, a < . . . < a . — o m
To show
i t s u f f i c e s t o show that
}) e X for a l l k < co.
n = 2 .(2m+l)-l.
I t s u f f i c e s to show that
Since h'TX]1
Let k < a) be g i v e n .
5 X, X h a s no l a r g e s t member.
So we c a n f i n d a _ , - , . . . , a e X s o t h a t a < a - < . . .
Q
m
Q,...,an})
as required. (iii)-Ki).
Let f be as in ( i i i ) , and set C = {X e CX]H°
Set
| f"[X]<8° c X}.
e X,
Then
71 Then C c C.
But C i s c l e a r l y club in [X]
. a
Lemma 1.6 Let
>
be an uncountable, f i r s t - o r d e r structure.
Then there i s a function f : [A] i s such that f fl [N]
N 6 [A] °
° -* A such that whenever
° c N, then N i s the domain of an
elementary substructure of 01 . Proof:
It i s standard that the s e t of a l l N e [ A ] 1 0 which are
domains of elementary substructures of 0t i s club in [A] ° .
Now
apply lemma 1 . 5 . D A function f related to a structure 01 as above w i l l be said to be (fc-skolem. Lemma 1.7 Let X be an uncountable o r d i n a l , and l e t 01 = < A , € , . . . . > be a f i r s t - o r d e r structure such that X c A.
Then the s e t
{N n X | N € [A] S ° & < N , € , . . . > - < « } i s club in [X] °. Conversely, i f C c [X] 01 = < X , € , ( f )
i s c l u b , there i s a structure
> such that for a l l N e [X] W °, i f
n n<6u then N e C. Proof:
The first part is standard.
For the second part, let
< Wo f : [X]
•> X c o n c e n t r a t e on C ( a s i n lemma 1 . 5
(iii))
and l e t
fn - fItxA Q Lemma 1.8 Let X,y be uncountable c a r d i n a l s , X < y. (i)
If C c [ y ] ° i s c l u b , then C [X * {XnX|X€C} contains a club in[X]H°.
72
(ii)
If C c [X]**0 i s c l u b , then C[p] = {Xe[y] W° |XnXeC} i s club in
Proof : ( i )
By lemma 1 . 5 , l e t f : [ y ]
° -»• y be such that
f"[X]
n+1
o
= o$ a
and set S -
U a11. n
inductively
n £ l f r n-,
Define g : [A] <W ° -> [X]" H ° by
g(a) = a n X. The s e t D = {X£[X]K° | g"[X] < K o
c CX]-W°}
i s c l e a r l y club in [X] ° . But i f X e D, then X n X = X.
Moreover, f"[X] < ° c X, so X e C.
Thus X = X n X e CTX. (ii) Trivial. U
Lemma 1.9 Let X,y stationary,
be uncountable c a r d i n a l s , X
SCy]
= {Xe[y] H °
i s stationary in [y] % * .
Proof : By lemma 1 . 8 ( i ) .
a
If S c [X]
is
73 §2. PROPER POSETS A poset
IP i s said t o be proper i f ,
whenever X i s an uncount-
able c a r d i n a l and S i s a s t a t i o n a r y subset of [ A ] ' ° , then N" (Note t h a t
stationary11.
"S i s
IP may c o l l a p s e X, or introduce new countable s u b s e t s
of X.) Lemma 2.1 All c . c . c . Proof:
Let
p o s e t s are proper.
IP be a c . c . c .
s t a t i o n a r y s u b s e t of [X]
p o s e t , X an uncountable c a r d i n a l , S a .
Given C e V
such t h a t
M-p "C i s a club subset of [X]** 0 ", we must show t h a t '•p "C n S
+ *".
Pick f e V<3P) s o t h a t '•-jp "? : CX]< H°
Define h :
m
-• [A]H<S by s e t t i n g , for X c [ X ] *
h(X) = CJ u (By the c . c . c .
for
+ X c o n c e n t r a t e s on C".
{a e X |
||f(X) =
a |p
IP, h(X) i s c o u n t a b l e . )
> C}. C l e a r l y , for any
X e CX]
iHp
M
f(X) e hOO" .
Now, t h e s e t {X e [X]" 0
h"CX]
|
c [X]*0 }
O
i s c l u b i n [ X ] ' 0 , s o we c a n f i n d an X e S such t h a t h"CX] < 8 ° c [X] H » .
By (*) we have
*-jp " f " CX]
° c l "
.
\
74 Thus
as required. P Lemma 2.2 All countably closed posets are proper. Proof:
Let
IP be a countably closed poset, X an uncountable
cardinal, S a stationary subset of [X]
.
Let C e V
be such:
V
»h "C IP
is a club subset of [
We show that J'C n S t •"•
Let I € V ^ ' be such that n- "f : [X]* ° + X concentrates on C". IP For each n < w, l e t R be the r e l a t i o n on P x [X] n R n ( p , a o , a 1 , . . . , a n ) iff p H- "f ({a>l,. . . ,a n
defined by ao
Choose K ^ X regular with IP e H , and consider the structure 01 = < H , £, IP, (R Given p e IP we find a q < p and a set X e [X]"°
such that
X £ S and q
IK "f"[X]
which at once implies that »H "(3X)(X£C n S ) " . By lemma 1.7 we can find a countable submodel
such that p £ N and N n X e S.
<w
enumerate [X] °.
Let X = NnX, and let
Since &> < 61 9 we can pick conditions
75
p
e IP n N such that P
and for some a
" po " P l " ' ' * £ X,
*n £ n n ( i . e . R (p ,a ,a ) , where m = la 1I ) . Since IP is countably m n n n ' n closed we can find a q e IP, q < p for a l l n < w. Then, as required, q »H " f " [ X ]
< M
°
e x " .
O
We shall give a combinatorial characterisation of properness. First some definitions. Let IP be a poset.
A set D c IP is said to be predense iff,
for each p e IP there is a q e D such that p and q are compatible. If p e IP, we say D is predense below p p
iff every extension of
in IP is compatible with some member of D. Let IP be a poset, A an uncountable cardinal such that P e H..
Let N -< H,
be countable with IP e N.
A condition p e IP is said
to be QP,N) -generic iff, whenever D e N is predense in IP, then D n N is predense below p in IP. Lemma 2.3 Let 1P,X,N be as above.
A condition p e IP is OP ,N)-generic
iff, whenever D e N is dense in IP then D n N is predense below p in IP. Proof:
Since any dense set is predense, one direction is trivial.
For the converse, suppose p e IP is such that whenever D e N is dense in IP, then D n N is predense below p in IP.
We show that
76 p is
OP,N)-generic. L e t D e N b e p r e - d e n s e i n IP. E =
Set
{q e IP | (3d e D ) ( q < d ) } .
Then E is dense in IP.
Since P,D e N < H^, we have E £ N.
E n N is predense below p in IP,
Let q < p.
So
Then q is
compatible with some r e E n N. Now, H x * (3d€D)(r
Then
Since q ^ p was arbitrary, we are done.D
The following lemma might throw some light on the notion of OP, N) - gener ic ity • Lemma 2.4 Let IP be a poset, X a regular cardinal such that (POP)eH . A
If G i s V-generic on IP, then G i s H -generic on W and [H.] A
=H.[G].
Moreover, if N •< H.
A
is countable with IP e N
A
and if p e W is OP*N)-generic, then p e G implies that G n N is N-generic on IP n N. Proof:
Since (POP) e H. , it is easily seen that G is H.-generic A A
on IP, that X is a regular cardinal in V[G], and that [H^]
= H X CG].
For the second part of the lemma, let D € N
be such that D n N is a dense subset of IP n N,
Then, clearly,
N ** "D is a dense subset of IP". So as N -< H. , lf we have H ^ D is a dense subset of IP". A Hence D really is a dense subset of IP. Since D e N, it follows
77 t h a t D n N i s p r e d e n s e below p i n IP, i . e . (¥q
t h a t i n V [ G ] , DnNnG + cj>, and t h e r e s u l t
follows at o n c e . a We w r i t e N -< H. if N -< H, IP X X e v e r y p e IP n N t h e r e
i s a q e l P ,
i s countable, q < p ,
IP e N, and for
such that
q i s QP,N)-
generic. Lemma 2.5 The following are equivalent: (i)
P is proper;
(ii)
There is a regular cardinal X such that
P e H. and the
set
contains a club; (iii)
For a l l regular cardinals X such that {Ne[H,]^° X
]P e H., the set
| N -< H. } ip X
contains a club. Proof: (i)-*(ii) .
Choose X regular so that IP e H,.
S = {N € [HJ*° | P e N -< X H, X Now,
the set
& N -fc p XH,h
Let
78 is club, so if (ii) were to fail, S would be stationary. us suppose this were the case.
Let
Define p : S •* H^ by letting
p (N) be some member of IP n N, no extension of which is ( 1P,N)generic.
By lemma 1.4, there is a stationary set S c S such
that p(N) = p for all N e S, where p e IP is fixed. V-generic subset of IP containing p.
Let G be a
In V[G], define
C = {N € [H V ] H ° | N-( H X V & for all DeN which are predense in IP there is a q e G such that q e D n N } . Clearly, C is club in [H, V ] H ° (in V[G]). A
So as IP is proper
V there i s an N e S n C. Notice that N e V and N -< EL . q £G, q < p, so that q Ih "N e C".
Choose
IP
We work in V from now on.
We have: o
V
V
For a l l pre-dense D e N, q 11- "(:}reG)(r e D n N) M . So, i t must be the case that: For a l l pre-dense D £ M there i s a q1 < q such that q1 < r for some r e D n N.
Hence q i s
( ]P,N)-generic.
so we have a c o n t r a d i c t i o n .
This proves ( i i ) .
But q < p = p(N), (In fact we
have proved ( i i i ) , but we s t i l l need to know that ( i i ) •> ( i i i ) . ) (ii)+ ( i i i ) .
Let X be as in ( i i ) .
We show that
Let JJL be regular with IP e H
^ C = {N £ [H y ] ° I N -
contains a club s e t .
If y - X, there i s nothing to prove.
There are two remaining c a s e s . Suppose that ji < X,
Let
By lemma 1.8(i), C- contains a club in [H ] °.
Let
79 C 2 - {N € [H ]H° | N -< H }. Then C« is club in [H ] °. [H ] H °.
Hence C
n C 2 contains a club in
But clearly, C
n C 2 c c,
so we are done in this case. Now suppose that y > X.
Let
C x = {N € [H ]H° | N n H^ -< By 1.8(ii), C
contains a club in [H ] °.
H x >. Let C« be as above.
u
Then C
n C 2 contains a club in [H ]c°
and C. n C« 5 C, so again
we are done, (iii)-»(i) .
Let S c [X] * be stationary.
We show that
II- "S is stationary". o (JP) Let C € v be such that II- "C i s club in [X] ° " . P We prove that IH
Pick f € V
v
n°
1
C n S f
it
<j> .
so that 0
v < ttft Y
IHp"f : [X]
°
•*• A concentrates on C".
If we can find an X e S such that lH j "f"[X]
By assumption, the s e t
E = {NeCH ] H ° I X,f € N -< H } contains a c l u b . Let p € IP be given, and set
80 E Then E
= {N € E | p € N}.
contains a club, so by 1.8(i) the set E X = {N n X | N € E } P P
contains a club.
N n X e S.
Thus E n S + 4>. P
Choose N e E so that P
We finish the proof by showing that if X - N n X,
then for some q < p, q Ih " ffICX3
Pick q < p to be (JP,N)-generic.
Let a e [X]
° and set
11
D = { r e P | Ga<X)(r«~"f(a) = a )}
w
Then D i s d e n s e i n
IP.
Moreover, s i n c e
Thus D n N i s p r e - d e n s e b e l o w q i n
q' +"ho)
P.
I P , f , X , a e N -< H , D e N . Let q1 ^ q be such t h a t
= a".
IP
By extending q f i f necessary, we may assume that qf ^ r for some r € D n N.
Since r e D and qf < r we have r ihF "f(a) = a".
But r, E \ f , a « N •< #„.
Hence a e N.
I t follows that
q H-p "f(o) c X", so we are done, a We give a model-theoretic c h a r a c t e r i s a t i o n of properness. Suppose W £ H..
We say that a structure
A
is
IP-good iff N -< M & |N| * iSo implies N -< H, . j>
Lemma 2•6 The following are equivalent:
A
81
(i)
IP i s proper,
(ii)
If X i s a regular cardinal such that W e H , and if M = < H . , € , . . . > , then there is an expansion Mf of M A
which is IP-good. ( i i i ) There is a P-good structure. Proof: (i)->(ii).
By lemma 2.5 there is a club set C c [w ]' °
such that N € C + N -< H , .
IP X
By lemma 1.7 there is an expansion Mf of M such that N < M f & |NI = H o •> N e C. (ii)-Kiii). Trivial. (iii)->(i).
By lemma 1.7, the countable elementary submodels of
a IP-good structure form a club set.
So by lemma 2.5, IP is
proper, a
§3. PROPER ITERATIONS The following result is an immediate consequence of the definition of properness: Theorem 3.1 If IP is a proper poset and Q € V ||Q is a proper poset ||
is such that
= I , then IP 0 Q is a proper poset. a
We shall show that any countable support iteration of proper posets is proper.
We begin by considering the special case of
an iteration of length a).
The general case is basically just a
generalisation of this case, so this approach will not result in
82 any great duplication of effort. Theorem 3.2 Let <]P such that
| v $ (u> be a countable support iteration of posets
IP is proper and for each n < a) there is a Q
e V
OPn )
such that TP
I) and IP
i s a proper poset ||
= IP ® Q + - .
n
= 1
Then IP i s proper.
(By Theorem 3 . 1 ,
each IP , n < a), i s proper.) Proof:
We use Lemma 2.6 i n order to prove that
X a regular cardinal such that (POP ) e H. .
IP i s proper. Pick
We construct a IP -
A
(JL)
0)
good structure M =
.
We shall assume that for each v < OJ, IP consists of sequences q =
= IP
(where we
for convenience) and for each n < v, q(n+l) is an
dp ) element of V^n'
such that qfn+1 \\r
"q(n+l) e Q
", with the
ordering q' < v q i f f q f ( 0 ) < Q q(0) & (¥n
| n < ca> and
- | n < OJ^> as
Let ^* be a well-ordering of IL , and l e t A
M =
A
IP ,Q , < * , . . . > OJ
(of countable length) be any fixed
O
Q -good s t r u c t u r e .
For each n < w, l e t M
be an element of
V ^ 1 ^ such that
OP ) For each n < w, f
- is an element of V
n
such that
83
|f
- : [H.] < H o + H. i s an M -skolem function) \Wn = i n"»*l n^l A A
(in the sense of lemma 1 , 6 ) . Now, by lemma 2,4, i f G i s V-generic on IP , then H,V[G ] . A n
Hence, every element of H , V [ G n ] i s denoted A
by a boolean IP -name which l i e s in EL . n A n
of H.
[But notice that elements
may also have names which l i e outside of H
.
Indeed,
the "obvious" boolean name of many elements w i l l involve a refV erence to X, and no such name w i l l l i e in H. , of course.
It
w i l l l a t e r be observed that this i s the motivation behind our following
definitions.]
For each n < to, define a function
such
that whenever a = {x1 ,... ,x }
is a set of boolean
In
P -names
n
' ) such that ||x. e H.|| n = I, then I (a) is l A n a maximal pairwise incompatible subset of IP with the property (i.e. elements of V
n
that if p e I (a), then for some boolean IP -name x in H., n n A p Ihp "f n
. (a) = x".
And define a further function
by l e t t i n g h (p,a) = x for some IP -name x in H. such that A n n p N-_ "f (a) = x". Jrn n+1
The behaviour of both I
n
and h in any n
case not covered above i s irrelevant to our purposes. Let M be the structure M = <M , ( I ) ,(h ) >. x o' n n
We must show that N -<-p H. . 0)
So,
84
given some p e IP nN we must find a q £ IP , q < p, such that whenever D e N is dense in IP , then D n N is pre-dense below q in IP . Fix some enumeration
co
The idea is now to obtain q by defining q(n),n
induetively> at stage n ensuring that D nN will be pre-dense below q in IP . Since D is dense in IP , o OJ M t= (3pQ € But
DQ)
(p Q < p) .
IP ,D ,p e N -< M, so we can find a p
p < p. r o
e D
n N such that
We shall ensure that we choose q so that q < p
which will take care of the dense set D .
o
in IP , or
Dealing with the other
dense sets will be a little more tricky. Now, M
o
is Q -good and p (O) e N -< M , so there is a (£ ,N) o o o o
generic q(O) e Q
such that q(O) < p (O) .
Before we define q(l),
we need an auxiliary notion and a couple of technical results. For each n < w, if G
is V-generic on ]P , then in V[G ] we
n n can consider the s e t , N _ , of a l l elements of EL n+1 a boolean
.
n which have
A
3P -name inside N. n
for the set N
n
o
We let N
, be a boolean ]P -name n+1 n
The proofs of the following two results are
deferred until later: Claim If q € IPn is QPn ,N)-generic, then _ _ _ _ _1. n
Claim 2 m -_.-•-__«-that
I f q € IP n
QP ) i s OP , N ) - g e n e r i c , and i f qf € V ^ n / i s such n
85 iS
Vl then q ^ < q f >
i s (JP
i
,N)-generic.
"
A
Now, q(l) will be an element of V °
such that
q(0) »h^ l f q(l) € Q, " • So, in order to define q(l) it is permisso ible to proceed as follows. Pick any V-generic set G on IP containing q(0).
Inside V[G ] , construct "q(l)tt as required.
Then back in V apply the maximum principle to make the actual choice of q(l) as a name for the element constructed in V[G ] . (Of course, this could all be done without any mention of generic sets as such, but for clarity we shall proceed as indicated.) So pick G V-generic on IP with q(0) e G now. in V[G ] until further notice.
We work
Let
E 11 = {r elPo I (3s < po)[s e D 1. & sfl= r ] } . K
V
Clearly, E is dense below p 11 in IP . L
O
Moreover, E c H .
O
1
A
v Indeed, since we have IP ,p ,D e N -< H. , we have E e N. Let O
A. c E- , A. £ P .
O
J.
A
1
V, be the <*-least maximal antichain below p [l in
Thus A. e N.
For each r e A , l e t r* be the <*-least
r* € D- such that r* < p
and r*fl = r.
Notice that
r e N -* r* e N. Claim 3 . GQ n (A^ n N) + (j> Proof:
We have : A- e N i s pre-dense below p (0) in IP and
q(0) < p (0) and q(0) i s OP ,N)-generic. of q(0) i n IP
Hence every extension
i s compatible with some member of A-n N.
i s V-generic on IP
and contains q(0) .
But G
Hence G n (A. n N) + <{>.A
86 Let r e G here.
n (A. n N) .
Clearly, r is uniquely determined
Let p- = r* for this r. P
l " P o ' Pl e Dl
n N> P
(But notice that although p upon G .
We have
l ^€G o '
e V, the actual choice of p
Different sets G may give rise to different conditions
By claim 1, since q(0) e G , we have N -< M . p- e N, so p-(1) e N-.
(Q-,N-)-generic. 1 1 OP )
Returning to V now, we pick q(l) e V ° ent q(l) just defined.
(There i s no
uish these two "q(l)-s".) (q(O),q(l)) is
Moreover,
So, as ML is Q -good there is a
q(l) £ Qn, q(l) ^ p-(1), q(l) 1 1
2,
depends
a name of the elem-
sense in trying to disting-
Then (q(O),q(l))£
IP
and, by claim
0P 1 ,N)-generic.
To choose q(2) now, we l e t G be V-generic on IP- with Let G = {rfa I r e G } . qf2 £ G, and work inside V[G,]. 1 l o o Then G i s V-generic on P and q(0) e G . o o o chosen as above for t h i s G . o
Let p1 be the condition l
Set
E 2 = { r £ l P 1 | ( 3 s < P l ) [ s £ D 2 & s f 2 = r) } . E 2 i s dense below p- \2 in W. and E« e N.
Let A« 5 E 2 be the
< * - l e a s t maximal antichain below p- \2 i n P - •
Thus A« £ N.
For
each r € A 2 , l e t r* be the < * - l e a s t r* £ D2 such that r* < p and r*f2 = r .
So r £ N •* r* € N.
Claim 4 : G n (A2 n N) 4 4. Proof: We have p- fl e G • Hence s « p- (0) A q(0) £ G . l o 1 o over, s i n c e sA(qf2) ^ q t 2 , sA(qP2) i s OP^N)-generic.
More-
But A2£N
87
is pre-dense below p f2 in IP
and sA(qf2) < p \2, so every
extension of sA(qp2) in P . i s compatible with some element of A« n N.
Since sA(qT2) e G. , the claim follows immediately. A
Let r e G- n (A2 n N) and set p 2 = r* for t h i s unique r . Then P
2 ~ Pl'
P
2
€ D
2
n N
'
P
2^2
€ G
l-
By claim 1 now, since qp2 e G-, we have N« -< M«. P 2 € N, so P 2 (2) e N 2 ,
Moreover
Thus as M2 i s (Q2-good there is a
q(2)€
We continue in t h i s fashion to define
q(3) , q ( 4 ) , • • • , q ( n ) , • • • ( n < u ) •
In general, q(n+l) is chosen in
V[Gn] so that q(n+l) is (Q n + 1 » N n + 1 )-generic, and q(n+l) < P n + I (n+D, where
?n+1
< p n , p n + 1 e D n+1 n N, and Pn+1fn+1 e GR (the actual
choice of p - being dependent upon G ) . n+l n
[Notice that although
Claim 3 was a special case (since q(0) < p (0)), Claim 4 generalises to any n+l.
In order to show that G
n (A
2
n N) + <j>, where
n > 0, we u t i l i s e the facts that (p
- Fn+1)A(qTn+l) e G and, n+l n since (p n + 1 Tn+1)A(qpn+2) < qfn+2, ( p n + 1 fn+l)A(qrn+2) is generic] Having defined q =
compatible with some member of D n N. qf
For n = 0 we have
e D n N, so there is nothing to prove.
We deal
88 with a l l of the remaining cases in V[G], where G i s any V-generic subset of IP
containing q 1 .
This c l e a r l y causes no l o s s of
g e n e r a l i t y , as we are dealing e n t i r e l y with members of V. advantage i s t h a t , working in V[G]
The
from now on, i f we set
G = {rpn+1 | r e G}, then G i s a V-generic subset of IP
contain-
ing qfn+1, for each n<cu, and there i s a well-defined sequence p > p. > p o > . . • p > . . . ( n « d ) , as described above, and moreover o l z n since q(n)^
p (n) for a l l n<w, we have, for each n
for a l l m > n.
Let us look at the case n=l now.
i s pre-dense below p fl in ]P
Now, A e N
and q(0) i s (D? ,N)-generic and
q f (O)
But then p
= r*.
Hence, since
we have q f (m) ^ q(m) < p- (m) for a l l m > 0 , we conclude that sAqf < p .
But p- e D- n N, so we are done in t h i s c a s e .
other cases are handled in a similar fashion.
All
For instance,
looking at n = 2, we know that A^ e N i s pre-dense below p [2 in ]P1 and qf2 i s (tt^ ,N)-generic and ( s i n c e qf f2 < (p 1 T2),(q'P2) and (p- f2)A(q f f2)
€
G^) (p x f2) A(qf 1*2)
e G , so there i s an extension,
s , of (p p2)A(q f p2) in G such that s < r for some r e A« n N. Then r e G-, so r* = p«, and s A q1 < p e again done. course.)
D 2 n N, so we are
(The case n = 2 i s t y p i c a l of any case n > 2, of That proves that q i s OP ,N)-generic.
We are l e f t with the proof of claims 1 and 2. claim 1. that
So l e t q e IP be Ot ,N)-generic. n n
We begin with
We want to show
89 fl q H^ Nn+1 < Mn+1ft. n t o show that
It suffices
x
So, g i v e n
ft t-
Ill-il
-I ^
"ft
IP - n a m e s x . , . . . , x e N s u c h t h a t l l x . £ H. II n l m l A
n
= 1
for each i , we must show that q It, l f f - ( { x - , . . . ,x }) e N ^ , " . Jrn n+1 l m n+l Now, s i n c e x- , . • . ,x e N and N -< M, we have I ({x- , . • . ,x }) e N. 1 m n l m Moreover, I ( { x , , . . . , x }) i s a maximal pairwise incompatible subn l m So, i f we are given any qf ^ q, then by the (JP , N ) n
s e t of IP . n
g e n e r i c i t y of q t h e r e i s a q" < q 1 such that qlf ^ r for some r e I ({x1,...,xm>) . x = h (rnil
{x-,...,x
m
Since r e I ({x }) i s d e f i n e d .
,...,xffl}),
We have
So a s qM < r , x
But r , x . . , . . . x
€ N -< M,
m})
=x"-
Hence x e N, and we conclude that
n Since q" < q1 and qf < q was arbitrary, this completes the proof of claim 1. We turn to claim 2.
qi
e
yOPn)
b e
s u c h
t h a t
q 1h
Let q e IP be OP ,N)-generic, and let n n t»qt
£
^
i s
(Qn+1,Nn+1)-generic11.
We show that q ~ < q f > i s (JP + , N ) - g e n e r i c . Let D c N be a dense subset of IP _ . n+1 q1^
We show that i s
if
compatible with
some member of D n N. QP ) Let D« e V n be such that whenever G
. i s V-generic on IP ,
90 then in V[G ] , n Do = {r e Q - iQseG )l s ~
IP - ,G ,D e N •< IL , we h a v e D o e N. n+1 n A 2.
||D2 i s dense in Q n+1 1| To verify t h i s we argue in V[G ] .
n
°
IP , , G ,D i n t h e H. , s o n+i n A We show t h a t
= I
Let r € Q
be given.
Pick s e G so that s It. "r € Q ". Then s ~
"rff < r & r" e D 2 ". n.
Hence the s e t {s"
I s" H-jp n i s dense below s in IP . But s e G . n n r" < r, r" e D 2 , as required. e
P
n
j Hence there i s an
Now, s i n c e q- ^ q, we have is
But q- \h
f
" q
< q1 ".
Hence as D. 6 N,
n |h
q
»»Qr
€
n
2
ih
]P
n
"r
€ D
+
)(r -
q1f).ft
OP ) and an r e V n n N such that
So we can find a q 2 < q q
D2 n N
2
& r
~
q
lfM-
Let D. = { s € p
"r k D 2 n )or (-3s1 >s)t s f ^
I (s ih n
We claim that D- i s dense in P . i n i s an s f
< s such that s f
«hp
l!
For suppose s e W . n r k D 9 ! f , then s f
If there
e D- and we are
91
done.
Otherwise, s Ibp "r e D 2 ", so by d e f i n i t i o n of D 2 , n s H- "Gs'eG ) [ s f ~ < r > d ) ] l f . Hence there i s an s" < s and an s1
ur n £ IP
n s u c h t h a t s " Ih,, " s f e G & s f ~ < r > e D " .
11
n
" s f € G n " , we must h a v e s f l < s \
s " il-jp
Since
n
we must i n fact have s f ' s < r > eD.
And i f s " l ^
Hence s" e D - .
"sf~
Since s"<s,
we are again Now, D- done. i s definable from IP - , D o , r in H. and P ^ . D ^ U i n n , , l n+l z A n+l z A Hence D- e N. that q 9 ~ s .
So, as q« < q < q there i s an s e D- n N such l!
Now, q 9 It-
Z
Z
r £ D " so s VjL "r i D 9 l f .
iir
Z
Ur YI
So as
Z
n n f s € D- there must be some s > s such that s 1 ~
But s , r , D , P sf
. e N -< H .
Hence we c a n f i n d a n s ' e N s u c h t h a t
> s and s f ^ < r > e D . Now, s f e N and r e N , s o s
f
^ < r > e D n N .
q 2 ~ s , s o a s s f > s we h a v e q 2 ~ s 1 .
u < q 2 , u Ifu »hp
* n
"r ~ q f .
"v ^ r,q_ f ff . •>•
s'^
We know t h a t
Pick u ^ q 2 , s f . GPn )
So we can find a v e V
Since
such that
But q 9 < q , so we have u ^
Thus q 1 ^ < q , l >
z
1
1
1
is compatible with s f ~
is in D n N, and we are done. • We are now ready to prove the general iteration lemma. Since this i s basically just a generalisation of the above proof, we shall simply give an outline of the alterations required. Theorem 3.3 Let <1P
| v £ X> be a countable support iteration of
posets such that IP is proper and for each v < X there is a
92
IP +
Proof:
such that |Q V + 1 is a proper poset||
= IP 0 G)+ ..
Then F y
By induction on v.
v
=1
is proper for all v < X.
For v - 0 there is nothing to prove,
and successor steps follow directly from Theorem 3.1.
Suppose
now that v is a limit ordinal and the result holds for all x < v. 8
(IP )
GP }
For a<3
=P
® Q . . W e
By the induction hypothesis, fix a system of structures ^M
||Q
is proper || *
| a < 3 < v^> such
Q
as in Theorem 3.2 so that M
ft
is ]P -good for all $
Q -good ||
a
= 1 for a l l a<$
Ot
Define^f^ | a<3
and then define a structure M =
V
again much as before in order to incorporate the necessary (forcing) information about the functions f .
In order to ensure
that the language of M is countable, however, we must include the indices a,3 as variables
rather than regarding them as meta-
mathematical indices. So, instead of including functions 8 8 I (a), h (p,a), a<3
Let y
To show that M is IP -good, we commence
Let N -< M be countable. =
We want to prove that
sup(Nnv), and l e t
ing sequence from Nny
which is cofinal in y.
The idea is to
proceed along the sequence <1P , , | n
93 $J/
\
) more or l e s s a s i n Theorem 3 . 2 .
S i n c e each o f t h e
indices y(n), n<w, lies in N, our present approach of treating these indices as variables in the functions of M does not cause any problems.
Hence we can finish just as before. •
§4. COLORING LADDER SYSTEMS Let S c Q. n =
A ladder system on S is a sequence
| 6 e S > such that r) is a strictly increasing ursequence
a
o
with limit 6, for each c = < c
x
| 6 e S >
coloring r u ( i ) . )
6 eS.
A coloring of n i s a sequence
such that c r e
W
2.
(We think of c . ( i )
as
A unif ormisat ion of c is a function f: w- -> 2
such that for each 6 e S there is an n e u such that m > n -> cp(m) o = f(n6(m)). If MA + -iCH is assumed, every coloring of every ladder system is uniformisable.
If 0 ( s ) holds, any given ladder system on S
w i l l have a non-uniformisable coloring.
If 2
< 2
and S
contains a club s e t , any ladder system of S has a non-uniformisable coloring (See [DeSh] for d e t a i l s . ) Shelah has shown (see [Ek])that if there is a stationary set S c Q and a ladder system q on S such that every coloring of n is uniformisable, then there i s a non-free Whitehead group of order H1 .
This makes the consistency of the aforementioned hypothesis
with GCH of great interest.
In this section we prove the
following result: Theorem 4.1 Assume GCH.
Let S c Q be stationary and costationary.
Let
94 n " ^^x | 6 £ S ) be a ladder system on S.
There is a poset IP
such that:
(i)
M - B2;
(ii)
P satisfies the ^ - c . c ;
(iii)
P is a-dense;
(iv)
||GCH|1P = 1;
(v)
||S is stationary || = 1
(vi)
||Every coloring of n is uniformisable ||
=1. O
P will be the limit of a countable support iteration, <1P |v
Since we shall have | IP | - W, for all v
(ii) will be immediate, and (iv) will then follow by virtue of (iii).
Each IP will be proper, so (v) will be immediate.
Hence all we need check is that (iii) and (vi) hold, and that each IP is proper. We begin by describing the basic forcing poset.
Let
c -
Clearly,
||The coloring c of n is uniformisable||
= 1.
Indeed, if G is a V-generic subset of IP(c), then in V[G], UG is a uniformising function for c on n»
Lemma 4»2 P(c) is proper.
Moreover, assuming GCH,
95 We u t i l i z e Lemma 2 . 5 .
Proof:
Let X be r e g u l a r w i t h IP = P ( c > € H . .
Let C be t h e s e t of a l l c o u n t a b l e N -< H. s u c h t h a t IP c N and A
N - \^J N , where n<0) P € N Q < Nx -< N2-< . . . < N and o)n n N In
e N for each n
We show t h a t N -< H.
C l e a r l y , C i s club i n [ H . ] °. A
f o r a l l N € C.
S o , g i v e n N e C and
p € IP n N we must produce a q e ] P , q < p , such t h a t q i s QP,N)In f a c t we do more:
generic.
we produce a q € P, q < p , such
that t h e s e t G = {r e W n N | q < r }
i s N - g e n e r i c on W n N ( i . e .
i f D £ N i s d e n s e i n IP, t h e n Gn(DnN) +
a
n
Let a • u. n N.
< a. < . . . < a , where 0
1
= w- n N • 1 n
Case 1.
a i S.
Let G be any N-generic subset of IP n N containing p.
Set q = uG.
By N - g e n e r i c i t y
(and remember that N - < H . ) , q e
Since
a i S, q € IP.
So we are done, q being as required.
Case 2
2.
a e S,
Let < D
n
I n < ^enumerate all dense subsets of IP lying in N.
By discarding various of the N fs if necessary, we may assume n (after reindexing) that D € N for each n<w, and that p € N . n n o Now, there are only f i n i t e l y many i
< i < ...< i • O
1
to a condition p c
l j
Working i n N •< H , we can extend p
K.
O
A
s u c h t h a t dom(p ) = n ( * - i ) + l
for j = 0 , . . , , k .
find a c o n d i t i o n p
Then a s D
e l P n N , p
e N
< P , p
and
P (n
^ ) -
i s d e n s e i n IP we c a n £ D .
Similarly,
96 working in N
i^ a
< a
^ ^a^
now, we can extend p
and i'
^(T^U)) and
= c a (i)
to a condition p. such that for a11 i such that
then extend p. to a condition p
e D n N .
Continuing thus w e define p / n<w, so that p e D n N , p ^ pn ^ n n n n o 1 p o > . . . , a c dom(p ^_ ) , and p .. (n ( i ) ) = c (i) for a l l i such z n — n+i n+i ot ot that a
n
< n (i) < a _. a n+l
Let G = {q e PnN | Gn) (p
G i s N-generic on PnN and q = UG € all i £ i . o
a
2.
n
< q)}.
Put q(n (i)) = c (i)
Then for
Hence q e IP, and we are done. Q
An immediate consequence of the above proof is the following result: Lemma 4»3 P(c) is a-dense.
Q
The iteration sequence
| v < u ) 2 ^ is a countable support with the colorings c chosen in the
usual fashion so as to ensure (vi) of Theorem 4.1.
Sof in
particular, for each v < 0)2 we shall have an element c such that
e V
v
||cV is a coloring of nll*^ = 1 and P v + 1 = !>v® ]P (c V ).
It will then be an immediate consequence of Lemma 4.2 and Theorem 3.3 that each P
is proper, v < u>2.
But wait a minute, will it
not be the case that because of all of the boolean names which occur in a forcing product P very quickly) ||p
| > #-?
® P (c ) that eventually (or indeed Well, it all depends upon the precise
definition of the product poset.
According to the definition
adopted in §0, already P- could be too large. not run into this problem, since for P
However, we shall
we shall not take all
97 of P
® p (c ) , but rather a certain dense subset of this product.
This ties in with our proof of part (iii) of Theorem 4.1, the only part left to prove now.
It suffices to show that P
is
a-dense for each v < u>2. Let < P |v < ui^y be the iteration sequence defined in the standard fashion, with P
. £ ]P ® P (c ) for each v < co2.
Let
3P , v <
and
The ordering of IP
is
p < q iff dom(p) = dam(g) & (V£ e dom(g)) (p(O = q ( O ) . Clearly, P v < u)2. in P
v
c ]p
for all v <
Moreover, | P
| = H. for all
We shall finish our proof by showing that P
for all v < CDO and that each 2
P
v
is dense
is a-dense.
Lemma 4.4 (A)
P
i s dense i n
p* < p f such t h a t (B)
P
Proof:
is
P
, a n d i n d e e d i f p e 3P
f o r s o m e Seti
- S,
there i s a p * e P
(VC€dom(p*))(dom(p*(C))
,
= 6).
a-dense.
By a simultaneous induction on v.
Once we have (A) for
v, we can work with conditions of the type p* in order to repeat the proof of lemma 4.2, Case 1, in order to show that P a-dense.
is
So what we need to prove is that if (A) and (B) hold
below v, then (A) holds at v.
Well, it is easily seen that if
(A) and (B) hold below v, then whenever p e P
and £ € dom(p),
there is a q € IP , q ^ P|C# and a function q(C) / such that
Let N =
P , |h >, where X i s large and regular, and v Fv
l e t
Let e c u
1
be club in w, with N. n 10, = 6 for a l l 1 6 1
6 e C, and l e t 6 be a l i m i t point of C which i s not in S. (Since S i s c o s t a t i o n a r y in w , t h i s i s in order.)
Let <6 |n<eo>
be a s t r i c t l y increasing sequence of members of C c o f i n a l in 6. Now, N~ -< H^ for each n < a), so we may make repeated use of the n extension property mentioned above in order to pick conditions p e Nx n IP , such t h a t p £ p > p _ > . . . f and ordinals n c v o l n _ 3 e dom(p ) , so that p Tft e E>o / P T$ ^ "p(3 ) = x" for some n n n n p n n n n function x, 6 c dom (p _ (3 ))/ and such that each element of n n+1 n VJ dom(p ) will be 6 for infinitely many m < to. n
It is easily
seen that p* = ^ p is as required now. O ^ n<0) *n §5.
THE PROPER FORCING AXIOM Let PFA (Proper Forcing Axiom) be the following statement:
if P is a proper poset and 3f is a collection of W, many dense subsets of
IP, then
3P has an 3-generic subset.
Since every c.c.c. poset is proper, PFA is a generalisation of MA
•
As we shall see, it is much stronger than MA
.
We commence by establishing the consistency of PFA with ZFC + 2 ° » W
.
The obvious idea is to commence with a
model of ZFC + GCH, and iterate proper poset forcing in a manner
99 analogous to the usual consistency proof for MA 3.1 and 3.3, the iteration will be proper.
.
By Theorems
So, in particular,
stationary subsets of to will remain stationary, which means that K
will be preserved.
(Since 8- is the only parameter involved
in PFA, it will not matter if other cardinals are collapsed in the iteration, so long as we end up with a model of PFA.)
The
problem with PFA, however, is how long the iteration needs to be. For MA
, if we start out with GCH, then an iteration of length
w 0 suffices.
This is because, although MA
refers to all c.c.c.
posets (of which there is a proper class, of course), by a simple LBwenheim-Skolem argument it can be shown that the full MA consequence of MA K-.
is a
restricted to posets of cardinality at most
Hence in our iteration, we need only deal with these posets,
of which there are only 2*° = B 2 > even allowing for the new posets which arise during the course of the iteration. have no such reduction.
But with PFA we
Certainly, if we only require PFA for
proper posets of cardinality at most K. > then we can achieve this by a straightforward iteration entirely parallel to that for MA
, only using a countable support iteration, of course.
[There is a little difficulty encountered in proving that the iteration satisfies the Ko-c.c.
Even if we start by assuming
GCH, which we do, this is not quite immediate, for if < IP
| v ^ a)2> is the countable support iteration obtained by
iterating all proper posets of cardinality Ji., then it is not the case that \ W
| s K, for all v < o^* because of all the
100
boolean names which we allow in product posets.
However, i t is
easily seen that if
IP is a proper poset, any countable set of ordinals (say) in V( IP) is contained in a countable set of ordinals of V, and using t h i s fact i t is easy to prove, by induction, that for every v < 0)2 »IP has a dense subset 1R of c a r d i n a l i t y at most S, •
Now the proof that IP
1
has the
^2
H 2 " c c . is straightforward.]
But for a full PFA we require some
device to restrict our attention from the proper class of all proper posets to some representative set of such posets.
This
can be done, but at the cost of a large cardinal assumption (and a significant one at that).
Moreover, it can be shown that some
such assumption is necessary, since PFA implies the existence of inner models of set theory with various large cardinals (measurable, and beyond ) • Recall that a cardinal K is supercompact iff for every X>K there is an ultrafilter U on [X]
(i)
the ultrapower V[A]
such that:
/U is well-founded; [X]
and, if M is the transitive collapse of V (ii) [ M ] X c M;
/U, then
( i i i ) if j : V -< M is the canonical embedding, then j \v
= idjv
and j (K) > X.
(We refer to such a U as a super-compact u l t r a f i l t e r on [X]
,
and the embedding j as a (K,X)-embedding. ) The following r e s u l t of R. Laver is the key device we need in order to get a model of PFA:
101
Theorem 5.1 Let K be a supercompact cardinal.
Then there is a function
f : K •* V such that for any set x, if A > |TC(x)|, then there is a supercompact u l t r a f i l t e r on [X]
such that if j is the
associated (K,X)-embedding, then [j(f)3(ic) = x. Proof:
Suppose not.
Then for each f : K -> V there is a least
ordinal Xf such that there is an x with |TC(x)|<Xf and <x,X f > Let v exceed a l l the X^fs and pick a
a counterexample for f.
supercompact u l t r a f i l t e r U on Cv] Let $(g,S) be the statement that for some cardinal a, g:a-*V , and 6 is the l e a s t cardinal for which there is an x with |TC(x)| < 6, such that for no supercompact u l t r a f i l t e r Ug on [6]
v
is the collapse of V
Let U
K
rV i < K /U . v
be the projection of U
v
onto K.
Then U
K
is a measure
ultrafilter on K and there is a canonical embedding h
. : V /U K, A
< V K
/U .
Consequently, we see that there is a
V
set A e UK such that for each a e A and each ff a Xff < K such that
(Because the same is true of K.)
Define f : K ->• V by recursion, as follows. and f
= f fa is defined.
: a -*- Va there is
Suppose a < K
Let f(a) = <>| unless a e A and f :a->V .
In t h i s case there is an x e V witnessing $ (f ,X^ )• a be such an x, and define f (a) = x .
Let x
102
Notice that, if j
is the (K,V)-embedding associated with
U , then: v
C j v « f o | a € A » ] ( K ) - f; [ J v ( < X f | a € A » ] ( K ) = Af; a [jv«xa | a e A»](K) = [jv where x i s a witness to $(f,A-) in M , and hence in V. Let IL
be the projection of U onto A,..
A *-
V
The following
I
diagram commutes and is elementary (where h.
V
embedding of M. into M ) : V Af
We have h,
- idTx., |TC(x)| < A.;
A-V
I
I
so x 6 M.
is the canonical
and h,
A-
(x) • x.
A^v
Thus: [j,
(f)3(K) = C(h. J^iajjf))^))
= (h. ^(x)
- x.
But this contradicts the fact that x witnesses $(f,A f ), so we are done, til We are now ready to prove our consistency result for PFA. Theorem 5.2 Con(ZFC+ "there i s a supercompact cardinal") •*- Con(ZFC+ Proof:
2 H °= H2
+ PFA).
Let K be a supercompact cardinal, and let f: K •> V be
as in Theorem 5.1.
We call a pair ( IP^S) suitable i f
IP is a
103
proper poset and 5 subsets of
is a sequence of length less than K of dense
P.
We shall construct a countable support iteration sequence
|a ^ K ^ of proper posets as follows.
Let
P
poset for adding a Cohen generic subset of w. defined and f(ct) e V v
aJ
P
+-
= P
f(a) is not of the above form, let o Ih "2 °
a
H
If IP
is
is such that tt- " f(a) = 0P,3 ) is a
suitable pair", then let
show that
be the usual
=K
® P. +.
P
If P
= P
® P
is defined and .
We shall
& PFAf!.
A routine A-system argument shows that Hence K remains a cardinal after o | P | = K, we have W-p "2 ^ K
P
P -forcing. .
has the K - C . C . Moreover, since
Since P
is proper, CD-
K
remains a cardinal under P -forcing. 8 2 °
So, as PFA -*- MA
->•
> oj , it remains only to show that W- "K
suffices to show that if G is V-generic on P
It
and ( P,S) is a
suitable pair in V[G], then there is a filter F c P i n V[G] such that F n D + <J> for every D e
For, this clearly means that
And moreover, on the assumption that K > o^
in V[G], it implies that there is a function from o>- onto OJ2 in V[G], which is absurd,
(Consider the proper poset of all count-
able maps from a)- to co^, together with the appropriate collection of dense sets.) So let G be a V-generic subset of P , and let OP,*) e V[G] be (in V[G]) a P
suitable pair.
is some cardinal X.
We may assume that the domain of
Let P,§- be P -names for P,£, and
104
suppose that IK "OP^) is a suitable pair".
(As usual this
K
causes no loss of generality.)
By the choice of K,f, we can find
a (K,A)-embedding j : V -< M such that [j(f)](ic) = (P,&) . Now, by construction, P 5j OP ) •
IP 5 V .
Hence jflP
= idflP and
Moreover, by elementarity, j OP ) i s constructed from
j ( f ) in M exactly as IP i s constructed from f in V. Ej(f)](a)=f(a)
for a l l a
and if we s e t j OP ) jOPK) s
But
J (
IP K 0 J O P K ) ^
^ = {pp(j
(K)
| p e jOP^)}, then
(K)-K)
.
Since M c V, G i s M-generic on P . M[G]-generic on j OP )
So if we choose H to be
, then G 0 H will be tt-generic on j OP ) .
Claim : The embedding j : V -< M extends to an embedding j * : V[G] < MEG 0 H]. Proof of claim :
Let x e VEG].
a Q , . . . , a n e V, x = T TMEG
Then for some term T and some
(aQ,...,an,G).
0 H ] ^ ( a )>###>j(a )
9 G
0
H)#
That
Let j * ( x ) = j * i s well-defined will
follow from our proof that j * i s elementary. Suppose VEG]* $(T V ^ G ] (? o ,G),...,T n V [ G -l(x n ,G)), where x\ c V. %
o
$- o
Then t h e r e i s a p e G s u c h t h a t p lhv $ ( T (X , G ) , . . . , T ( X , G ) ) . p o o n n
So applying j , j (p) * ^ ( s > )
$
( T o ( J ^ o ) , G) , . . . ,T n (j (x n ) ,G)) .
p € IP , so j ( p ) = p e G c G 0 H.
But
Hence
MCG 0 H] * $(T o M C G 0 H ] (j(x" o ), G 0 H ) , . . . , T n M [ G 0 H ] ( j ( x n ) , G0H)), i . e .
105
MEG 9 H > » ( J * ( T
VCG:l
oo
(x
,G)),...J*(T
VCG3
n
(X
n
,G))).
S i n c e we c a n
carry out exactly the same argument with -i$ in place of $, the claim is proved. A Now, j fx c M and | j f x | = X. jfA € M. (1P,3),
Thus j [X e
M[G 0 H].
Hence as [M]
c M, we have
Consider the action of j [X on
Clearly, j \\ embeds W in j*QP) (in an order preserving
fashion), and for each v
into the set j*(3) .
(This i s an immediate consequence of j * : V[G] -< M[G ® H] and j*fx » j[*X.)
Thus, using an obvious terminology, which we shall
assume implies the s u i t a b i l i t y of both p a i r s , M[G 0 H]fc"jfx embeds (f9$)
in (j*(P), j*(5)) f ! -
Now l e t K be any M-generic subset of j (JP ) . R = G ® H^, where G is M-generic on P on jflP )?^K\
Since j Tv
= id tv
G is in fact V-generic on P .
Then
and IL, is M[G ]-generic
and [M]A C M, i t follows that Hence the above argument i s
s t i l l valid, and so M[K]*MJrx embeds (p f 5) in (j*flP) , j * ( ? ) ) " . Thus as K is arbitrary as above, ) But 0P,|) = Cj(f)](K).
Hence
(i)
M ^ Cj(f)](K) is a pair of members of v
(ii)
»-?(F)
If
[j(f)](K)
is embeddable in ( j O P ) , j ( « ) n .
( )
Hence: (iii) and
M * (^a<j(K))[[j(f)3(a) is a pair of members of V I K ^ vlf[j(f)](o) is embeddable in (j (JP) ,j (£))"
106
Applying j : V -< M gives: dp ) (iv)
V *= (3a
" f(a) is embeddable in 0P,£)" ] . K
(where we have used the fact that (IP ) K
Pick some a
- IP ) . a
a
Now, since we have adopted the
convention that embeddability implies the two pairs are suitable, by (iv) we have ^hp"f(a) i s a suitable pair". But f(a) is a pair of members of V
a
, so clearly,
*- "f(a) is a suitable pair11. Hence by construction, G
- G flP
is V[G ]-generic on (f(a)) .
In V[G] now, let ir : (f (a)) -* 0P,5) be an embedding. exists by virtue of ( i v ) , )
Then, clearly, TT[G
f i l t e r on IP, and we are done. Ga
] is an Sf-generic
[Notice in particular that since
meets every dense subset of (f(a))
in V[G ] , i t certainly
meets every member of ( f ( a ) ) - , and hence IT [G member of J . ]
(Such a TT
] meets every
D
We now give a few sample applications of PFA.
We commence
with two applications in tree theory, both due to Baumgartner. Let T be a tree of height CJ- and cardinality H, .
We say T
is special if there is a function f : T •* u> such that whenever s , t , u e T, if f(s) • f(t) • f(u) and s < t , u , then t < u.
In
case T has no o> -branches, this is equivalent to saying that there i s an h : T •+ u> such that h(s) = h(t) implies that s and t
107 are incomparable in T.
[One implication is totally trivial.
For the other, given f as above, each set f
Cn] necessarily
consists of totally incomparable chains, each of which will be countable, so we can partition f
[n] into a countable family of
antichains, and thereby obtain h.] Lemma 5.3 Let T be a tree of height oo- and cardinality Si .
If T is
special, then T has at most ft. OJ- -branches. Proof:
Let f : T -> to specialise T.
If B is an co^-branch of T,
we can find an n e to such that {x e B I f(x) = n} is uncountable. Choose any t e B such that f(t) = n.
Then, clearly,
B = {x € T|(x < t) or (x > t and (3y>x)(f(y) = n))}. Hence B is determined by t and n.
The lemma follows. D
Lemma 5.4 Let T be a tree of height to- and cardinality 8-.
If T has
at most V>- to.,-branches, then there is a c.c.c. poset Q such that IK "T is special". Proof:
By enlarging T if necessary, we may assume that T has
exactly H- to -branches, and that every element of T belongs to an CD -branch.
Let < B
|a
For each a
not in ^
and set S = {s(a) | a < u>-}.
for all a < OJ .
Thus |B
n S| < H
Bg,
In particular, there is no OJ -branch through S, where we regard S as a subtree of T. Suppose that f : S ->• OJ is such that if f(s) = f(t), then s
108
and t are incomparable in S.
Then we can define g : T •* u) by
the following.
Let t e T.
Thus t > s ( a ) .
Let g(t) - f ( s ( a ) ) .
g specialises T.
Let a be least such that t € B . I t is easily checked that
Hence i t suffices to find a c . c . c . poset Q
such that IK" f : S -*
Let p ^ q iff
Providing i t s a t i s f i e s the c . c . c , Q will clearly be as So assume not, and l e t J = {p |a
a b l e , pairwise incompatible s e t .
W.l.o.g. we may assume that
|p | = n for a l l a<0L , where n i s minimal such.
We may assume
further that if ot^B, then dom(p ) n dom(p ) =
< t , then in fact s < t . (Notice
that since p and p o are incompatible, there always are such s , t . ) a
p
Let U be a uniform u l t r a f i l t e r on u>- . dom(p ) = {s , . . . , s ot
-}.
n*"l
o
For each a e ai., l e t
For each a there will be i ( a ) , j ( a ) < n
such that { g e
u
l I si(a)<Sj(a)}
eU
'
Furthermore, there must be i , j < n such that A = {a | i ( a ) = i & j(a) = j} e U. ct 6 Let a- >ot« € A. Then for some 3 > a-,a 2 * s. < t . for a = a-,a«» a 3 a2 1 Thus s . and s. are comparable. Thus {s? | a e A} determines an col-branch through S, a contradiction.
The proof is complete.o
109
Lemma 5.5 Assume 2
> IL.
c a r d i n a l i t y H-.
Let T be a t r e e of height CD- and
Let F be a countably closed poset.
Then
frjp "All w -branches of T l i e in V". Proof:
Suppose, on the contrary, that v
v
W~ "B i s an ok-branch of T and B 1 V", Since tt^"B i V", i t is easy to construct, inductively, elements p f of IP, x f of T, for f e 2W, such t h a t : (i)
f c g •* p g < P f &
(ii)
Pf
(iii)
xf.
^
Xf
<
Xg;
" X f e B";
xf#xl>.
Since IP is countably closed, for each f c 2 we can find a p f e IP such that p f < p f K for a l l n < a). p f Ihp "{ x f f |n
Then
so for some p f f ^ p f and some y f e T, we
have p ! IK "y yf
e B & (¥n
Theorem 5.6 Assume PFA.
If T is a t r e e of height u)- and cardinality
f?l, then T i s s p e c i a l .
Hence every tree of height co- and
c a r d i n a l i t y K-j has at most H-, w-. -branches. Proof.
Suppose T has X u)--branches, where X > H. .
Let IP be
the poset of a l l countable maps from u)- to X, ordered by reverse inclusion.
Then IP is proper and \h "|X| ~ H, = h^11*
So by
lemma 5.5 (which is valid because PFA •* MA -> 2 ° > H2)> ^p " T has K-, OJ--branches".
In V w , choose Q a c . c . c . poset so that
110
*- "T is special", as in lemma 5,4. ||
=1.
Now, ||Q is a proper poset
Hence 1 = P 0 Q is a proper poset (in V).
for some I e V
Moreover,
,
tt- "f : T -> to specialises T". For each t e T, let D = {r e H I r \\- "f (t) - n",for some n e w } . Clearly, D is dense in 1R.
Let
$ = {Dt | t € T}. By PFA, let G be an J-generic subset of ]R.
Then we can define
a function F : T -»• to by F(t) = n
(§r€G)(r ^ f I f ( t ) - n ") .
iff
Clearly F specialises T, a Theorem 5.7 Assume PFA. Proof : tree.
Then there are no u^-Aronszajn trees.
Suppose, on the contrary, that T were an ou'Aronszajn Let IP be the poset of all countable maps from to- into
a>2> ordered by reverse inclusion.
Thus ^"l^ol '
Since the levels of T have cardinality at most H-i > the same proof as in lemma 5.5 (more or less) shows that lh "T has no branches IP
v
of type o) ".
QP)
In V
, let h be a s t r i c t l y increasing, cofinal
map of a) into o^* and let S - ^
T,, v.
Then, in V
tree of heighto)- , cardinality S, , with no (^..-branches. be the poset defined in lemma 5.4.
, S is a Let Q
Thus there is an f e (V
) ^
such that, with boolean value 1 , f maps S into OJ in such a way that if f(s) = f ( t ) then s and t are incomparable in S (hence T).
111 Let
1R = P 0 Q.
OP) c.c.c. in V ^ ,
Since IP is countably closed and ]R is proper.
Q has the
We shall apply PFA to 1R.
For each y < OJ^ (working in V from now on) , let I
- w1 ° 5 t 0 T .
For a, i < u^,
let
ff
h(a) = y &
Set
Clearly, 3 is a set of dense subsets of 1R. on 1R,
Let G be ^-generic
Define a function H : u). •* a)2 by
H(a) - Y
iff (§P€G)(p ^ n h ( a ) = y " ) .
Clearly, H is strictly increasing (though not, of course, cofinal!), If we now set U • ^
F(u) - n
iff
T«, x , we can define F : U -> a) by
(3P€G)(p ^
"£(u) - n " ) .
Clearly, i f F(u) = F ( u f ) , the u and u f are incomparable in T. Now choose x e T , where 5 let b
€ Tu/ 01
N
s
sup(ran H).
be such that b
rl vOty
< x. Ot
Then for each a < w-, For some a < 3 < to-, we
1
must have F(b ) • F(b R ), of course.
1
This contradiction proves
the result. Q Our third application is indirect, in that it follows from the previous two.
If two is due to Baumgartner, though the
present proof involves ideas of Todorcevic. Theorem 5,8 a Assume PFA + 2 Then 0(E) holds.
1
- H2«
Let E • {cKo^ | cf (ot) • OJ,}.
112
Proof
:
W c a2 ot —
We must f i n d a s e q u e n c e <W
| a e E> s u c h t h a t 2
jW | < B-, , and f o r any f e a 1
2 t h e r e i s an a e E s u c h
t h a t f pa e W . For e a c h a < u u , l e t < f ^ | £ < a^ > e n u m e r a t e 2
f €
2*
For e a c h
2 , we c a n d e f i n e F f : to -* o)2 by F f ( a ) - min{£Jf [OL - f ® } .
S i m i l a r l y , i f a < au and f e a2 we d e f i n e F f : a + 1 •*• w 2 by
Ff(g)
= min{?|fr3 -
fj}.
For (j>,^ € ^On, say <^ < if; iff
| (a|<J)(a)>^(a)} | < H- .
-< is a well-founded p a r t i a l ordering of
2
Clearly,
0n.
We define W , a e E, by recursion on a.
So l e t a € E be
such that Wo is defined for a l l B e E n a* P
Let 3 denote the set a l l trees T with the following properties: W
(i)
T is an i n i t i a l segment of
2 of cardinality K-. 5
(ii)
every element of T is contained in an ot-branch of T;
(iii)
if f € T then (Vy
(iv)
if y < a and f^ e T, then if £ < £ i s such that t1\6 \ W.), then f^ e T, If 3 + (J) we define T = f)J . Clearly, T is a tree Ot
Ot
Ot
satisfying (i), (iii) and (iv). (ii), whence T e J .
that b° | T . g
= Ub € a 2 .
We show that T also satisfies
Let f e T .
a-branch of T° containing f. b° c T for all T e 3
01
we are
Let T° e'Jand let b° be an
Set g° - Ub° e a 2 .
done.
Now, if
Otherwise choose T e J so
Let b 1 be an a-branch of T containing f. Set Using properties (iii), (iv), and (i) we conclude
113
that for some y
< a, y < 6 < a •+ F (6) < F (6). [Otherwise 8 g 1 1 1 o there are a r b i t r a r i l y large 6 < a so that F (6) = min{£|g ^6 =
^
| g i r 6 - fp X
*9
\>
F
< gO"|
6
so b
)>
y C 111 )
and
< iv )> g o T
6
€ T1
for a r b i t r a r i l y large 6, so by ( i ) , g ffi e T1 for a l l 6 < a, giving b° c T , a contradiction.] If b c T we are done. 2 1 2 2 Otherwise choose T e *f so that b ? T and l e t b be an a-branch a
of T
containing f.
-t
Set g
= Ub
e
2.
Then for some y« < a,
y 9 < 6 < a -> F
(6) < F (6). And so on. If this process ever 8 2 1 terminates we are done. Otherwise, if we set y = sup < y , then Z
8
y < a, since cf(a) = w .
But then we have F
1
8
F 8 (y) >..., a contradiction. 2 If 3
= $ we set W W
a
Thus T a
(y) > F 8
o
(y) > 1
e3 • a
= {f°! | £ < a}.
Otherwise we set
= {Ub I b is an a-branch of T }. ' a
By Theorem 5.6, in either case we have |w | < K,. the definition of < W
That completes
| a e E>.
We show that <^W otherwise, and let f
| a 6 E ) i s a ^(E)-sequence.
Suppose
e 22 be such that (VaeE)(f pa 4 W ) , but if
03
g €
2
2 is such that F -< F
Let S = {f["a | a < a)2 & f €
, then (§a£E)(gpa e W^). W2
2
& (VacEMfpa 4 W^)}, an
i n i t i a l part of ^ 2 such that every element is contained in an o) 2 -branch.
(By our i n i t i a l assumption,S \ $.)
Let N -< H be of cardinality H-. such that a = Nnoo2 e E and f o , S , <Wa I a e E>,«f[?£ I ? < aj2o> I 3 < a>20> e N. Set T = SnN. Thus T i s an a-subtree of a 2 of cardinality Iv. • It is easily seen that T e U . a
Thus T c T c N. a -
114
Let b be an ot-branch of T , and set g
= Ub e
the choice of f , the a-branch determined by f of T .
So for some y < a, y £ 6 < a -> F
a
g
as b e f o r e . ]
Let g1 = g TY»
a
2.
Now, by
i s not a subset
(6) < F o
(6).
[Argue
o
Then g- € T c T c N.
Let U = {h € ^ 2 | [ h c g x ] or [gx c h & (Vg€E)(hfB { Wp) & & (y ^ 6 < a)2 •* F h (6) < F f (6))]} .
<**
°
U is an initial segment of *^2.
Moreover, U e N, since all
parameters involved in the above definition are in N. Y ^ 6 < a + g f6 e U, N * "U has height u) 9 ". U really does have height djn. h e U n C < F
2. (6).
Since
Hence as N « H
,
Let y < 6 < o)2> and let
Let ^ - min{ c [ h • f }.
By definition of U,
Thus |U n 6 2 | < V> for all 6 < o>2.
By Theorem
Z 5,7, U has an o)2"-branch, say d. Let g = Ud € 2. Then F (6) < F (6) for all 6 > y, so F X F . But d c U, so g 8 o o (VB€E)(grB i W D ) , so this contradicts the choice of f . The P o proof is complete. O
For many more applications of PFA we refer the reader to [Ba].
§6.
HISTORICAL REMARKS As we said in the introduction, the theory of proper forcing
was worked out by Saharon Shelah.
As far as I am aware, the
first appearance in print of the general notion occurred in [Slfl. Much of the more advanced theory of proper forcing - which I have
115 not mentioned in my present account—was developed in a series of "open letters" from Shelah to E. Wimmers [S12], photocopies of which have been distributed to "the cognoscenti".
Some of
this material is promised to appear in [W],
References [Ba]
J.E, Baumgartner. ation.
Some Applications of PFA.
[DeJo]
K.J. Devlin and H. Johnsbraten. The Souslin Problem. Springer Lecture Notes in Mathematics 405 (1974).
[DeSh]
K.J. Devlin and S. Shelah. Follows from 2 ° < 2 V 29 (1978), pp. 239-247.
In prepar-
A Weak Version of 0 Which
Israel Journal of Mathematics
[Ek]
P.C. Eklof. Set Theoretical Methods in Homological Algebra and Abelian Groups. University of Montreal Press (1980).
[Je]
T.J. Jech.
Set Theory.
[S11]
S. Shelah. 573.
Independence Results.
[S12]
S. Shelah.
Letters to E. Wimmers.
[Sh]
J.R. Shoenfield. Unramified Forcing. in Pure Math. XIII (1971), pp.357-381.
[SoTe]
R.M. Solovay and S. Tennenbaum. Iterated Cohen Extensions and Souslin's Problem. Annals of Math. 94 (1971) pp.201245.
[W]
E. Wimmers. The Shelah p-Point Independence Theorem. Israel Journal of Mathematics. (To appear).
Academic Press (1978). JSL 45 (1980), pp.563-
A.M.S. Proc. Sym.
The Singular Cardinals Problem Independence Results
Saharon Shelah Institute of Mathematics The Hebrew University, Jerusalem, Israel
Abstract:
Assuming the consistency of a supercompact cardinal,
we prove the consistency of 1) $*
strong Iimit52
™ a) strong limit, 2
= )$
= yr
3) ft strong limit, cf 6 = &
, a < a), arbitrary;
, a < u)2 arbitrary; , 2
arbitrarily large before
the first inaccessible cardinal; for ft "large" enough. Our work continues that of Magidor [Mg l][Mg 2 ] ,
The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research by Grant 1110. The author thanks Menachem Magidor for patiently listening to some wrong proofs and to the present one.
117 Notation: Let i/j/CX/g/Y/S/? k e ordinals, 6 a limit ordinal, X,y,K,x cardinals (usually infinite) £,k,n,m, natural numbers. Let P
(A) = {t: t a subset of cardinality < K } .
t,s denote members of P
We let
(X) .
Notation on forcing: PyQ,R denote forcing notions, i.e. partial orders, R c Q means every element of R is an element of Q and on R the partial orders are equal.
Let R < Q mean R c_ Q, any two elements of R
are compatible in R iff they are compatible in Q and every maximal antichain of R is a maximal antichain of Q. Let C O 1 ( X , < K ) = {f: f a partial function from (K-X) X X to K, f(a,i) < a, f has power < X}.
We let IT,a denote members of P, q,r members of Q or R. We say i\,o are compatible if they have a common upper bound and equivalent if they are compatible with the same members of P. Note that TT,G are equivalent iff for any generic G, iteG <=> o£G* For any forcing notion, let 0 be its minimal element. §1. 1.1. Framework: n < ca, X
< X
In our universe V, K is X -supercompact for , moreover the X -supercompactness is preserved
by any K-directed complete forcing notion (see Laver [ L ] ) . R
is a K-complete forcing notion.
_ , ||- "X = X ". n So if we force by R , K is still X -supercompact (more exactly |X |-supercompact, as maybe X
R < R
was collapsed), so there is an
R -name E of a normal fine ultrafilter on P (X ) = {t: t a n ~n
of power < K } .
Note that P
(X )
belongs to V,
is included in it and is P < (X ) ; as forcing by R
does not add
sequences of ordinals of length < K . But the members of E (which are subsets of P
(X )) are not necessarily from V.
118 Let for t € P
(X ) , a <X , a(t) be the order type of a n t.
n strongly inaccessible] possible as we have assumed
and t n K
H~Rn "An= V Let t c
s
mean
t £ s, |t| < K ( S ) .
We let C be an FR -name for every n, i.e. C is an R -name for ~ n ~n ~n every n, and |k R ~n n 1.2 The forcing notion:
The forcing notion P^ we shall use is
defined as follows: An element TT of P has the form:
r e RQ
B)
t =
t
C)
f =
f^ an R £ + 1 -name
D)
Let K 0 = K(t 0 ) - (which is the order-type of K n t ) , J6
€ I , t
c t
Jo
Jo
then f Q e C o M J ^ , < K ^ , f% e Col U n ( t n ) + , K £ + 1 ) for 1 < I < n and f e Col(X (t ) , < K) (i.e. those things are forced, but ~n n n f
is a name of an element of V, so we omit the ~ if we know the
value and write f e V(f
e V).) Jo
E)
A = , A> is an R.-name of a member of E.
F)
G =
Jo
domain
* X>
-name of a function with JO T J~
I , and G (t) e Col(X (t) ,
#x
' Jo
We w r i t e n = n[iT], t .
Jo
= t . [ f r ] e t c . , or n = n , e t c .
119 1.2(A)
The order on P:
The order is natural: TT < a A)
r* < r° (in R Q )
B)
n77 < n a and t* = t*
C)
fJcfJforA-0
for £ = n
iff
for I = I,...,]/
n* ( i . e . r* |h R " ^ 1 g ) "
+ 1,. ..,n is
1.2 (B) Claim: subset of P^ 1.3
forced
The set of IT € P_, such that f* e V is a dense So usually we deal with such it only.
Technical Definitions on the forcing conditions:
1.3 A
Definition:
of
IT if
a)
TT < a [
J
For IT,a e P. we call a a j-direct extension
for j < A < n^
"G(t A ) = f £ [ a ] " for n71 < £ < n a
O
q t a ] |h R
d
>
S£[TT] = A£[a]
for
£ > na
e)
G£CTTj = G £ [ a ]
for
£ > na
Convention: We omit j when j = n 1.3 B Definition:
For TT,O e P, we call a a j-length preserving
extension of TT if (a)
TT < a
... TT a (b) n = n
(c)
+ 1.
fJ = fJ for I < j
120 Convention We omit j when j = n
+1
1.3.C Definition: 1) For ir,a e P, we call a an R-extension of TT if TT < c, r
= t a ,f* = f°, A* = A°, G* = G°f or at least if r*
forces those inequalities. 2) For 7r,a e P^ we call a an R-constant extension of TT if IT a TT < a, r = r . 1.3.D Claim and Definition: If j < a), IT < IT2 then there is a unique TT such that ir is a j-direct extension of TT ,TT2 is a j-length preserving extension of TT, and rCir.] = rC7T2-' ^ a n d rC^J = rCir ]]. This unique TT is called the upper [lower] j-interpolant of TT ,TT . 1.4
If j = n[TT2] + 1 we omit j ,
The Inner Model: The forcing P^ gives too much, e.g. it collapses all
cardinals which are both < £ X and > K, and maybe also K. But n we shall use an inner model. Define some P-names: t ,K are — ~n ~n t ,K(t ) (for every large enough TT in the generic set), F
= u (f 0 : TT in the generic set}, and C as an R -name is a
P^-name. For a generic G <=_ P^, we shall be interssted in the inner model V C < K 0 C G ] , F [G]:£
< o)>, C C G J ] ; let V
be a P-name of this
class. 1.5 Automorphism of P: The proofs of the following are well known.
121 1.5
A Claim:
Suppose H is an automorphism of P, then it TT
induces naturally a permutation of the set of P-names a -> a and if a , ...,a are P-names, <j) (xn , •. . ,x ) a first order formula, "^1 ^n Jn
IT e p , t h e n TF II— "(|)(alf . . . , a n ) " iff H(Tr)||- p "cf> (a^, .. . ,a£) say that H preserves a if ||1.5 B Claim: that TT 11— V
If TT||-
We
"a = a ".
"a e V " then there is a P-name b, such
"a = b", and every automorphism of P_ which preserves
(i.e. preserve K , f , C) preserves also b.
1.5 C Claim:
Let H be a permutation of
u A , which maps n<0)
A (n < 0)) onto themselves, H [V = the identity: 1) H induces an automorphism of Py which we denote by H too, as follows: for t £ I (n < co) let H(t) = n H(TT)
r
{H(i):i € t}
II
= r
: t
2) Note also that 1I1IR
n
"{t e I : H(t) = t} e E " and that H n ~n
preserves V . 1.5 D Definition:
We call a a V_-name if it is a P-name and is
preserved by any automorphism of P^ preserving V_, and
ll-p "a « V 1.5 E Claim: A
Suppose H is a permutation of UA , mapping each
onto itself, H [ K = the identity.
122 Then for any n e P, H(TT) are compatible.
and H * (TT) =
,f H (7r) ,A7rrG7T>
Moreover suitable increasing of the A makes
them equivalent. Proof:
By 1.5 C(2), remembering that the E "s are ultrafilters.
1.6 Claim: Suppose a) ir,a e P, n U ) = n(a) = n, t
= t ,f
= f
and
is in V, A71 = A a and G* = G°. b) every r e R
compatible with r[a] is compatible with rCir]
c) a is a V -name. ~ ~f Then if TT 11— Proof:
"a = a" (for some a 6 V) then o\\-
"a = a".
Easy.
1.7 Definition.
Good Cardinals for P
A cardinal y is good for R and there a r eforcing y -complete and Q
n
or for n in short, if y = y —
notions Q # Q , R = 0 * C!...# Q i s nn nn m m '
satisfies the y -chain condition (i.e.
II" ^ "Q^ satisfies the y -chain condition") . Remark: 1) Really Q? * Q > R is sufficient. in. Zxa m 2) Note that 0 R
= Q
x Q , 0
is not required to be K-complete, so
y -complete, 0
satisfying the y -chain condition
is sufficient for this definition. 1.8 The Main Lemma: A) If K is good for every n ^ n , then in V . K is strong limit.
Moreover every subset of X (t ) belong to LC
x (t ) hence 2
= X (t ) n n
^
l
(the + is in V. and in V too) . f
123 B) If y > K is good for every n ^ n , IIO R then in V. y
is still a regular cardinal.
"y is regular",
n Moreover for any
function g from y to ordinals, for some A 6 V, V |= " | A | = y" and Range(g) ^ A. The proof is broken to a series of claims. 1.9. Notation: For m < a), let P -m
= {
€ V and n
= m} •
m For any y, K < y < X we define an equivalence relation « on P ^ :
«
has y
n y = t. n y, f
= f , and
which is the identity on y,
1 2 and maps t 0 onto t 0 (all for 1 < & < m) . equivalence classes, and we can find
< t i / j , f i / j > (i < y < K ,j < y + ) such that: < t i f j , f i / j > /-* depend on i only, every « -equivalence class is represented by some
Claim: Suppose TrePy m < w , g a V -name of a function from y to
ordinals, ir ||-
"t
t_ y".
Suppose further that y is good for m
and y = y Then there is a length preserving extension a of TT, such that if a < a1 e P_, i < y, a1 ||-p "g(i) = a", mEa'J = m then also the upper interpolant a U ^ of a and a 1 force this.
Moreover,
for some set A of ordinals, |A| < y (A £ V of course), for every
124 R-extension a of the lower interpolant of a and a 1 , if 0
||- "g(i) = a", then a £ A .
Proof of Claim 1.10; So R = 0 * Q / and R^ < R , hence there is a pair m nn £m 0 m e Q * Q , such that every extension of it in Q * 0 is compatible with r
(which belong to R ) .
We now define for i < y, ordinals a.. < y , and for j < a. a condition TT . ., such that J-/ 3 A) r[ir .] = < q ° ,q .>, q ° 1
1J
if] ~J- fj
< q°
when
t,/^
i/j
1 < 5 or i = 5, j < <;) . B) for each if {q. . : j+1 < a.} is a maximal antichain 13+L 1 ( f
° Sm}
(i e
- -
C) tCir.
. .] = t
1 / j
, f[ir. . . ] = ^'^nC-rr.
. ] = m (see Claim
1.9A).
D) IT
is a direct extension of TT .
E) For each a e t m
either i\. ... determines (i.e. forces) a 13+1
value for g(a) or there is no length preserving IT ' > TT . . which ~
i/ 3
does so. F) F o r I > m, | | - R
(Vt £ A t C ^ i f
j+1
G) F o r I > m, G CTT . . ] i n c r e a s e s , ^36
then
||-
] ) (*£'3 £ t ) . i . e . if
< i , j > < f
^
" i f (1) t e A [IT
(2) i = j = O t h e n G 0 CTT. JO
] a n d t e A [TT
. ] ( t ) c G 0 [iT r
1 , 3
~J6
r ] t, r Q
II-
i , j
" i f t £ A .[IT.
] or
(t)".
H) G increase unnecessarily, nu [TT. .] does not x,
t £ Io
1 / D
i.e.
. ] o r j i s n o t a successor o r q. . i s n o t i n
t h e g e n e r i c s e t t h e n G [TT. . ] ( t ) i s t h e u n i o n o f G [TT Jo
1 / 3
<£ t,> < < i , j > , t £ A[TT ~
s/ h
] or i = j = O".
36
c , , C,
]
125
this
Note that G.LTT. . J(t) i s increased only when tt 1 ' 3 c t so J ~£ i] m ~ m
is X (t) -complete, so we can continue to define.
K)
So there is
36
no problem to carry the construction by induction on
satisfies the y -chain condition). In
the end we have to define a.
For r[a], note first that
{q. .:i < y,j < a.} has an upper bound q e Q as Q is y 1,3 l y y y complete (and by A)), and r[7rj, (q ,q ) are compatible by the choice of (q ,q ) , and let rCa] be any upper bound of r, (q ,q ) . ~
y /N/
Obviously, t a = t71, f ° = f^.
Now A^ = {t e I : t e Ao (.TT) and r J
^ 36
36
<^36
for any i < y , j < a., t '"* c t implies t e A 0 [TT. r /
1
^
*^36
.
] . Moreover
1 , 3 ""i
if s /cx/ t, s « m t1'11, f 1 ' 3 c K (t) x K(t) , H a permutation of y m u A. which is the identity except interchanging t 1 ' 3 and s then i 1 t = H(t) €H[A&r. ]].} A is an R -name as each A [ir. .] is, and ""36 i, X °"^ ^ ' ~^ i / D is forced to be in E
by the normality of E 36
(a conclusion of it,
36
more exactly). Now G0 Ccr] ( t ) ~36
i = j =0.
i s t h e u n i o n of Go ETT .
3 6 i , j
. 3 (t) , t
e ACTT .
. 3 or
~ l , j
I t i s easy to check everything, because g e V , (and
use 5D, 6) . 1.11. Claim; Suppose IT € P, g a V -name of a function from i\ to ordinals, II
it
TT
- "for every m > n , rv t j^ y" and y = y P m — Then there is a length preserving extension, a of IT, such
that if a < a 1 e P_, y good for n[a'3, i < y , i e t p a1
||-
^[a 1 3, and
"g(i) = a" then also the upper interpolant a" of a and a 1
126 forcesthis.
Also i^f for some
a < a1 e P_,
y is good for R r , -., i e y n t r , So1 ],
Ql
ll"p "9(i)
= a
" then a e A. .
Hence if y is good for
arbitrarily large m, if a < a1 £ P_5af ||- "g(i) = a" then for some direct extension a" of a 1 , the upper interpolant a"1 of a, a" forces this. Proof:
Repeat claim 1.10 u) times.
1.12 Proof of the Main Lemma 1.8B.
™
Quite easy from Claim 11, because ||- "y c u t " and — n<0) lhp " f ° r every m > n , t £_ y"} is a dense subset of P_.
1.13 Claim:
Suppose IT E P, m > n , ||- "g e V_, g a function
from y to ordinals". Then there is a e P, -n < o such that A) 7T and a are identical except that possibly G [a] is not equal to G [IT], ~m B) Suppose o 1 e P, n[a, ] = m, rlo 1 > r[a], tQ e I. for 1 < £ < m,
for m < I < Suppose further a 2 is an (m-1)-length preserving extension of o^., nCa 2 ] = n[a 1 ] = m, tEa ] = t[a J, f.Ca ] = f [a ] for I < m, A[a 2 ] = A[a 1 ] f G[a2J = g C ^ ] , i e t m n y, a
2 H"p "g(i) = a""
Then < r [ a
2]'
i
^
g
1
1
forces this too. Proof: ————•
Just note that for each t m
the number of
127
The parallel claim to 13 holds for m = n .
1.14. Corollary.
Suppose g is a V.-name, y good for every
m > nlir], g a function from y to ordinals.
Then there is IT., , a
length-preserving extension of TT such that: if ir < a 6 P, i £ t CaJ n y, a II- "g(i) = a" then also .—.
m.
JL
* ii
p
i+u
"" •
*
a ||-
" g ( i ) = a" where a
i s d e f i n e d by
r[a*J = rCa] nCa J
= nCa], t [ a ] = t CaJ f o r Z = l , . . . , n [ a ] f [ a ] = f CcrJ f o r I = O , . . . , m - 1 f.Ca
] = G0CTT 3 ( t ) f o r I = m , . . . , n [ a ]
36 A 0 [a ^^36
/^»X/ JL 36 ] = A.CTT.J for nCa] < £ < co '>"'36 JG [ a ] = G [TT ] f o r nCa] < i6 < a). '^"'36 '*°36 J-
Proof:
Use Claim 1.11 and then 1.13 for all m.
1.15 Claim:
Suppose IT.., g, n = nCir^.] are as in corollary 1.14,
y < X(t ) and m < n[ir ], and g is a function from y to {0,1} (i.e. g e v[< K [G],f [ G ] : n < I
TTj, < TT 2 , t [ i r J = t [ i r 2 ] ,
g [ i r ; L ] = GCTT2J
B) I f a i s an m - d i r e c t e x t e n s i o n of TU, a < y , i < 2 a
lhp "g(a> = i" then also ^2 forces this-
128 Proof: Let W
Fix a. = {
t* = t, f > = f £ V } . For every w = (t,f) € W, we define an R
-name V (t,f) of
an ordinal < 3: for r e R, r ||- "V (t,f) = I" iff A) I < 2,
or B)
1=2,
andf o r A
n o r
l
< gk(tk)>,
f
r < r
I
€ R , ^
l
< 2
G(vCk+l,a))>
Aftk+1,0)),
||-p " g ( a ) =
I".
Now for every k < a) we define by downward induction on I < k, for every (t,f) e W
an R
< 3 and B (t,f) of a member of E
-name V Ct,f) of an ordinal .
For £ = k we have defined V (t,f), and let B,(t,f) = I
.
Now let (t,f) € W , I < kf V (tf,£') is defined for (t1,?1) e W
. Then V, (t,f) is the unique i < 3 such that: XJ "rJL
/X
'X
(t =
(note all the names in the expression above are R
-names)
and B (t,f)
= {t
€ I
: V, ( t , f )
= V (t A
>, fA
Now we can d e f i n e A CTT ] : A [IT J = {t £ I : for every ( t , f ) and k > I,
e W . , such t h a t t
t e B ( . t , f ) , and of course t e A [TT ] }
c t,
129 This is fine for a, and as there are "few" Ci.e. JA) a's there is no problem to prove the claim.
§2.
Applications
For this section we make the following hypothesis. 2.1 Hypothesis:
There is a universe V satisfying ZFC + G.C.H.,
in which K is supercompact, moreover the supercompactness is preserved by K-directed complete forcing. Remark:
We can weaken "K is supercompact" by "K is A-super-
compact" for A suitable for each theorem, bur as long as we cannot get inner models with supercompact this is not so interesting, and anyhow clearly we get by our proof the expected results (or almost, replacing A by X ) . Similarly for assuming G.C.H. - it is expected that violating C.G.H. is "harder" so we do not lose generality, and so though it seemed that we can get rid of it, there is no point in doing this. Notation:
( # a ) + 1 = # a + ± so A*1"1 = A + .
2.2. Theorem: 1) For any a < i).. , there is an extension V f of V in which K is
.B* , is strong and 2
= $*
2) Moreover in V^ there are f. e TT H r in * n
... for i < H
n, ot+1
f.,i.e., {n: f. In) < £ .(n)}is co-finite.
Proof: 1) For a < a) this is done in Magidor [Mg 1 ] , so let a = 6 + k, 6 a limit ordinal, let 6 = u D , D finite, increasing. Let Q be any K-complete forcing K n adding n subsets to K and satisfying the K -chain condition e.g. Q = {f: f a partial function from K
to {0,1} of power < K } .
130 Let T n = {(3,y): O < g < y < <$ and (V? e D n ) -i (3 * 5 ^ y) } = (n C o l ( K + e f < K + Y )) x Q . te,Y)£T n . +a+2 X = K
R
n
n Now clearly: Fact A: R is K-directed complete, II- 2 < K , hence for -j n K n w every t e I , A (t) < K(t) . Fact B: K
is good for R if for some y, g = y + 2 ,
y + l e D
or g > 6 + 1, hence every y + 2 < a + l i s good for R for every n large enough. X
K
Fact C:
In V - (from §1) K is jf , 2n < x t2
• • "
K
r
< K, where x
n
=^
n
0)
and n
= A (t ) so K is }j and strong limit.
Now the theorem is immediate. 2)
Just change Q to add f .:K -> K, (i < K
) , such that
f ± <* f. for i < j . 2.3. Theorem: 1) For any a < i>2 there is an extension V f of V in
^ ^ 1 L^
which K is yS , and is strong limit, and 2
= jy
2) For any fixed £ < GD_ we can assume that V v f H ( # ) = H(/f ) (H(X) is the family of set of hereditary
cardinality <
X) .
Proof: 1) Just amalgamate the proof of 2.2 and of Magidor [Mg 1 ] §5 (see [Mg 3] too) . 2) Just l e t in §1 f
£ Col(iv
, <
131 Remark:
By Magidor [Mg 4 ] (improving Galvin and Hajnal) if
Chang's conjecture holds, JJ w
2
l
tt
< jv
.
is strong limit, then
So using a method which gives 2.3(2) , we cannot
improve 2.3(1). Remark: a > (2
By [Sh 2 ] , we cannot improve 2.2(1) result for ° ) + if the method gives 2.2(2) too.
2.4. Definition: 1) For a monotonic function f from ordinals to ordinals we define a function f
from ordinals to ordinals by
induction on i: fC0j(a) = a fCa+l3(a) =f(fCi3(a) fC6](a) -
u
f [ i J (a)
2) For a function f from ordinals to ordinals we define a function f
from ordinals to ordinals f*(a) = f C a J (0)
3) For a class C of ordinals we define by induction on a a function Sue
from ordinals Suc°(i) = min{£: E, e C, £ > i}
Sue (i) = C
4)
u Sue (i) C
In 3) if C is the class of infinite cardinals, then we omit
it. 2.5. Lemma:
Suppose X has cofinality J J O / and for x < ^# n
Sue (x) < A. Suppose further y > X but there is no weakly
< w
132 inaccessible cardinal K, X < K ^ y. D,D
c_D
, u D
Then there are
^{x:A<x-P/X
a
cardinal}, and
Sue* (K) > y. n Proof:
We prove^
by induction on y 3 the existence of
y = X.
We let D (y) = {X}, and there is no problem. Case II;
y = X
Let D D
n+2 ( l l )
=D
even x < y - *$ • X
U D
*'
(y)(n<0)) is increasing, with union =>_ "£K: ^ -
(x)
Sue
or
= {K: X < K } , D 0 ( M ) = {K: k > \i}, D][(y) = D
n(X>
SoD Sue
f ° r some Xr
- X/ hence Sue
(X) >
j$
K
- y}/
hence is > y, so
(X) > y for n > 1; for n = 0,1 this is true too by the
definition. Case III:
y is a limit cardinal, but for x < y# -^ < y» X Let \i = I. )i. , x ~ cfy, y. < y, y. increasing continuous, i<x
*
Let D Q (y) = D 1 (y) = {K: K > y } , ^ n + 2 (y) = (x: for some i, ~
x
< y
i+l'
X
£ D
n(ui+1)
and
^i
e D
n(^\} *
The
checkin<
?
is easy. 2.6. Theorem:
1) For any y > K, y
= y, smaller than the first
inaccessible cardinal > K, there is a forcing extension V f of V, in which K is Sue (ft ) , is strong limit, 2
= y, and no cardinal
in [K,y] is collapsed, except possibly successors of singular cardinals.
133
2) Moreover we can assume t h a t i n V t h e r e a r e f u n c t i o n s f.
1
€
II K ( i < y where K = £ K , K < K ,..) such t h a t for n n+1 n
i < j , f± <* fjfc Proof:
Similar to the proof of 2.2., using lemma 2.5 (so for
t e I n , Sucn(K(t)) > X(t)).
134 REFERENCES
[GHj
[LJ
F. Galvin and A. Hajnal, Inequalities for cardinal powers, Annals of Math., 101 (1975), 491-498. R. Laver, Making the supercompactness of K indestructible under K-directed closed forcing, Israel J. Math>,29 (1978), 385-388.
[Mg lj M. Magidor, On the singular cardinals problem I, Israel J. Math., 28 (1977), 1-31. [Mg 2] M. Magidor, On the singular cardinals problem II, Annals of Math., 106 (1977), 517-549. [Mg 3] M, Magidor, Changing cofinality of cardinals. XCIX (1978), 61-71.
Fund. Math.
[Mg 4] M. Magidor, Chang conjecture and powers of singular cardinals, J. Symb. Logic, 42 (1977), 272-276. [Sh lj S. Shelah, A note on cardinal exponentiation. Logic. 45 (1980), 56-66.
J. Symb.
[Sh 2] S. Shelah, Jonsson algebras in successor cardinals, Israel J. Math., 30 (.1978), 57-64. [Sh 3] S. Shelah, The singular cardinals problem independence results and other problems, A.MS. Abstracts. [Si]
J. Silver, On the singular cardinals problem, Proc. of the International congress of Mathematicians', Vancouver, 1974, Vol.1, 265-268.
TREES, NORMS, AND SCALES
David Guaspari
§0.
Introduction
A. In the past few years descriptive set theory has changed from a miscellany to a subject. The main cause has been the study of various determinacy hypotheses - which seems to require a unified, axiomatic approach. This article describes three unifying principles, sets out their relations, and derives from them the best known elementary facts of classical descriptive set theory. (The approach is due primarily to Moschovakis and not all to the author of this paper.) Sections 1 - 3 consist essentially of one hour talks on each of the notions: trees, norms, scales - the leisure of print allowing the inclusion of a few digressions and, more important, a slightly more abstract approach than is practical in a talk. Paragraph B of this section fixes notation and some basic terminology. Paragraph C needn't be read immediately. It contains a definition or two that will be referred to later, but consists mainly of a chatty example which, though not necessary to the sequel, may help orient the beginner. B. H , the set of reals, is a) - the set of functions from a) to a). We're interested in the descriptive set theory of the points paces 3R m x a) ; and will use X, £/, Z for pointspaces and x, y, z, ... for their elements. For the elements of R we'll also use a, $, y (ordinals will be denoted by Greek letters from the end of the alphabet); and for elements of a): i,j,k,m,n. Topologically we think of u) as a discrete space and B as the topological product of countably many copies of U). To avoid any spurious pretence that this article is completely self-contained we'll immediately declare: the reader
136 1 rl must at least know the definition of II , [_., etc. and have a nodding acquaintance with the elementary facts of recursion theory. From the points of view both of topology and recursion theory there are, really, only two pointspaces: ]R and a). For if m > 0, then ]R x a/1 is homeomorphic to 3R via a recursive function; and, of course there are also recursive bijections from a)n to a). The spaces homeomorphic to ]R are of type 1_, those homeomorphic to a) of type 0_. A pointset (denoted by A,B,C) is a subset of some pointspace, and a pointclass (denoted by r,T') a collection of pointsets.
The pointclasses of interest will contain subsets of
every pointspace.
Examples: the (pointclass of) closed sets;
the recursive sets; the II^ sets.
The examples to keep in mind,
for our purposes, are II , 7 , and A . We will devote attention primarily to subsets of 3R . To be at all interesting a pointclass must possess some closure properties.
A notational convention: if Op is a logical
operation, then Op T is the result of applying Op to all pointsets (or combinations of pointsets) in T.
For example, the sets
in vr are obtained as follows: If A £ x^ij and B c yxz are in V then C = {(x,y,z)|(x,y) e Av(y,z) e B} is in vr. We'll use Q
,V ,V
to represent, respectively: bounded
numerical quantification (both existential and universal), universal quantification over a) , and universal quantification over 3R .
(Final example: if A £ T, then
B = { (m,x) |v*n.< m(n,m,x) e A} is in 0
r.) F is closed under
op if op r c r. Here are some closure properties of F if F is one of vl 1 1 I , II , A : (The grouping is for future reference.) I a) closed under A , V, Q b) closed under recursive substitution; i.e. if f: X -* if is total recursive and A c (/is in F, then {x|f (x) e A} is in F.
137
II a) closed under vw, b) closed under V
if T is IT under -3 n
if T ; under L n
neither if A . n Definition F is adequate iff T contains all recursive sets and is closed under the operations in I. Not only are the IT , etc., adequate, but so also are: 2 , I ,...; II , £ /...; the collection of pointsets recursive in some higher type obj ect ; etc. One last piece of notation. £ is the relativization of T. Its elements are those sets of form {x|(a,x) e A} for a e ]R and A e T. If T is adequate, so is £; and we get the same collection of relativized sets from r whether we fix several co-ordinates or just one. C. Introductory examples (having little to do with the sequel). The theorems of the sequel have the following look: If V has certain closure properties (e.g.< is adequate) and certain structure properties (examples follow), then.^.. (something follows). Here is an example of a structure property: the parametrization property. Notation : If B c x*y, then for any x e X, B is {y|(x,y) e B } , Definition B is X-universal for T subsets of y iff 1) B € T and B £
2)
{A
6 T|A £ y] =
{ B |X
€ x}
If B is universal and A = B we call an index for A (although it should properly be called a B index for A ) . Definition F is X-parametrized iff for every y there exists a set which is ^-universal for r subsets of y-, and, simply, parametrized, if X.-parametrized for some X. One of the elementary facts of descriptive set theory (which will not be proved here) is that:
138 Each of II , \
, II , J
(but not A ) i s a)-parametrized.
Each of 11°, Y°, II , 7 (but not A ) i s ]R-parametrized. ~n zji ~n &n ~n In the abstract, we can tidy up parametrizations as follows: If F is adequate and X-parametrized, then T is 3R parametrized if type (X) = 1, 0)-parametrized if type (X) = 0 . Proof.
Say type (X) = 1 , and that B c_ x\y is an X-parametrization
of the T subsets of y. Let f: 3R < — — > X be recursive and put B 1 = { (a,y) | (f (a) ,y) e B} . Since the map (a,y) I—> (f(a),y) is recursive and V is adequate, B f e V. And clearly {Bf |a £ 3R } = {B |x £ X> . Similarly easy to prove is the fact that: r adequate and parametrized = > so is £. The usual parametrizations of, say, the II sets, have important uniformity properties (the s.m.n theorems). For example, there is a recursive f: w •*• co such that if n is a II index for A and m a l l index for B, then f (m,n) is a TI index for AnB. The definition of "parametrized" does not require that the various universal sets be related in any such nice way, but: If r is adequate and parametrized, then there is a collection of universal sets {B^|y a pointspace} having all the uniformity properties one could hope for. For simplicity let f s look at spaces of type 1. For each y of type 1 let f.n\ y < > JR be the "obvious" recursive map. Let B-£ be X universal for the T subsets of B and obtain all the others by: B^. = {(x,y)|(x,f^(y)) e B } . Since the fy's are nicely related so are the B f s . Dealing with adequate parametrized pointclasses, then, we are on familiar ground. We can use all the indexing tricks familiar from recursion theory. One example, Theorem
(Hierarchy theorem)
If F is adequate and parametrized, then
*1r<£_Y.
Proof. The usual one. Let T be X-parametrized. Let B be X universal for r subsets of X and A = {x| (x,x) ^ B} . The adequacy of r guarantees the adequacy of "ir, which in turn guarantees that A £ IT. As usual, A | T. For If A e V and y is an index for A, (y,y)
{ B <=> y £ A <=> y £ B v <=> (y,y) £ B
139 The reader may simply shrug at what has after all been nothing more than an axiomatization of the well known details of a well known proof. Indeed, the example was chosen for practice, to go down easily. But is has the virtue of focusing the attention in the right places and of avoiding a wild and forbidding notation. And it helps single out the notions to be discussed in the next three sections.
§1.
Trees
A. It will be convenient to define "tree on £", "tree on , etc., rather than to provide a single definition "tree on the set X." For any X, FS(X) is the set of finite sequences from X. We'll use s,t,... variously sub or super scripted for finite sequences. If f is a finite or infinite sequence and n € oo, then f(n) is the finite sequence f ^ n --i.e., (f (0) , . . . ,f (n-1) ) . Notice that f(0) is the empty sequence and that n > every element of domain(f) implies that f(n) = f. T is a tree on Z iff 1) 0 ^ T £ FS(£) 2) for all n £ a) and all s, s £ T = > i (n) £ T. A path through T is a function f: co -> E, such that for all n, f(n) £ T. Finally, [ T ] is the set of all paths through T. Trees grow downward and a path through T corresponds to an infinite descending chain in T. (Precisely: partially order T by s ^ t iff t £ s.) Since a path through a tree on E, is an element of £, a path through a tree J on £xri ought to be a pair (f ,g) with f £ ^C, g £ ^r), and f and g running through J side-by-side: J is a tree on £xn iff la) 0 ± T c_ FS(?)*FS(n) lb) (s,t) £ T = > length of s equals length of t 2) for all n £ 0) and all s and t, (s,t) £ T = > (s(n),t(n)) £ T. A path through J is a pair (f ,g) £ ^ x ^ n such that for all n e a), (E(n),g(n)) £ T. Again, [j] is the set of all paths through J (and (s,t) K (s',t') iff s < s' and t < t') • Etc. It is easy to check:
140 Lemma 1.1. A £ ]R
is closed iff A = [ T ] for some T on a)
A c ]R
is II iff A = [ T ] for some recursive tree T on 03 • vl B. Let A be a K subset of UR . Then A is the projection
of a IT subset B of B B
(and conversely) . For any real a let
be the a-section of B: B
= {31 (a,3) e B}. So
a 6 A < = > 3B(a,3) € B <=> B
^ 0.
We can rephrase the above in terms of trees. Let T be a recursive tree on coxa) such that B = [ T ] . Then for any a the closed set B^ is [T(a)], where T(a) is the following tree on a): {s| 3n(a(n) ,s) e T} . So, a € A < = > 3B(a,B) e [ T ]
<=> [T(a) ] f 0 < = > T(a) is not wellfounded (under
^)
Abstracting from this: Definition 1.2 If T is a tree on then p[T] = {a|T(a) is not wellfounded} where T(a) = {s|3n (a(n),s) e T>. Since wellfoundedness is absolute (for transitive models of enough set theory) we immediately have: Theorem 1.3. "a € p[T]" is an absolute predicate of a,T. Corollary 1.4. I predicates are absolute for transitive models of (enough) set theory. (To be a little more explicit about the corollary: Given an index for a £7 set there is an absolute - in fact, recursive way to obtain a suitable tree T,... etc.) Notes: Here K is any ordinal, though often only its cardinality matters. Say that A is K-Souslin if A = p[T] for some tree T on K. We've shown that co-Souslin is equivalent to ^ . Trivially, if K is the cardinality of ]R , then every set of re*aTs is KSouslin; and we'll indicate a proof (later) that in ZF one can prove outright that all ^ s e t s ^ r ^ Jf -Souslin.
141 C. Perfect set theorems. A c_ M is perfect if it empty and closed, and every point of A is a limit point T, a tree on GO, is a perfect tree if every s e T splits i.e., has at least two incompatible (with respect to ^C) sions in T. It is easy to see that A is perfect if and A = [ T ] for some perfect tree T.
is nonof A. in T extenonly if
A perfect set contains a topological copy of 2, so has the cardinality of the continuum. One of the early programs of descriptive set theory was to prove pieces of CH by showing for larger classes T that T had the perfect set property: PSP(F) =££ every uncountable element of T contains a perfect subset. (There exist sharper, more effective, versions of the PSP. See below.) vl ) - a classical result. We'll show below that ZF proves PSP() Godel observed that V = L impliesfPSP (ITr) . A theorem due independently to Mansfield and to Solovay says that PSP (III") is equivalent to the statement: for every a only countably many reals are constructible from a. Let's briefly consider the classical proof of PSP(closed sets). Start with A = [ T ] and try to find a perfect tree T* S T - f ° r then the perfect set [T*] will be contained in A. We attempt to obtain T* by pruning T of its non-splitting nodes. For any tree J on w let J' be {s| s splits in j } . (Notice that J is perfect iff J' = J.) One such pruning may not suffice, for it's possible that some node which splits in J no longer splits in J'. Put T° = T; T ^ + 1 = (T^)'; and for limit A, T\ = Q T^ . For some n < # , this stops: T n = T n + =, say, T*. If T* = 0, then, presumably, we can show that [ T ] was countable to begin with. Proof: We've reduced [ T ] to 0 in countably many steps. All we need to know is that at each step only countably many paths were discarded (i.e., [T?]\[T^+ ] is countable). But all the paths in [ T ^ ] \ [ T ? + 1 ] are isolated points of [T^]. Here is an ultramodern version of that argument: Theorem 1.5. Let M be a transitive model of enough set theory and T, a tree on cox?/ be an element of M. Then, p[T]
142 Proof Suppose A is a £_ set and T is a tree coxa) with A = p[T]. Let M be a countable transitive model of set theory with T e M. Then if A is uncountable p[T]<£_M - and so A contains a perfect subset. Corollary 1.7 (Mansfield; Solovay) vl Any £2 s e t containing a non-constructible real contains a perfect set.
vl 2 s e t i-s P^T^ f ° r some T e L. (This is nonShow that any £2 l h tree t i question ti i usually ll called l l d the th Schoenfield S h f i l d trivial. The in is tree of the \z set.) Note: From 1.7 it follows fairly easily that PSP (JO < = > PSP()O < = > for any real a, LCa] n 3R is countable. ~ Proof of 1.5 For any tree J on u)x£ say that (s,t) splits in J if there exist (s ,t ) and (s,,^) in J such that (s.,t.) -< s
and s., are incompatible; and let J
1
(s,t) and
={ (s,t)|(s,t) splits in
J}. Suppose now that T e M is a tree on 0)x£; and let T T
V+1
V
= (T )'; and, for limit A, T
J !—> J
1
= ^
V
T .
= T;
Since the operation
is absolute the sequence of T^'s is definable in M
(through the ordinals of M ) . fying T
X
= T
. Then n 6 M.
Let n be the least ordinal satis(Proof: M must satisfy "the
sequence of T 's is eventually constant"; so for some n* e M, y\-k
71*4-1
M satisfies "T ' = T ' T
= T
". But then it m u s t b e true that
. So n < n*-) n 5^ (jb
Consider two cases:
Case 1_. T
It f s easy to check that p[T ] is a subset of p[T] containing a perfect set. (There is no reason to believe that pCT1"!] is itself closed.)
143 Case 2.
Tn = 0
We'll show p[T] <=_ M.
Suppose a e p[T], and choose any f such
that (a,f) £ [T]. Since (a,f) is an element of [T ] but not of [T1^], there must be some v satisfying: (a,f) e [T ]\[T Since (a,f) i [ T
].
V+
] there is some n for which (s,t) = _(a(n), f (n)) cit £ T , which means that (s,t) does not split in T . Then: \) (1) If (s,t) and (s.. ,t, ) are elements of T extending (s,t) , s
and s 1 must be compatible.
(2) Every initial segment s 1 of a°
satisfies: there exists t 1 such that (s',t!) £ T (s',f) z< (s,t).
1
1
So a = ufs'l^t'Es ^ ) £ T
V
and
and (s' ,t •) -< (s,t)]} ,
But this definition is absolute and therefore a £ M. A closer look at the argument gives: Theorem 1.8 The "M" of 1.5 can be taken to be the least admissible set containing T. (Note: The proof of 1.5 presented above is a little too crude. If M is merely T-admissible it is not, it turns out, guaranteed that n £ M, but only that r\ < On n M. That, however, is good enough to carry through the rest of the argument.) Corollary 1.9 (Harrison) (A "lightface" perfect set theorem) rl If a h set contains a non-hyperarithmetic real, then it contains a perfect subset. Proof Let T be an infinite recursive tree on 03x0) and apply 1.8 in light of the following facts: The least T-admissible set is L cK. The reals in L cK are precisely the hyperarithmetic reals. (Note: The proof presented for 1.9, like that for 1.6, is much more high powered than necessary.) D.
A Miscellany of other applications. 2 The Kunen-Martin theorem: If A £ 1 is a wellfounded relation and a K-Souslin set, then the ordinal length of A is less than K .
144 One corollary of this is that every vl i wellordering has length less than $2. So, if there exists a J^ wellordering of ]R , then CH holds. (Mansfield has proved the much stronger result that if 3R admits a ^ wellordering, then all reals are constructible. On the other side, Harrington has shown that it is consistent with ZFC that the reals have a A^ wellordering but that CH be false.) Generalizing the Souslin-Kleene Theorems: A is K-Borel if A is in the least class containing all clopen sets and closed under unions and intersections of fewer than K sets. (So "Borel" is " ^ -Borel".) The classical results (and proofs) that
i) Borel £ j} i.e.
ii) disjoint £- sets can be separated by Borel sets; ~ i) u)+-Borel <=_ co-Souslin
ii) disjoint o)-Souslin sets can be separated by OJ -Borel sets; become i') K+-Borel c_ K-Souslin ii 1 ) disjoint K-Souslin sets can be separated by K+-Borel sets. Since o)+-Borel and oo-Sotfslin are equivalent to A and K it is natural to look for other correlations between the hierarchy of Souslin operations and the analytical hierarchy. Nontrivial results have been obtained from AD. §2.
Norms
A_. From the representation theorem for K sets we know that for any Ilir set A there is a recursive tree T such that x € A <==> T(x) is wellfounded. I assume that the reader has, at some time in his life, seen the succession of coding tricks which transforms this into the basic representation theorem: Theorem 2.1 A is II iff for some total recursive function f into M , x € A <==> f(x) codes a wellordering. By "a codes a relation B on co" we mean that B = { (m,n) | a (<m,n>) = 0 } , where <,>: oo^ < — > OJ is some fixed recursive pairing function. For future reference, the relation coded by 3 will be denoted by < ; WO = {$|fB codes a wellordering}; and for 3 e WO, |$| = the ordinal coded by 3 = the order type of
145 The basic representation theorem provides us with a natural way of attaching ordinals to the elements of a 3T_ set. In the setting of Theorem 2.1, define a: A — > On by a(x) = If(x)I. Such a map induces a relation < on A — a prewellordering — b y (*) a < 3 < = > a (a) < a($) . Definition 2.2 A map from A into On is a norm on A. A relation < a obtained from a norm a as in (*) is a prewellordering of A. (Alternatively: a prewellordering satisfies all the requirements for a wellordering except, possibly, anti-symmetry.) Of course every set admits trivial norms. The interest lies in those norms whose associated prewellorderings are simply definable. The most important property of the pwo's provided by the basic representation theorem is this: Not only are they II "• relations, but also their initial segments are A in a uniform way. Theorem 2.3 (pwo theorem for II ) 1 Let A be II , a the norm obtained from the basic representaI relation < and a £ tion theorem. Then there is a I T- relation < and a V., relation
x e A and a (x) < a(y) <===> x <^ y < = > x < n y.
* (In future we'll refer to (^) as 'the set-theoretical requirements on <^ and < '.) Proof of 2.3 Let f be as in the basic representation theorem.
y
Here's
y
L
how to define £ : x
into < . . ) .
Notice that if y e A, then < .
is a
wellordering and so the existence of such an a guarantees that ^,_, . is a wellordering (hence that x e A) and of length less than or equal to that of <
..
If y i A we don't care in
the least which x's satisfy x <^ y.
The definition is \\ because
the expression in parentheses involves only recursive operations
146 and number quantifiers. "Hot
For the record: x <
y iff x £ A and
(a embeds < . . as a proper segment of < . . ) .
We'll now abstract from this. (Use Y to stand for the dual class of T - what we've previously called *lr.) Definition 2.4
r A norm a on A is a r-norm iff there exist relations < r and <
in
in T such that
for all y £ A and all x,
r x £ A and a(x) < a (y) < = > x < <=> x <
y y.
Definition 2.5 PWO(D 2/
iff every set in V has a V norm.
Some consequences of PWO
Reduction and separation Red(F) says that given any A ,A £ V there exist A 1 ,A' £ V satisfying: A': /// o A^: \\\\\\\; i.e., A! c A . ; A'nA' = 0; and A'UA.1 = A UA- . 1 —
l O l
O l
O l
Sep^T) says that given any disjoint A,B £ V there exists C £ r n r satisfying:
; i.e., A c c and BDC = 0. The following fact is a consequence of propositional logic: Theorem 2.6 Redd 1 ) = > Sep(F) . Our first application of PWO is to prove Red and Sep. Theorem 2.7 adequate and PWO(D = > Red(T), hence S e p ( D .
147 Corollary 2.8
lJ), Sep(^) . Proof of 2.7 Let A ,A_ £ F. x to A A
1
1
or to A* ?
The problem is: if x £ A
n A
do we assign
We'd like to say: take norms a. and put x in
just in case a (x), "the ordinal which puts x in A ", is less
than a..(x).
The A.1 would satisfy the desired set theoretical
requirements but needn't be elements of Y because the pwo property tells us nothing about comparisons between different norms. a simple trick.
Let B = {(O,x)|x e A } u {(l,x)|x e A } - thus
B is a r subset of a)X]R - and let a be a T norm on B. x £ A
1
Apply
<=> x £ A
and (x e A.. = >
(0,x) <
Put
(l,x))
x £ A' < = > x £ A. and (x £ A = > (l,x) < (0,x)) . o 1 o a In order to illustrate a characteristic point about pwo's let's show in detail that A' e V. o x £ A^ < = > x £ A Q and
"](x e A 1 and (0,x) {
< = > x £ A Q and
1(x e A± and (l,x)
< = > x £ A Q and
T((l,x) ^
(0,x)).
The point is that: The statements "(0,x) $ "(lfx) <
(l,x))
(l f x)" f
(0,x)" are not necessarily equivalent to one another
for all possible x.
However, if x £ A , then the statements •
o
— — —
"x £ A, and (0,x) £ (l,x)", "x e A. and (l,x) < (0.x)", and 1 ^ o 1 a "(l^x) < F (0,x)M are equivalent.
(To show that A' is V it is
most convenient first to establish the following fact: let x <
y iff a(x) < a(y). Then the initial segments of <
uniformly elements of T n F) . Propogation of PWO Theorem 2.8 Suppose T is adequate and V Then, PW0(D = >
^
r ^ r.
are also
148 Proof Suppose that A = {x|9$(x,$) e B> with B e T. Let a be a F norm on B and define a norm \p on A as follows: iMx) = inf{a(x,B) | (x,3) e B} . Then it is easy to check that if; is an -3 V norm on A. Corollary 2.9
It is now tempting to suppose that one could obtain PWO(TI^) by repeating the last trick with "sup" in place of "inf". It is midly instructive to try that and see where it fails. The full story is: i) If T is adequate and parametrized (see §0.C) then PWO(D and PWO(f) cannot both hold. In particular, it cannot happen that both LV 1 and IT1 have PWO. n n ii) (Addison) V = L = > P W O ( p ) for n >^ 2. iii) (Addison-Moschovakis; Martin) Protective determinancy implies that PWO holds for each n^,.. and Y^ odd ^even iv) (Harrington) Any pattern for PWO consistent with PWO(IIr-) and theorem 2.8 is possible (in some model of set theory). Boundedness and the lengths of pwo's w
Notation: Abbreviate V n T by A; £ is the boldface version of r. Theorem 2.10
(Boundedness theorem for subsets of H )
Suppose that T is adequate and V r <=_ T. that A 6 F\A; and let a be a T norm on A.
Let A e T be such
Then every £ subset of A is bounded in a; i.e., if B e A and B £ £, there is an a € A such that 3 e B ==> a(3) < a(a) . Proof Suppose B is a counterexample. Then we get an immediate contradiction by providing the following V definition of A: r
a 6 A < = > 33(3 € B and a
<
3) .
Notes: There is an analogous theorem for subsets of a) (or any other point space) . The requirement that V <= V cannot merely be omitted, for the boundedness theorem is not true of lz.
149 Corollary 2.11 If B is a £
subset of WO, then sup{|a||
Proof We need to know: (1) that WO is 1^ but not A-]_, (2) that the map sending a to |a| is a 11^ norm on WO. (2) is proved by consulting the proof of 2.3 (the pwo theorem for 11^-) • If WO were A^ the basic representation theorem (theorem 2.1) would immediately imply that every n^- set is A , contradicting the hierarchy theorem. The boundedness theorem, simple as it is, nonetheless allows us to get a handle on the particular ordinals which appear in norms. To be able to talk sensibly about such things say that the length of a norm is the order type of its range. (So if, as we can without loss of generality do, we require a norm to be onto an ordinal, then its length is its range.) Notice that WO is a complete II^ set in the sense sens that every 11:: set is a recursive preimage of WO (and every II:r set a continuous preimage) . The natural IT^^- norm on WO has length len }% and so all norms on II sets obtained from the basic representation r e p r n t i o n theorem have length ^ /£]_• In fact, every III" norm on WO - or on any complete n^set - has length exactly /$-, ; and every III" norm has length i We'll state the facts solely for IL , but in a way that suggests the correct generalizations: (1) length of any II norm on a complete IL subset of 3R = longest possible II-J- norm = sup{£| 5 is the length of a A
pwo of M }
(2) length of any II norm on a complete subset of u> = longest possible 11. norm on a subset of OJ = sup{?| £ is the length of a A
pwo of u)}
(= a) ) 2
determinacy
The definitions of "game" and "determinacy" are assumed known. We'll show how to deduce PWO (Ili) from the axiom of determinacy (in fact, from the assumption that A^ games are determined). Here "3" is a generic odd integer: The proof axiomatizes, and so, in conjunction with 2.8, we obtain the pattern mentioned before: PD implies that PWO holds for II^
150 and for £ 2
?#
(This result is due independently to Martin
and to Moschovakis and Addison. Addison is presented below.)
The proof of Moschovakis and
Suppose that A is J 2 and B = {x|va(a,x) e A } .
Let a be a
£ 9 norm on A.
As mentioned before, setting x < y to mean that 1 sup{a(a,x)|(a,x) e A} < sup{a(a,y) € A> doesn't yield a H 3 norm
on B.
The idea of the proof is to effectivize this.
We'll put
x < y if given any a, there is a strategy for producing a 3 with a(a,x) < a(3,y). Consider first the games G defined only when x,y € B. x,y *Player I plays a, player II plays 3, and II wins iff (a,x) < a (B,y). Now put x < y iff x,y e B and II has a winning strategy in G
. Finally, let x < y == x < y and T y < x. x,y i •* df ~ -^ ~ ASSUME: A determinacy.
From now on
Subl emma (). For any x,y e B, G
is determined.
For any x,y e B, x < y iff I has a winning strategy in G y,x Proof Let x,y e B.
It's trivial to check that the set of winning
plays for I is L . SoG is determined. For the second part, * J ~2 x,y £- ' note first that: I wins G if II does not win G iff l y < x. |jr y,x ~ So we immediately have that x < y ==> I wins G . Suppose, y,x on the other hand, that I wins G . Then 1y < x and so to y,x ~ complete the proof that x < y we need only check that :x < y i.e., that II has a winning strategy in G Winning strategy for I in G in G
. S o let T be a
and consider the strategy for II
indicated by the following diagram.
The wavy lines
represent plays made in accordance with strategy x. the game G f
The run of
indicated on the right half of the page is merely
player II s scratch paper while he pursues his strategy for G x,y
151
nA —
—
n
The game goes like this: I plays n . that play, consults x, and plays m
II temporarily ignores
= T (0) , the opening move of
T in the game G I next plays n.. II now enters onto his y,x scratch pad I's first move. He copies n into the game G i.e., he presents T with the sequence of plays <m ,n >, then takes T f s response, m. = T(<m ,n > ) , and plays it in his own game. Etc.
The result is that (a, 3) is played in G and (3, a) x,y
is a play of G
in which the first player has throughout em-
ployed strategy T. So ($,a) is a win for the first player in G - and therefore, because x and y are elements of B, y,x (a,x) < ($,y); which means that player II has won his play of G
x,y
Sublemma 1 < is a prewellordering Proof a) For all x e B, x ;< x. proof: A winning strategy for II in G is: Copy the last move X X of I. ' b) x < y and y < z
=> x < z
proof: Suppose that II has winning strategies in G and G * x,y y,z The diagram indicates a winning strategy for II in G x, z
152
yn
_ _ _ _ _ _ _ _
A Since (a,3) and ($,y) are winning plays for II in, respectively G and G we must have (a,x) < ($,y) ^n (j,z) # Thus II x,y y, z a o has won the play of G c) For all x,y £ B, x < y or y < x. Proof: Each of the following sentences Gimplies the next. G . . T x < y. II does not win Gx,y . I wins Gy#x . y < x. y < x. ' ~d)* < is wellfounded. xy yx J Proof:
Suppose that x
> x, > x 2 > ... with each x. e B. Then
I must win each of the games G
•
CNotice that we're not
using sublemma 0.) So consider the plays of these games indicated by the diagram.
£
3
C
C
D
153 Since I wins each of these games we must have a (x ,A) > a (x.. ,B) from G
; z a(x..,B) > a(x«,C) - from G ; etc. Therefore X 0/Xl 1'X2 is not wellfounded, which is a contradiction, X
< a
Sublemma 2_ < comes from a m
norm
Proof Let 8 be a norm associated with <. First notice that < is, outright, a II :r set - hence we can take ]_
<
to be £.
For,
x < y < = > x,y £ B and II has a winning strategy in G ~ x,y < = > x,y £ B and I has no winning strategy in G < = > x,y £ B and Vx-33 ((x*$,x) < (3/y)) where x*3 is the sequence of moves of player I which results when he uses the strategy x and II plays 3. (To be exact we should say that " T " varies not over strategies but over reals which encode strategies in some effective way.) We can hardly expect that < also be ).^-
To
obtain the
"J-side" of <, consider the games H defined for all x and y X y by: ' I plays a, II plays 3, and II wins iff 1) (3,y) i A or 2) (ot,x) , (3,y) e A and (a,x) <
(3,y).
The following facts are easy to check: (a) If y £ B, then II has a win in H . (b) If y e B and x i B, then I has a win in x,y H . (c) If x,y £ B then H is the same as G . These rl x,y x,y x,y guarantee first that each H is determined, and second that x,y H = _ {(x,y) I II has a winning strategy in H } satisfies the set theoretical requirements for < £3. 0 The proof of the sublemma is concluded by making the routine check that H ±s_ in fact £_. What this argument has shown is:
154 Theorem 2.12 (Martin; Addison-Moschovakis) Assume determinacy for A sets and that V is adequate. Then,
^ T §3.
£ r and PWO(D = > PWO(VED .
Scales
A. We're going to glue together the notions of §§1 and 2 to produce the stronger notion of a scale. The idea came from analyzing the argument behind the classical proof of the n^- 2 uniformization theorem (which says that every II subset of ]R has a III selection function), and is due to Moschovakis. We f ll begin by considering a simple selection problem: If T is a tree on some ordinal £, how can we select in some canonical way an element of [ T ] ? One answer: Choose its leftmost branch, h T . I.e. h T (0) = least n < £ such that for some f e [ T ] , f(0) = n T
h (n+1) = least n < £ such that for some f e [ T ] ,
g iff f ^ g and, letting n be the first input at which they differ, f(n) < g(n).
Although < is not a wellordering of w £ (nor, usually, of [ T ] ) x ex nonetheless h T is easily seen to be the least element of [ T ] . T Notice also that the definition of h is absolute: For any s let T[s] = {t € T | t is compatible with s}. Then h (n) = the least n such that T[h (n) 'n] is not wellfounded. If T is a tree on a)x£; we can easily extend the same trick to A = p[T]: For any (a,f) let ^a/TNread "a shuffle f" be the sequence (f(0), a(O), f(l), a(l),...). There are technical reasons for beginning with f(0) rather than with a(0). We choose an element of p[T] by first choosing (a*,f*) e [ T ] S O that q*/~f'** is lexicographically least, then dealing out a*. When examining a we needn't survey all (a,f) for all possible f, but only those pairs of form (a, h ^ ) , where hj is the leftmost branch of T(a) . We want to define a* "internally" - as we previously showed how to generate the paths h T . This is straightforward, but it is in fact convenient to be slightly roundabout. By means of the h a 's we're provided with a sequence of norms o = (o ,o.,...) on A as follows:
155 (**)
a±(a) = h*(i)
We now define: £ = least ordinal in range (a ) o o K = least n e o> such that for some a e p[T], (a (a) = £ Q and a (0) = n) ; and inductively,
inf{O
m+l(a)l
a 6A
m+1 }
k m + 1 = inf{a(m+l) | a e A m + 1 and a m+1 (a) Here, a (a) is the sequence (a (a) ,a., (a) , . . . ) . Now it is trivial to check that a* = (k,k_,...) and This sequence of norms also has an interesting limit property. To state this it will help to introduce one piece of terminology. Say that a sequence converges in norm to "t = (XO,A^,...) if for each n the sequence
a £ A
2) a (a) <]ex t. (We'll call conclusion (1) the closure property and conclusion (2) the semi-continuity property.) Proposition 3.0 The sequence a defined in (**) has the limit property. Proof _^ Suppose that
to "£. We'll show a £ A by showing that (a,A) £ [ T ] . S O , letting n £ a), we want (a(n), ) e T. For some large enough 0 n-1 i, a. (n) = a(n) and = . I _^ o I n-1 i o n-1 Since (a.,a(a.)) = (a.,hT ) is a path through T, (a.(n),
,a
,(a.)>) e T and we're done. That n-1 i
156 a(a) ^
X is immediate because a(a) is (by definition) the lex leftmost branch of T(a).
Definition 3.1a A scale on A is a sequence of norms on A which has the limit property. (Note: in some discussions a scale is supposed to satisfy the stronger requirement that "a (a) be pointwise less than or equal to A".) Suppose, conversely, that we start with a scale a on A. We can then associate with that scale a tree T such that A = p[T]. Definition 3.1b If a is a scale on A, the tree associated with a is { (a (n) , ) |n € u) and a e A } . Lemma 3.2 If a is a scale on A and T its associated tree, then A = p[T], Proof If a e A,_^then by the definition of T each proper initial segment of (a,a(a)) is in T, so a(a) is a path through T(a) and a £ p[T]. Suppose now that a e p[T], and choose some path f through T(a). By the definition of T there is for each n a real a such n that a (n) = a (n) and f (n) =
157 B^. Digression. This section is not necessary to what follows. It discusses a few obvious ways of varying the definition of "scale". Let f
158 3) whenever two sequences from A converge in norm to the same sequence of ordinals they converge topologically to the same real, Proposition (iii) Any scale on A can be gently perturbed to a nice scale on A. Proof Use the previous trick, at the same time shuffling in a: Given a, let v n (a) = < C^a/ala^1 (2n)> C. Definable scales; the uniformization theorem. Definition 3.5 a is a F scale on A iff a is a scale on A whose norms are uniformly T norms; i.e.
r
r*
There exist subsets < ^ and < of a)XX*X which are, respecively, elements of T and T, such that For all y e A, all n e co, and all x,
r x £ A and a (x) < a (y) < = > <=> Definition 3.6 Scale (T) =
(n,x,y) e <^ (n,x,y) e <
Every set in T has a T scale.
1 rl Fact (proof deferred) : Scale (II.) and Scale (} ?) . Granted the Fact we'll employ Scale (IL ) to prove the well known basis theorem: Theorem 3.7 Every nonempty II. subset of ]R contains a II singleton. Proof Let A be 11^, a a II -scale on A, T its associated tree and a* the canonically chosen element of p[T]. We'll show that {a*} is a nj- set. Go back to the "internal" definition of a*, marked (+) on page 1551. We want to be able to rephrase that definition so as to refer (in a II way) only to reals, not to both reals and ordinals. The key is the semi-continuity property, which guarantees that, in the notation of (+), ( ? , ? , . . . ) = a(a*). (Proof: Note (1) to lemma 3.2 - which is a consequence of semicontinuity - points out that 5"(a) is the leftmost branch of T(a). But by construction £ is the leftmost branch of T(a).) Let P(x) be the following predicate:
159 x(0), x(l),... are respectively k ,k, ,... and a (x), a, (x) , ... are £,£.,... Then a* is the unique solution to P. The check that "x e P" is II1 is tedious, but routine. Here it is. (We1 re using "a = B" to n abbreviate "an(a) =on($)"r etc.) Paraphrase of P(x)
Translation
x e A and
VY 1 (Y
(Y E £ x = > x(0) <
x(0) = k
Y(0))
o
and (
[Vi < n
(Y
=\ x and Y(i) = x(i)) ] X
V
TT
and
y x(n+l) = k
Vi < n+1(Y =. x and) and
n+1
Vi < n Y(i) = x(i) ] = > 1 3R < Y(n+D Note: We've repeatedly used the fact that IT. is closed under V . Definition 3.8 Unif (T) = For every A e V, A c_ x*y, there exists B e T, B c xxy which is a selection function for A. By relativizing the proof of theorem 3.6 we easily show that Unif(IlJ). Abstractly, Theorem 3.9 Suppose that T is adequate and closed under V Scale (D = > Unif (D .
3R
. Then,
Notice that 3.7 does not allow us to deduce Unif(^) from Scale (£1) . That is no problem since, trivially, Theorem 3.10 r is adequate, Unif(D = Corollary 3.11 Unif (nl) and Unif (J^) •
^
160 Note: It takes slightly more effort to demonstrate than the corresponding fact about pwo, but the scale property propogates from F to 31*F (when T is closed under V-^ ) . Hence, under V = L and PD the patterns for Scale (so, for Unif) are the same as those for PWO. These are not the only existence theorems. E.g., a theorem of Martin and Solovay says that if all the sharps exist, then every ni set admits a III scale (of length u = the u)*-*1 uniform indiscernible) . We'll conclude with a proof of Scale (II, ) . Theorem 3.12
(Kondo-Novikoff-Addison-Moschovakis)
Scale (II*) Proof Let A be II and f: A -> M a recursive function such that x e A <=> f(x) € WO. For any n e w , put: x ^
y <=> |f(x)| < |f(y)| or (|f(x)| = |f(y)| and |f(x)f"n| < |f(y)f"n|)
where for any g, $ f"n = { (i,j) | i < j < n} . P
P
(Equivalently, letting < , > again be the lex-order isomorphism u> < — > d) , a (x) = <|f (x) | , |f (x) Is n|>.)
It's left as an
exercise to check that the a are, uniformly, II.. norms.
(Had we
defined a(x) to be simply |f(x)^n| it wouldn't follow that each a
is a n| norm.) Checking the limit property: Suppose that <x.|i
m <_,
)n is to say precisely that :
|f (x±) /Sn| < |f (x)i/^n| !; Semi-continuity: More or less by definition, (a) |f(x)| = sup{|f(x)Tn| + 1: n £ 0)}. We also have (b) n > sup{X'
|n e 0)};
161 for, if i is large enough, |f(x.)| = r] and |f(x.)f"n|= X1 . Since we've just shown that the map n -> Xf is order preserving on <
it (x)
, (c) |f(x)f n| < X1 for each n.
n
Putting (a) - (c)
together we obtain: f (x) < r\. Using this and, once again, fact (b) we have an (x) =
On the regularity of ultrafilters
Karel Prikry*
It was shown in [6] that in Gfldel's construetible universe L, every uniform ultrafilter over w^ is regular. This involved a new combinatorial principle stronger that Kurepa's hypothesis. Chang [2] and Jensen [4] have generalized this principle to higher cardinals and extended the regularity result of [6] to all u)n, (new). Benda and Ketonen [1] have generalized and simplified the result of [6] by making use of a weaker combinatorial principle. In the present note we show how to generalize the ChangJensen result along similar lines. Kunen asked in a discussion whether this can be done. We start by formulating the transversal hypothesis in a customary form: TH(A,v) : There is a family F c_ v such that |F| = X and for all f,g e F, if f ^ g, then f (y) ^ g(y) for all sufficiently large y e X. The following consistency results are well known. Theorem 1, (Solovay, see [3]). In L, T H ( K ,K) holds for all infinite cardinals K. In the opposite direction we only give a special case of Silver's Theorem [7]. Theorem 2. If ZFC + there is a Ramsey cardinal is consistent, then ZFC + i T H t a ^ w ) is consistent. The author received support from the NSF Grant MCS 74-06705.
163 For A < v+. approach TH(o)2,co) disjoint
regular A it follows trivially from TH(.A,v) that An obvious attempt at generalizing the Benda-Ketonen to, say 0)2, leads to considering statements like . But there does not exist a family of even o)-j_ almost functions from o^ into a), let alone 0)3 such functions.
Definition 1. A uniform ultrafilter U over A is (A,v)~ regular if there are sets a a e U, (aeA), such that for all I £ [ A ] v , n{a a : ael} = 0 . U is regular if U is (A,0))-regular. Benda and Ketonen showed Theorem 4, [1]. If T H ( K ,K) holds, then every uniform ultrafilter over K + is (K + ,K)-regular. As a special case, if TH(o)^,o)) holds, then every uniform ultrafilter over o)^ is regular, which is an optimal result. By Theorem 1, this holds in L as well. We shall now introduce a modified form of the transversal hypothesis: MTH(K,V)
: There is a family F such that each f e F is a
function f : [ K ]~ •* v, |F| = K , and for every f £ F, and every G c_ F, if |G| < K and f ft G, then there is a e K such that f(x) ^ g(x) for all g e G and all x e E K ] ~
satisfying a e x.
. Remark. Note that if we require only |F| = K, rather than K , in the formulation of MTH, then we can prove the resulting statement for all infinite K and v. If we contrast this situation with the preceding remark concerning TH(0)2,0)), it becomes clear that formulating MTH is a step in the right direction. We proceed to prove the claim in this remark. This is non-trivial only when v < K. For each x e C K ] ~ V let g x be a^n injection of x into v - {0}. Now for each a e K , a £ 0, define g x (a) if a £ x
0 Let a,3 £ K, a ^ 3/
if a i x a,3 t 0.
Our main result is
Then if a £ x, f (x) ^ fo (x) . ex p
164 Theorem 5. If M T H ( K , V ) holds, then every uniform (K,V ) regular ultrafilter over K is (K,V)-regular. We shall first of all derive some corollaries, in particular the Chang-Jensen result. Kurepa's hypothesis, K H ( K , V ) is the statement: There is a family S c_ [ K ] K such that |s| = K + and for all X£[K]
V
,
| s h x | < v, where S/*x = {snx : seS}.
Let K H ' ( K , V ) be K H ( K , V ) + "S is K-almost disjoint".
A slight modification of Jensen's proof of K H C K , V ) gives K H ' ( K , V ) . More precisely, Theorem 6. K H ' ( K , V ) holds in L whenever K is regular, not ineffable, and v < K. Proposition 1: For regular K, K H ' ( K , V ) implies M T H ( K , V ) .
Proof; Let S a [ K ] K be as in K H ' ( K , V ) . Now define for each s 6 S, f s (x) = snx, C X £ [ K ] ~ V ) . Since there are only v sets of the form x n s where s ranges over S, we can regard f s as a function into v. We set F = {f : seS}. Let G £ F, |G| < K, f € F-G. Let R = {reS : f r £G}. Let a e sQ - R. Then f o o disagrees with all g e G on those x which contain a. The Chang-Jensen result now follows from Theorems 5, 6 and Proposition 1: Theorem 7. The following holds in L: If K is regular, not ineffable and v < K, then every uniform (K,v+)-regular ultrafilter over K is (K,V)-regular. Corollary. regular, (new).
In L, every uniform ultrafilter over con is
Proof of Theorem 5: Let a a e U, (CL£K) , O ( a a
: ael} = 0 if 111 = v . V
we find U <_„ U where U is over [K]~~ . h : K •> [ K ] ~ U = h*(U).
V
by h(p) = {a£K : p£a a ).
As usual,
Namely, define Hence |h(p)| < v.
Then for each a e K , r a = { X £ [ K ] ~
now suffices to show that U is (K,V)-regular.
V
: aex} e u.
We set It
165 Let F be as in M T H ( K , V ) .
If f, g e F and f ^ g, then f ,g
differ on some r a , and thus mod U* Let -< be the ultraproduct ordering on F. Since |F| = K , there is some f e F which has K predecessors in -< . Let them be f~, ( £ £ K ) , f^ distinct.
We can now define the (K,V)-regularising family for U. Suppose that b^ £ U, (£ < n) / are defined, where n < K . Consider the family {f^. : £ < n} £ F, and f - not in the family. By M T H ( K , V ) we can fix a £ K such that f (x) ^ frCx) for all x £ r a , all 5 < n.
Set
b^ = r a
n {X£[K]-V
: f^ (x)
< f(x)}.
We claim that b^, ( ? < K ) , is (K,V)-regularising for U. Suppose not and let I £. K be such that |l| = v, and n{b£ : E,el} ^ 0. Pick x o in this intersection. It then follows that f^(xo), del), are all distinct. Indeed, if £ < r), £, n £ I, then x Q £ r a , and f n (x Q ) ^ fp(xQ) follows from the choice of a n . On the other hand, f^(xQ) < f(x Q ) < v for all 5 £ I, and |l| = v. This contradition completes the proof of Theorem 5.
University of Minnesota Minneapolis, Minnesota.
166 REFERENCES 1.
M. Benda and J. Ketonen, On regularity of ultr^filters, Israel Journ. of Math. 17 (1974), 231-240.
2.
C.C. Chang, Extensions of combinatorial principles of Kurepa and Prikry, Theory of sets and topology, ed. G. Asser et. al., Berlin 1972.
3.
Keith J. Devlin, Aspects of cons true tibi lity, Springer*Verlag, New York 1973.
4.
R. Jensen, Some combinatorial principles in L and V, an unpublished manuscript.
5.
Richard Laver, A set in L containing regularizing families for ultrafilters. Mathematika 24 (1977), 50-51.
6.
K. Prikry, On a problem of Gilman and Keisler, Annals Math. Logic 2 (1970), 179-187.
7.
Jack Silver, The independence of Kurepa's conjecture and two cardinal conjectures in model theory. Axiomatic Set Theory, Proc. Symp. Pure Math., Vol.XIII, Part I, ed. D. Scott, Amer. Math. S o c , Providence 1971, p. 383-390.
MORASSES IN COMBINATORIAL SET THEORY
A k i h i r o Kanamori Baruch C o l l e g e C i t y U n i v e r s i t y of New York New York, NY 10010 Jensen invented the morass in order to establish strong model-theoretic transfer principles in the constructible universe.
Morasses are structures of
considerable complexity, a culminating edifice in Jensen's remarkable program of formulating useful combinatorial principles which obtain in the constructible universe, and which moreover can be appended to any model of set theory by straightforward
forcing.
Gfidel's Axiom of Constructibility V = L is surely the ultimate combinatorial principle in ZFC, and the morass codifies a substantial portion of the structure of L. As set theorists looked beyond the well-known ^ and D for applicable combinatorial principles, i t was natural to consider extractions from the full structure of a morass.
168
This paper is an expository survey that schematizes the higher combinatorial principles derivable from morasses which have emerged in set theoretical praxis.
I t is notable that most of these principles
were formulated by combinatorial set theorists to isolate salient features of particular constructions, and shown by them to be consistent f i r s t by forcing.
Then,
the specialists in L established how they hold there, in ad hoc fashion using the full structure of the morass.
The sections of this paper deal successively
with Prikry's Principle, Silver's Principle, Burgess1 Principle, and finally sequence reflects
limit cardinal versions.
This
the historical development of ideas,
the progression toward further
complexity, and coinci-
dentally the author's series of papers [Ka2][Ka3] [Ka4].
The cumulative layers of sophistication pro-
vide an illuminating approach to the full morass structure, whilst at the same time providing a hierarchy of principles which, seen in this scheme, will hopefully
find wider application in the future.
The
emphasis will be on shorter, illustrative proofs
for
the casual but interested reader, with adequate
ref-
erences for the more persistent researcher. The set theoretical notation is standard, and here is a short litany: The f i r s t Greek letters
169
a,3,Y,..« denote ordinals, whereas the middle Greek letters K,A,]J,.., are reserved for infinite cardinals. If x is a set, |x| denotes its cardinality, Cx] K denotes the collection of subsets of x of cardinality K, and if f is a function, f"x = {f(y) jy e x}. Finally, ^x denotes the set of functions from y into x.
170
§1. PRIKRYfS PRINCIPLE
In the f i r s t
three s e c t i o n s , K w i l l always denote
a successor c a r d i n a l , with K" i t s predecessor.
The
general s i t u a t i o n w i l l be considered in the l a s t s e c tion.
P r i k r y f s Principle i s the following proposition:
(P )
There i s a c o l l e c t i o n
{f
t h a t whenever s e [K ] K |U This f i r s t
< K|Va€s(f a (5)
|a < K + } _C KK SO and < J >e
t *(a))}|
approach t o the system of
K, we have
< K. approximations
which comprises a morass says roughly t h a t there are K
functions:
K -> K such t h a t : i f
guesses are made at
possible values for any K~ ijiany of them, then sufficiently
for
large £ < K, a t l e a s t one guess i s
dered c o r r e c t a t £.
ren-
Although i t i s not made e x p l i c i t ,
notice t h a t we can assume t h a t the f ' s are pairwise distinct of P
and have range = K, since easy applications
show t h a t only < K~ of these functions
have these p r o p e r t i e s .
Historically,
P
do not
was the
of the higher combinatorial p r i n c i p l e s t o be
first
formulated
In a ground-breaking paper, Prikry [P] devised h i s principle
and e s t a b l i s h e d i t s
consistency with the GCH
by a method of forcing with side conditions. 2
Assuming
= K, a simple diagonal argument provides K func-
tions s a t i s f y i n g
the conclusion of P ; P r i k r y f s
171
argument yields K many, and in fact can provide a r b i t r a r i l y many in a cardinal-preserving extension.
forcing
There was no particular emphasis laid on
p o s s i b i l i t i e s in L, but at any r a t e , the f i r s t
result
about L in this whole context was due to Jensen, who showed that if V = L, then P
holds for every succ-
essor cardinal K, using the morass structure that he invented, Prikry was answering a question of Erdtts, Hajnal and Rado [EHR] in the partition calculus, and this was the f i r s t example of the phenomenon of relative consistency r e s u l t s , rather than outright demonstrations, in this area of set-theoretical research.
To recall
the relevant case of the polarized partition symbol of [EHR], X
y
K
v
means that whenever F: Xx« -* y, there are X e [> ] y and Y e [ K ] V such that F"(X*Y) ^ y.
To denote the negation
of this proposition, -> is replaced by / - .
Besides [EHR],
see the secondary source Williams [Wi] Chapter 4 for background.
Prikry educed P
from the negative p a r t i -
tion relation that he wanted to show consistent, which i s equivalent to the ostensibly weaker version where the < > | e SK just range over the constant
functions:
172
K
(If
{f
| a < K + } i s as provided by P , s e t F(a,g) = f a (3)
t o get a counter-example.) P
has s e v e r a l other a p p l i c a t i o n s .
Prikry himself
in [P] provided a consequence about an old problem of Ulam's on m e a s u r a b i l i t y with r e s p e c t t o a sequence of measures.
In a r e l a t e d development, Szymanski CSz]
formulated the following concept in order t o e s t a b l i s h some Baire Category-type theorems for U(aK)f the space of uniform u l t r a f i l t e r s
over 00-. .
For any
infinite
c a r d i n a l X, a matrix {An| n < 0 ) , a < X } i s a
X-matrix
iff (a) if m < n and a < X, then A
<=_ A ,
(b) LJ{An| n < a)} = a)1 for each a < X, and (c) for every infinite s ^ X and $ e
S
u),
a € s} | < GO-,. A basic clopen set for U(a)-,) is a set of form {u e U(o)-.) I A e u} for some A £ 03. ; and a G, closed s e t i s a countable i n t e r s e c t i o n of b a s i c clopen s e t s . Szymanski e s t a b l i s h e d the following equivalence: A X-matrix e x i s t s i f f
t h e r e i s a family of X G~ closed
and nowhere dense subsets of U(o)-.) such t h a t the union of any i n f i n i t e
subfamily i s dense in U(o)-,).
connection with P
is 1
clear:
The
173
If P
, then there i s an co--matrix.
(Let g: 10, •> GO be any subjection, (fj
and given
a < a)2l as provided by P ^ set A* = U | g- fa(£) ^n}.
Incidentally, Baumgartner has established t h a t the following are equivalent
(see [Ka2] for a proof):
(a) an oo-.-matrix e x i s t s ;
(b) an co-matrix e x i s t s ; and
(c) there i s a subset of
GO of cardinality co., without
an upper bound in ^co under the ordering of eventual dominance. Delimiting the ranges of the functions further
composing with a fixed surjection
yields the (P~)
in P
even
g: K -> 2
following:
There i s a collection
K
{f
I a < K+} C K2 SO that
Ot •f
K~~
whenever s e [K ]
S
and <|> e 2, we have
|{5 < K| Va£s(f a (O ^ (f> (a) ) } | < K. Even this weakened principle has its uses: Balcar, Simon and Vojtas asked ([BSV] Problem 20b) whether the following is consistent: whenever X is regular and uncountable and U is a uniform ultrafilter over X, then there are X sets in U such that the intersection of any infinitely many of them has cardinality < X. Probably, this is true in L, and the proof will depend heavily on the structure of ultrafilters.
But at
least, one can affirm the case X =00-,: If P , then
174
for any uniform ultrafilter over K there are K in U such that any K cardinality < K.
sets
of them has intersection of
(The proof is immediate.)
Perhaps a more substantial application of P. is to the Hajnal-Mate Principle, formulated in the study of set mappings in combinatorial set theory, in Hajnal Mate [HM]: (HM )
+
There is a collection {hr | £ < K } c
K with
hc-(ot) ? a for every £ < K and a < K
such that:
_i_ ^ —
whenever s e [ K ]
, we have
|{£ < K| Vaes(h^(a) / s) } | < K. In the most general s e t t i n g , a set mapping on a set X is a function
f from a subset of P(X) into P(X)
such that xnf (x) = 0 for every x in the domain. subset H £ X is free with respect to f iff for every x £ H in the domain.
A
Hnf (x) = 0
The general problem of
when large free subsets for set mappings exist was extensively investigated through the f i f t i e s
by
classical combinatorial means in Eastern Europe.
(See
the secondary source Williams [Wi] Chapter 3 for background; a timely application of this theory is found in Galvin-Hajnal CGH].)
Forcing and in particular
Prikry's method of forcing with side conditions extended the realms of p o s s i b i l i t y ,
and Hajnal and Mat£
d i s t i l l e d t h e i r principle with the following implica-
175
t i o n in mind: If HM and t h e r e i s a K-Kurepa t r e e
(see
the next s e c t i o n ) , then t h e r e i s an f:
such
[K ]
•* K
t h a t no s e t of c a r d i n a l i t y K~ i s free with r e s p e c t t o f.
Note t h a t HM i t s e l f
mappings and free s e t s .
i s a p r o p o s i t i o n about s e t F i t t i n g i n t o the p a t t e r n ,
Hajnal and Mcite e s t a b l i s h e d the consistency of HM by forcing,
and then i t was shown l a t e r t o be t r u e in L,
t h i s time by Burgess [Bu2], who f i r s t
established a
s t r o n g e r p r i n c i p l e in L t o be discussed in §3. Theorem 1 (Kanamori) : j-
Suppose t h a t
{f
P
-> HM .
| a < K } i s as provided by P .
For each a < K , l e t \p : a -* K be i n j e c t i v e . define _v: K -*- K
for ^ < K by: i f a = 0,
h (a) =
{ ty "(f
Finally,
(£))
if
else
t h i s i s defined,
and
otherwise. Thus, we took care t h a t hu (a) ^ a for every a < K . To v e r i f y HM , suppose t h a t s e [K ] the l e a s t element in s - { 0 , 1 } , Define
a
|
(3).
.
Let 3 be
and s e t t = s -
(3+D .
Then | h ^ ( a )
4 s) }
and hence by P , this last set has cardinality < K . -|
176
§2. SILVER'S PRINCIPLE In order to formulate Silver's Principle (and also Burgess1 Principle in the next section), it will be helpful to establish once and for all some conventions regarding trees.
If T is a tree, Ty will denote
the members of T at its £th level; if x e T
and
£ < C, then TIV (x) is the tree predecessor of x at level 5•
Let us assume that trees are normalized at
limits, i.e. if 6 is a limit ordinal and x ^ y are both in the 6th level, then there is a ^ < 5 such that Tiv (x) ^ TIV (y) • A K-Kurepa tree is a tree with height K + 1 such that |T | > K, yet |T r | < £ for every £ < K. (This is congruous with the usual definition; it will be convenient to identify cofinal branches with a top level.) (W )
This settled, here is Silver's Principle:
There is a K-Kurepa tree T and a function W with domain K
such that:
(a) for each £ < K , we have W(£) £ ^ T r ^ K
(b) for any s € CT ] K , there is a y < K such that whenever y < E, < K , we have
Like P , W
meets requirements mandated by K"
size subsets of K
(in essence, as | T | > K ) at all
177
sufficiently large stages.
The new feature in the
ascent towards the morass is the K-Kurepa tree structure: the system of approximations has small initial stages.
This combines with the potency provided by
the function W, which plays a role somewhat akin to the sequence of distinguished subsets in A . Notice that if T is any tree of height K + 1 , even if T were not necessarily a K-Kurepa tree, as long as IT | > K" and there is a function W satisfying (a) and (b) above for this T, it is not difficult to see that 2 K
= K
must be satisfied. The formulation of W evinced an evolution from K
P
in the focus of attention as well.
Upon seeing
some consistency results constructed by Hajnal and Juhasz in set-theoretic topology, Silver extracted W K
from the morass in order to effect these constructions in L.
So, unlike P , W was formulated with ramifiK
cations in L in mind.
K
That many, long-winded combina-
torial emanations from a morass actually follow from W
is a testimonial to Silver's insight.
Incidentally,
the consistency argument for W through forcing is not K
difficult.
To the usual notion of forcing for ad-
joining a K-Kurepa tree one appends further clauses reminiscent of Prikry's side conditions [Bui] for an exposition).
(see Burgess
The following proof i l l u s -
178
trates the constructions possible with W : Theorem 2
(Silver):
W
+ P.
K
|-
K
Let T and W be as provided by W ;
well assume that T
= K by renaming.
we might as Since W im-
plies 2 K = K , for each 6 < K and s e W(6) , we can enumerate
S
K as {h|| 6 < £ < K } .
T O take care of more
and more of the h^'s, for each £ < K we shall define functions g~ : T^. •> K such that: (t)
Whenever 6 < n < £ , s e W(5), and TT^'S e W(6), there is an x € s such that g, (x) = h* (TT^ (X) ) . Once this is donef the proof can be completed by
defining f : K •*- K for a < K by:
To v e r i f y
P , l e t s € CK + ] K
and < > f e
S
K.
By h y p o t h e s i s ,
t h e r e i s a 6 < K such t h a t 6 ^ 5 < < i m p l i e s TT^"S e W(5) / where we can take 6 s u f f i c i e n t l y so t h a t TTr, i s i n j e c t i v e on s . h
r.s
= ^^ ## TT TT 5 = 5
.
(Here, TT." (Here TT"
.
large
For some n ^ 6 , we have iis c l e a r from t h e c o n t e x t
as that inverse of TT. whose range is the top level.) Then for any 5 such that n < 7T
x e IT "s such t h a t g J x ) a = 7 r ? " 1 ( x ) / then f a ( £ )
= h
E, < K , there is an "S
(irg (x) ) .
But i f
= g^ (x) = <>| (a) .
A l l t h a t remains i s t o define t h e functions as t o s a t i s f y
(t).
g
so
But doing t h i s i s e a s y : Fix £ < K,
179
and l e t {<s ,6 , n > | ^
^>
C < K"} enumerate a l l t r i p l e s
r>
<s,6,n> where 6 <
n < £, s e W(5), and 7Tr"s e W(6).
Now define exactly one value for g». in each of K stages inductively: If c < K", since only £ values have been determined before the £th stage and s c a r d i n a l i t y K , there i s an x e s not yet been defined. 6 = 6
and r\ = n r -
has
such that g r (x) has
Set g^ (x) = h
(7T.(x))f where
Finally, after K" stages extend q~
a r b i t r a r i l y to a l l of T^.
This completes the construc-
tion of g^_, and the proof i s thus complete.
*\
Let us now turn to the work of Hajnal and Juhcisz alluded to e a r l i e r .
Pondering the existence of special
topological spaces of large c a r d i n a l i t y , Hajnal and Juhasz realized in the early 19 70's that concrete constructions readily follow from certain e x i s t e n t i a l principles concerning matrices of s e t s .
The following
proposition i s the strongest form of these p r i n c i p l e s , and can be appropriately dubbed the Hajnal-Juhdsz Principle: (HJ )
There i s a collection {f I a < K + } C K 2 SO that whenever p < K~ and s: K~xp •* K i s i n j e c t i v e , there i s a y < K such t h a t : if x £ [K-y] {e|x
f
s(a,T)
2, there i s a a < K" with f o r e v e r y
T <
w
and
180
In set-theoretic topology, HL and HS are acronymic for hereditarily Lindeltff respectively,
and hereditarily
separable,
and an L space is an HL space which is
not HS, whilst an S space is an HS space which is not HL.
There is quite a l i t e r a t u r e on the study of these
spaces nowadays, particularly in connection with Martin's Axiom, and a good but older reference is M.E. Rudin [Ru] Chapter 5.
An i n i t i a l version of HJ
considered by Hajnal and Juhasz with the to just p = 1.
was
restriction
Taking the concrete case K = oo. (other-
wise, we would have to frame the discussion in general terms around K~-Llndelttf and K~-separable), they show [HJl] that this r e s t r i c t e d principle implies the existence of normal S spaces of large cardinality, the socalled HFD spaces, and establish i t s consistency by forcing.
Then Devlin [D] established t h i s
principle in L, directly using morasses.
restricted As Hajnal
and Juhasz l a t e r realized, the full principle HJ
im-
plies the existence of normal, strong S spaces of large cardinality.
(A strong S space i s a space X such that
Xn is an S space for every n e GO.) Kunen [Ku2] has shown that under MA + -iCH, there are no strong S spaces. Concerning L spaces, Hajnal and Juhasz early on [HJ2] formulated the following principle to construct
181 L spaces of large cardinality: (HJ~)
There is a collection {f a | a < K + } C_ K 2 so that whenever p < K ~ and s: K~xp -> K
is injective
and <J>: K""xp -> 2, we have |
|
(f ,
T>
(5) 7^ *(a,T))}|| < K.
(Actually, they had a further condition on {f | a < K } to insure good separation properties for the space constructed, but this is the crux of the matter.) HJ~ immediately implies P ; that HJ~ follows from HJ is not unexpected, and details are provided in [Ka3]. The proof of the following theorem is also given in full in [Ka3]; it can be culled [HJ1][HJ2] and especially [Ju] where a simpler conclusion is derived. Theorem 3
(Silver):
2
= K" and Vtf + HJ .
§3. BURGESS1 PRINCIPLE Although we saw in §1 that the Hajnal-Mate Principle follows from Prikry's Principle, Burgess [Bu2] originally established the Hajnal-Mate Principle in L from a more complicated principle.
The following,
asserting the existence of what he calls quagmires, can be dubbed Burgessf Principle.
Here, the notation
for K-Kurepa trees developed in §2 is still in effect.
182
(B )
There i s a quagmire, i . e .
a K-Kurepa tree T with
tree ordering <, equipped with a binary r e l a t i o n
and a ternary function Q such t h a t :
(1) y <3 x implies t h a t x and y are d i s t i n c t e l e ments on the same l e v e l , and
<* linearly
orders every l e v e l . (2) Q i s defined on t r i p l e s
(3) (Conmutativity) If y o x < x1 < x, then Q ( Q ( y , x , x f ) ,xf ,x)
= Q(y,x,x).
(4) (Coherence) If z o y « x < x, then Q(z,y,Q(y,x,x)) = Q(z,x,x). (5) (Completeness) If y o x e T ,then for some 5 < K , TT^(y) « TT.(X) and Q (TT^ (y) ,TT- (x) ,x) = y. The reader familiar with morasses will already see a growing resemblance, and as with morasses, he or she is advised to draw pictures to get the picture. B may seem a bit ad hoc, but it is really the next natural rung in the evolutionary ladder toward a morass.
Whereas W merely hypothesized K cofinal
branches and a system of approximations by K ~ size subsets, B endows a linear order on these branches which is moreover reflected in the <* orderings through the previous levels.
Thus, B incorporates an impor-
tant feature of morasses; the main ingredient which
183 must still be added to get the full structure of a morass is the limit continuity across levels.
Burgess
[Bu2] established the following result: Theorem 4 (Burgess): |-
If 2 K = K and B , then W .
The idea here is first to enumerate the powerset
P(T^) as {X- | p < K ) for each £ < K , using 2 K = K. For x e T
and £,p < y let S(5,p,x) =
{Q(y/7r r (x) ,x) | y < 7r r (x)
and y e Xr } .
Finally, for
y < K s e t W(y) = { S ( £ , p , x ) | £ , p < y and x e T } .
Then
t h i s f u n c t i o n W w o r k s , c a p t u r i n g more and more of t h e images of t h e K~ s i z e s u b s e t s of t h e t o p l e v e l as we move t o t h e r i g h t a l o n g o and upwards a l o n g < : The Completeness c o n d i t i o n
(5) i m p l i e s t h a t any
x € T h a s a t most K <»-predecessors. s e [T ]
Hence, given any
, t h e r e i s an x e T such t h a t y <3 x f o r
every y £ s .
Again by C o m p l e t e n e s s ,
f o r each y e s
t h e r e i s a £ (y) < K such t h a t Q(7rr , , (y) ,ir r , , (x) ,x) = y . K \Y) s vy; S e t 6 = sup{£(y) | y £ s } . Then an e a s y a p p l i c a t i o n of Commutativity shows t h a t f o r any g w i t h 6 < 5 < K and y £ s , we have TJV (y) « 7Tr(x) Finally,
a n d
Q (TT^ (y) ,TT^ (x) ,x)
l e t p be such t h a t X^ = ^ " s .
5 ^ m a x ( 6 , p ) , t h e above arguments confirm
=y.
Then f o r any that
7r "s = S(6,p,TT£ (x) ) £ W(5) / c o m p l e t i n g t h e p r o o f .
-\
Just as Silver f s Principle establishes Prikry*s Principle, Burgess1 Principle establishes an extended
184
Prikry's Principle, f i r s t
formulated by Rebholz [Re]
soon after morasses f i r s t
saw the light of day.
Rebholz' Principle i s the following: (R )
There i s a collection {f
| a < K } of
functions
with f : a -> a, so that whenever s e CK ]
and
< > | i s a regressive function with domain = s (i.e.
<>| (a) < a for a e s) , then }|
< K.
A , although 2 K = K seems to be
Using morasses (and
sufficient by a more complicated proof), Rebholz established that if V = L, then R holds for every successor cardinal K.
Clearly, R is equivalent to
the following principle if we compose each f with a bijection a <—» K for K < a < K : (R1)
There is a collection {g | a < K } of functions
K.
CX
+ K"~ with g : a •> K , so that whenever s e C K ] and
<(>: s -> K , then | {£ < f\ s | Votes (g (?) 7* <(> (a) ) } | < K . Also, it is easy to see, by considering {g [k \a < K }, that: R
-> K
P , K
Finally, like P , R has a consequence in the partition K
K
2 calculus. ever with
f:
To b e p r e c i s e , C X ] 2 -> y , t h e r e
Ux
X •* C y : v ]
means
i s a n X e CA]y
< O Y , and f"{<£fC>|
that
when-
a n d a Y e CX]V
5 € X a n d ? e Y} 7^ y .
185
2 Note t h a t the negation X /
[y:v]
2 implies X / - CvH-v] ,
and so provides a strong counterexample to the ordinary p a r t i t i o n symbol.
Rebholz formulated his principle
with the following immediate consequence in mind:
(If
{g I a > K } i s as provided by R1,
set
F(3/Ot) = g (3) for g > a to get a counterexample.) The conclusion here i s stronger than the negative polarized p a r t i t i o n relation entailed by P , and the difference
i s revealing: there, each f
need only be:
K -> K, and here, we must have an elongated f : a -> a. Incidentally, result, If
K
t h i s i s the best possible
limitative
since Shelah [S] established in ZFC + GCH t h a t :
> oo i s regular and y
<
K
, then
K
->•
(K+Y) 2 -
Rebholz [Re] also provides an application of R
to the
theory of free subsets for set mappings, answering a question of Mcite. The following derivation highlights the
lateral
approximations provided by o in B . Theorem 5 (Kanamori) : (2K |-
= K and B )•> R •
As mentioned in the proof of Theorem 4, by the
Completeness condition, any x e T o-predecessors.
Hence, as T
has a t most K
has cardinality > K, i t
must have a subset well-ordered by < in order-type K .
186
So,
by renaming and trimming, we might as w e l l assume
further
that:
(6)
T
= K , and for a , g < K , we have a <J g
K.
iff
a < g.
To prove the theorem, i t more t r a c t a b l e R 1 . W satisfying
S
= K, for each 6 < K and s e W(S),
K as {hf. | 6 < £ < K }, and define
g^: T>. -> K s a t i s f y i n g
the condition
j u s t as in the proof of Theorem 2 . define
f : a -*- K for a < K as
If
the
the clauses of S i l v e r ' s P r i n c i p l e for our
we can enumerate functions
to establish
By Theorem 4, t h e r e i s a function
Thus, by 2K
t r e e T.
suffices
Finally,
(t) , l e t us
follows:
C < a, by Completeness, t h e r e i s a p such t h a t
IT (C) < TT ( a ) .
Let p
be the l e a s t such p, and s e t
This definition underscores the importance of <; once an ordering of the top nodes is established, < reflects and completely approximates this ordering through the lower levels. To verify R 1 , let s e [ K ]
K
and (j) e S K .
There
is a 6 < K such that: (a) 6 < 5 < K implies TT>."S € W ( £ ) / and (b)
6 < 5 < Kand a < 3 e s implies a n d Q ( T T . ( a ) ,TT^ ( g ) , B ) = a .
i i y ( a ) o TT £ ( 3 )
187
H e r e , we c a n a c c o m p l i s h
(b) u s i n g Completeness and
C o m m u t a t i v i t y much t h e same a s i n t h e p r o o f
of Theorem
4.
6 < £ < K ,
N o t i c e t h e f o l l o w i n g FACT: I f C < H s ,
and t h e r e i s an a e s such t h a t
TIV(C)
<
TT>.
(a) , t h e n
f o r e v e r y g e s we a l s o have Trr(C) <* TTr (3) . T h i s i s ^ ^ s o by t r a n s i t i v i t y of
(4) i f
(3 < a . 7T
6" S *
Now f o r some n > 6 , we h a v e h E = (C < ^ s |
3aes3^
Q(7Tr (C) /TT^ (a) ,a)
(TT (C) < TT^ (a)
= C)}.
of t h e e x c e p t i o n a l
Clearly
-1 =
.
<J>»7TJS
and
| E | < K; E c o n s i s t s
ordinals:
Suppose t h a t C e (f^s
- E).
By Completeness and
t h e FACT, t h e r e i s a f i x e d p with n < p < K such p = p
for every a e s
(where p
course of t h e d e f i n i t i o n
of f ) .
t h e proof as i n Theorem 2 . g's,
Set
was defined in
that the
Now we can complete
By c o n d i t i o n
(t)
on the
t h e r e i s an x e IT "S such t h a t g (x) =
7T " S
h
But i f a =
(TT.(X)).
f
(C) = g (x) = (f)(a) Ot
TT
(X)
e s,
. This e s t a b l i s h e s
P
then R'.
-|
K
The following diagram summarizes the implications in the first three sections assuming the GCH:
R
K K
^
HJ HJ" I K P
K
188
I do not know whether any converses are true.
§4. GENERALIZATIONS This section considers versions of the various combinatorial principles also available at limit cardinals; perhaps the main interest in these generalizations lies in the consequent limitative results in the partition calculus which counterpoint the positive results available from large cardinals.
So, let me
provide the backdrop of historical context, first of all for the polarized partition relation. In general, the proposition K
K
K
K
(*)
seem to hold but rarely.
The earliest result along
these lines was due to Erdtts and Rado [ER] Theorem 48, who established (*) for K = oo. Hajnal [H] then established (*) for K a measurable cardinal; see also Chudnovsky [C] and Kanamori [Kal] for some refinements. Chudnovsky claims without proof in his paper that (*) holds for K a weakly compact cardinal, and proofs have since been provided by Wolfsdorf [Wo], Shelah, and Kanamori [Ka2]. For successor cardinals K, we saw in §1 that P denies (*) in strong fashion.
Unpublished work of
189
Laver [L] provides a p o s i t i v e consistency r e s u l t : Say t h a t a n o n - t r i v i a l ideal over a regular cardinal K > oo i s a Laver i d e a l i f f whenever X £ P(K) .
I
+
K
|X| > K , there is a Y e [X] Z e [y]
, then
O z / I,
- I with
+
so that: whenever
Notice that any measure
over a measurable cardinal K is dual to a Laver ideal over K.
Laver noted that the existence of a Laver
ideal implies (*) (where the 2 can be replaced by any ordinal < K ) .
Refining an argument of Kunen [Kul], he
then established the relative consistency of a), carrying a Laver ideal, by forcing over a ground model satisfying ZFC and a strong large cardinal hypothesis, the existence of a huge cardinal. The study of the even rarer (**)
K
-> (a)o ^ o r
also has a rambling history.
ever
Y ot < K
After years of partial
results and conjectures, Baumgartner and Hajnal [BH] established (**) for K = co, as a consequence of a more general result which they established in elegant fashion by using Martin's Axiom and an absoluteness argument.
Avoiding these tricks of the trade, Galvin
EG] provided a direct proof which is a combinatorial tour de force.
More recently, Tedorcevic has announced
further refinements.
It is not known whether (**)
holds for K a measurable cardinal; perhaps the best
190
p a r t i a l r e s u l t i s due to Laver.
He was f i r s t
to ob-
serve t h a t i f there i s a Laver ideal over K, then K
-> (K+K+l,a)2 for every a < K .
I established
Laver's r e s u l t without knowing of i t , i s provided in [Ka4].
and a full proof
Also in [Ka4] i s the r e s u l t
if
K i s a weakly compact cardinal, then
K
-> (K+K+1, [K :K
])2'
a
+ stronger than
K
technical statement somewhat 2
->
the known r e l a t i o n
[KIKD^J
(*)
which already follows
from
for weakly compact c a r d i n a l s .
For successor c a r d i n a l s , we saw in §3 t h a t R (**)
that
in strong fashion.
In the positive
denies
direction,
there i s again Laver's r e s u l t about a Laver ideal over 0).. , s t a r t i n g with the consistency strength of a huge cardinal.
Gray also has some p a r t i a l positive
Turning to the subject a t hand, j u s t as P R
results. and
deny p a r t i t i o n r e l a t i o n s for successor cardinals,
there are weaker versions which delimit the s i t u a t i o n for possibly limit c a r d i n a l s . (wP )
There i s a collection whenever s € CK ] | U < K| Va£s(f a (£)
(wR )
{f
| a < K } ^
and <J> e
S
K S O that
K , we have
* (f>(a))}| < K.
There i s a collection
{f
| a < K } of
functions
+ K
f : a ->• a so that whenever s e C K 3 and <j> is a regressive function with domain = s, then
|
0 l
f a ( £ ) ^
K.
191
There are versions of the other combinatorial principles with the requisite strength (one is stated for W in [Ka3]), but to discuss them would take us somewhat afield.
In direct analogy to previous results, we
have: K K
I established [Ka2][Ka4] the consistency of these propositions via a forcing which does not (too much) disturb the universe; e.g. for the stronger wR , Theorem 6 (Kanamori); If the ground model satisfies K
= K, then there is a K - C . C ,
forcing extension in which wR
holds.
(Furthermore,
properties like the Mahloness of K are preserved.) The proof involves a new and elegant kind of density argument, first seen in the work of Shelah.
By
itself, this is a piecemeal result, and to genuinely contrast the positive partition relations from large cardinals, the actual situation in L must be ascertained.
Recent and continuing work of Shelah and
Stanley CSS1][SS2] and Velleman [V] have made the formidable apparatus of the (K,1)-morass (a gap-1 morass at K) more tractable (at least for some) by providing a Martinfs Axiom-type characterization.
That is,
192
certain partial orders and collections of dense sets are described, and the existence of a morass is shown to be equivalent to the proposition that for every such partial order and every such collection F of dense sets, there is an F-generic filter in the usual sense.
The partial order used in the proof of Theorem
6 is a paradigm case of a canonical limit partial order, in the sense of Shelah and Stanley.
It was to handle
such orders that led Shelah and Stanley to extend their characterization of morasses.
They show how
canonical limit partial orders can be accommodated in a Martin's Axiom-type ^characterization for ( K , 1 ) morasses "with built-in 0 principle", when there is a non-reflecting stationary subset of K , i.e. an S £ K which is stationary in K yet Sf\ a is not stationary in a for any a < K.
They establish that such
morasses with built-in A principle exist in L, and, of course, it is a well-known result of Jensen [Je] that in L, a regular K > a) is not weakly compact iff there is a non-reflecting stationary subset of K. Velleman is also developing a scheme along similar lines, but with a more concise formulation.
Assuming
that the partial order used in Theorem 6 fits into either the Shelah-Stanley or Velleman scheme in its final form, we have the following characterization of
193
weak compactness in L: Theorem 7:
If V = L, then the following are equiva-
lent for regular K > oo: (i) K is not weakly compact (ii) wR (iii) wP,^
(iv) (v)
K+ fi EK:K]^
This i s the heralded counterpoint to large c a r d i n a l s .
REFERENCES [BSV]
B. Balcar, P. Simon & P. Vojtas, Refinement properties and extending of filters, to appear.
[BH]
J. Baumgartner & A, Hajnal, A proof (involving Martin's Axiom) of a partition relation, Fund. Math. 78(1973), 193-203.
[Bui]
J. Burgess, Forcing, in: J. Barwise, ed., Handbook of Mathematical Logic, North Holland (Amsterdam 1977), 403-452.
[Bu2]
J. Burgess, On a set-mapping problem of Hajnal and Mate, Acta Sci. Math. 41(1979), 283-288.
[C]
G. Chudnovsky, Combinatorial properties of compact cardinals, in: Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai 10, North Holland (Amsterdam 1975), 289-306.
[D]
K. Devlin, On hereditarily separable Hausdorff spaces in the constructible universe, Fund. Math. 82(1974), 1-10.
194
[EHR]
P. Erdtts, A. Hajnal & R. Rado, Partition relations for cardinal numbers, Acta Math. Acad, Sci. Hungar. 16(1965), 93-196.
[ER]
P. Erdtts & R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62(1956), 497-dftQ. 427-489.
EG]
F . G a l v i n , On a p a r t i t i o n t h e o r e m o f Baumgartner and Hajnal, in: Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai 10, North Holland (Amsterdam 19 75) 711-729.
EGH]
F. Galvin & A. Hajnal, I n e q u a l i t i e s for c a r dinal powers, Annals Math. 101(1975), 491498.
[H]
A. Hajnal, On some combinatorial problems i n volving large c a r d i n a l s , Fund. Math. 69(1970), 39-53.
EHJl]
A. Hajnal & I . Juhasz, A consistency r e s u l t concerning h e r e d i t a r i l y a-separable spaces, Akademie van Wetenschappen Amsterdam Proceedings A76 (or Indag. Math. 35)(1973), 301-307.
[HJ2]
A. Hajnal & I . Juhasz, A consistency r e s u l t concerning h e r e d i t a r i l y a-Lindelttf spaces, Acta Math. Acad. S c i . Hungar. 24(1973), 307-312.
EHM]
A. Hajnal & A. Mate, Set mappings, p a r t i t i o n s , and chromatic numbers, i n : H. Rose & J . Sheperdson, e d s . , Logic Colloquium ' 7 3 , North Holland (Amsterdam 19 75).
EJe]
R. Jensen, The fine s t r u c t u r e of the construct i b l e h i e r a r c h y , Annals Math. Logic 4(1972), 229-308.
EJu]
I . Juhasz, Consistency r e s u l t s in topology, i n : J . Barwise, e d . , Handbook of Mathematical Logic, North Holland (Amsterdam 1977), 503-
EKal]
A. Kanamori, Some combinatorics involving u l t r a f i l t e r s , Fund. Math. 100(1978), 145-155.
195
[Ka2]
A. Kanamori, Morass-level combinatorial p r i n c i p l e s , t o appear in the proceedings of the Patras 1980 conference.
[Ka3]
A, Kanamori, On S i l v e r ' s and r e l a t e d p r i n c i p l e s , t o appear in the proceedings of the Prague 1980 conference.
[Ka4]
A. Kanamori, P a r t i t i o n s of K , to appear.
[Kul]
K. Kunen, Saturated i d e a l s , Jour. Sym. Logic 43(1978), 65-76.
[Ku2]
K. Kunen, Strong S and L spaces under MA, i n : G. Reed, e d . , Set-Theoretic Topology, Academic Press (New York 1977), 265-268.
[L]
R. Laver, Strong s a t u r a t i o n p r o p e r t i e s of i d e a l s ( a b s t r a c t ) , Jour. Sym. Logic 43(1978), 371.
[P]
K. P r i k r y , On a problem of Erdtts, Hajnal and Rado, Discrete Math. 2(1972), 51-59.
[Re]
J . Rebholz, Consequences of morass and diamond, Annals Math. Logic 7(1974), 361385.
[Ru]
M.E. Rudin, Lectures in Set Theoretic Topology, Regional Conference Series in .Mathematics 23, A.M.S. (Providence RI 1975) .
[S]
S. Shelah, Notes on combinatorial s e t theory, I s r a e l Jour. Math. 14(1973), 262-277.
[SSI]
S. Shelah & L. Stanley, S-Forcing, I : a "black box11 theorem for morasses, with a p p l i c a t i o n s : super-Souslin t r e e s and gener a l i z e d Martin's Axiom, t o appear in I s r a e l Jour. Math.
[SS2]
S. Shelah & L. Stanley, S-Forcing, appear.
[Sz]
A. Szymanski, Some Baire Category type theorems for U(o)]_), Commentationes Mathematicae U n i v e r s i t a t i s Carolinae 20(3)(1979), 519-528.
II,
to
196
[V]
D. Velleman, Doctoral dissertation, University of Wisconsin--Madison, 19 80. To appear as a Springer Lecture Notes in Mathematics.
CWi]
N. Williams, Combinatorial Set Theory, North Holland (Amsterdam 19 77).
[Wo]
K. Wolfsdorf, Der beweis eines satzes von G. Chudnovsky, Archiv Math. Logik 20(1980), 161171.
A SHORT COURSE ON GAP-ONE MORASSES WITH A REVIEW OF THE FINE STRUCTURE OF L.
Lee Stanley Maths. Pures - Les Cezeaux Universite de Clermont II 63170 Aubi^re, France.
§1.
Introduction This paper is based on a series of lectures given at the
Cambridge Summer School in Set Theory in 1978.
Those lectures
were primarily devoted to gap-one morasses, but it seemed reasonable in the context of this paper to review the fine structure of L and collect together the various remarks and modifications circulating in the folklore which together constitute a serious rationalization of this theory as it first appeared in [6] or [2J.
1 should emphasize that there is little or nothing here
which is not already implicit in Jensen's original treatment, and that most of the cognoscenti have probably worked out similar improvements in private.
Finally, it's worth noting that the
approach taken here becomes practically indispensable when dealing with models other than L; in fact, Jensen and Dodd chose exactly this approach in [5 3; my exposition owes much to their treatment. My approach to morasses owes much to Jensen, of course, but also to Devlin, Silver and Burgess.
A notable feature is the
early introduction of a less complex structure, the coarse morass, which in a very real sense embodies the condensation arguments needed to prove combinatorial principles like ^
and
198 0
and which, like these principles, does not need the fine
structure theory.
As the title indicates, I shall only be con-
crened with gap-one morasses, thus, unless otherwise indicated "morass" means "gap-one morass".
This material has also appeared
in [2], and is introduced rather sketchily in Chapter 1 of [9]. The proof outlined in [9] that morasses exist in L will be fully developed in this paper.
It avoids the unnecessary computations
involved in proving {Q?) (cf. [2]), it brings out more clearly the way the morass properties arise as abstractions from the situation in L and thus as abstractions from the fine structure theory of L.
This proof has the added advantage of generalizing
relatively straightforwardly to
higher-gap morasses.
Jensen had a proof along these lines no later than 1972; the proof given was worked out in the course of an informal seminar on morasses in Berkeley in early 1972.
The basis "map-
splitting" lemma, (2.11), attributed to Burgess in [9] should probably be jointly attributed to Jensen and Burgess; the proof is new, more in the spirit of certain arguments of [4]. More recently, Silver and Richardson have developed an alternative construction of morasses in L via Silver's machines. The approach taken to adding morasses by forcing also was developed by Jensen and antedates 1972.
It is essentially that
presented in [9], in that the generic object is a structure simpler than a morass (Jensen called it a premorass) but which can be thinned in the generic extension to yield a morass.
In
the meantime a much simpler set of forcing conditions has been worked out by D. Velleman [10] who has kindly permitted me to present them here.
It seems unlikely however that there is a
generalization of Velleman-style forcing for higher gap morasses. I work in ZFC, except in §3 where, when working in L, I temporarily assume V = L for convenience.
Notation, terminology
etc. which is not standard is either intended to have a clear meaning (e.g. card (x) for cardinal of x, o.t.(x) for order type
199 of x) , is to be found in [6], or [2], or is introduced explicitly as needed.
I suppose the reader is familiar at least with §§1,2
of [6] (see the discussion of the beginning of §2). §2 is devoted to the fine structure theory of L; § 3 to morasses and the proof that they exist in L.
In §4 I give some combinatorial
applications of morasses; for others, see [l],[7] and the forthcoming [8], [lo] where it is proved that the existence of morasses is equivalent to certain Martin's Axiom-type principles, thus providing
a new and very systematic way of applying morasses which
is exploited in [8], In §5 I show how morasses can be added by forcing. §2
THE FINE STRUCTURE OF L, NOW THAT THE DUST HAS CLEARED. My goal in this section is to rework the theory of projecta
and standard codes as presented in §§3,4 of [6], My point of departure is §§1,2 of [6] which remains an excellent source for the basic properties of rudimentary functions, the J and the auxiliary S
hierarchy.
hierarchy
Since this paper does not pre-
tend to be self-contained, I shall assume knowledge of this material.
Nevertheless, before getting under way it seems
reasonable to point out a number of minor improvements and/or corrections which have been made in the meantime, and to summarize some particularly important points. First, the J 's are not only transitive, but closed for transitive closure; this is proved in a more general setting in [5].
Second, by slightly modifying the finite basis for the
rudimentary functions which gives the definition of the S 's, the S 's may actually be taken to be transitive.
Third, for a
correct account of the exact relationship between the J hierarchy and the L -hierarchy the reader should consult [3]; it will suffice here to recall that the subsets of J
in J
are
just those which are parameter-definable over J , and that if a = a).a then J
= L . Finally, I'll recall a number of facts
which figure prominently in all that follows.
200 a) For all a, all 3 < a, all n < u>, (S : n < a).3+n) ,
n
(J : y < 3) , (< Is : n < 0).3+n) £ J . Y J ri a b) There are Z formulas ip(£,a), ^'(f^a) which hold, respectively, iff (a e OR and f = (S : n < a)) and iff (a e OR and f = (< Is : n < a) . J 1 X]
c) For all a, (S : n < a).a), (Jo: 3 < a) , (< |s : n < a).a) , rj P J TI < |j are uniformly Z.. (J ) . d) There is a IU sentence 9 such that for transitive rudimentarily closed M, M f= 9 iff for some a, M = J . 9 can be taken to be e.g.: V3 3fi|>(£,B)AVx 3f 3 Y > 3 [i|;(f ,3) Axef (Y) 3. In what follows I'll designate 9 by "I am a J ". a e) The Z satisfaction relation \=
is uniformly I (M,A)
for transitive rudimentarily-closed amenable structures (M,A); further, uniformly Z uniformisation holds for such structures, when M is a J , and h ,, the canonical a a,A I^-Skolen function for (J ,A) is the uniform I. uniformisation of \= mention of it.
h
. When A = 0, I suppress yields minimal I,-elementary
substructures of (J ,A) when applied to sets of the form
CJXBX{X},
whenever Bu{x} c^ J
and B is closed for
some reasonable pairing function which, together with its
coordinate-wise inverses is I (J ,A) definable
in parameter x.
The range of h ^ on a,A
COXBX{X}
^S
just the set of denotations of Z -terms of (J ,A) in parameters from Bu{x}.
For non-amenable (J ,A) the Z,-
Skolem function can also be defined with all these properties except the Z -definability will Condensation Lem ma states that if Xover • < (J J ,A); then hfor some 1
Ot
Ot, A
also denote the Z^Skolem function in this setting.
The
201 3/X
= Jo.
For all a, there is a finite set of ordinals
P
a £ J map
such that J
then for
x
e dom
= h^ (u)Xu).ax{a}) .
If IT is a collapsing
TTDL TT (X) ^ T X , TT CJXI ^ T X .
I can now get started on the fine structure theory of L, (2-1) DEFINITION:
Let (J ,A) be amenable.
The £ -projecturn of
a and A, p , is the least p < a such that: a,A (*) :
There is a c_ j
such that anJ
\ J
and which is
E ~(J ,A) parameter definable.
, the Zn-standard parameter for a,A is the < -least p e J
p Ot , A
u
JL
OL
such that some a as in (*) is L (J ,A) in p. A^ , the 2 l a a,A l standard code for a,A, is [= nJ , where p = p ; that (J ,A,p) is, (i,x) 6 A
p
a,A
iff (j f A) [=
(<(>.: i < a)) is some reasonable fixed Gode"l-numbering of £,formulas.
(J ,A) is 1-sound iff J Ot
= h
" (o)Xo).px{p}) , i.e. iff
Ot / A
Ot
all elements of J
are Z -definable in parameters from co.pufp}. 1 1 1 In keeping with the above convention (e), I write p ,p ,A , 11 J is 1-sound" if A = 0. (2.2) Here is an example of amenable (J fA) which is not 1-sound. a
Let x £ P(u))\L be such that
S
=
S
r
and suppose
V = L[x]. Let a = GO +1, A = {oj^^+n: n e x } . Clearly (J ,A) is amenable, x is H_ (J ,A) in parameter co . Hence o = 1 and so l a 1 a,A (J ,A) is not 1-sound. In what follows, I do not always give the most compact proof possible if I feel that it would mask "what is going on". On the other hand, in order to develop the theory as harmoniously as possible, I've had no qualms about making heavy use of the Condensation Lemma even when it's not strictly necessary.
This
has the drawback, for the reader interested in models other than L, that it is not always exactly clear what generalizes and what does not, but forewarned is forearmed.
202 (2.3) PROPOSITION: 1
a) If X] < a and J
x € J , then there is a c_ j Z^ (J ,A) in x; hence n ^ P b) If p 1
.. (=a).p Ot
I f n < OJ .p
a
/xc^s
Ot , A
p = p
p
is a cardinal. Ot , A
then x e J T)
) ; hence (J ,A
Ot j A
" (o)Xa).nx{x}) where
Ot , A
such that anw.n ^ J / which is
< ot then J
Ot , A
c)
= h
Ot
••
iff x e J Ot
(where P
) is amenable, and more generally if Ot / A
x c_ S , T\ < oj.p, and x is £.. (J ,A) parameter-definable then x 6 J . P Proof; a) (this is essentially the argument of example (2) of p.256 of [6]). First, define, for £ < a>.a7(£) = the largest limit ordinal < £, and let U J = £-(£) . If U J = o, set (£) o = o, (S) x = (£); otherwise, let i,j be such that [£] = 2 1 . (2j+l) and set ( ^ ) Q = i, (O ± = (C)+J.
Then g(O = ((^)Q,(C)1)
is S (J ,A) (in fact, Z (J ) , in fact A (J ) ) , and for Y ^ a, g: aj.y -> ajxw.y i s a b i j e c t i o n . Set C e a i f f ^ € OR , h a / A ( ( C ) o ' (C) , , x ) i s defined and £ ^ h ((£) ,(£).. *x ) . C l e a r l y a i s as J_
Ot, A
required.
O
J.
The second assertion is clear from the first.
b) Suppose p = p _, n < P / f: w . n ^ c j . p , f £ j . Let a, A ot a £ P(o).p)\J be I (j ,A) in p . Then, since n < P / Ot
JL Ot
Ot, A
f""1Ca] £ J , because f" 1 [a] is E. (J ,A) in (p ,,/f). Ot
-L Ot
Since
Ot, A
f £ J ,f"f" 1 Ca] £ J , so, if f were onto f f ^ C a ] would be a, and would lie in J , which is absurd. c) (WARNING: CONDENSATION LEMMA)
Let x £ S and let 03.y
be the largest limit ordinal < n. Let Y = h " (ojxu).yx{x}) . Thus S
£ Y -^
J
i and
so
setting
TT:
J- -—> Y,
TT"1(.X)
= x. Since
J- = h_" (ajxa).yx{x}) , by a) a < cj.p: i.e. x e J . The remaining assertions are now clear. (2.4) PROPOSITION: If 1 < n < a) and x c j is I (J ,A ) then p ~n p a, A
203 Proof:
(this is Lemma 3.2 of [6]; the proof is the same as
there)
Let B = A
. It clearly suffices to show that if x c j
is £ o (J f B) then x is ^ ( J ^ A ) .
For this it suffices to show
that b(u) = Bnu is EO(J ,A) . There are two cases. ~z ex CASE 1:
There are Y < wp, g: y •> ^. , oia such that g is ' ^ ' cofmal * £, (J , A ) . In this case Jensen's trick is first to replace B by a B 1 which Is A (J ,A) and such that B is E Q (J , B ' ) . Thus, if x is I. (J , B ) , x is Z (J , B f ) , so it suffices to prove that b'(u) = B f nu is £ 2 (J ,A). This is easy since y = b 1 (u) <—> (Vz)(z£y <—> zeu A zeB 1 ).
The matrix is A (J ,A) and so
y = b'(u) is in fact IL^J ,A) . It remains to find B 1 with the desired properties. (9z € S
Set (v,y) e B 1 iff v < y and
^((Z/Y) € B) , where B is E (J ,A) such that
B = {yeJ : (-9zeJ )(z/y)£§)}. B 1 is A (J ,A) because the definition can be written out as: v < y and (•35) (3w) ("5 = g(v)"A"w = S "A(^zew) ((z,y)eB)) , or v < y and (Vw) (("C = g(v)"A" w = S ") = >
(^Z£W) C(z,y)£B)) .
Finally, B = {y: (3v < y)((vfy) e B')}. CASE 2:
There are no such y,g.
In this case we have a
replacement-like property for Z (J ,A) relations and elements of J : P If Rzy is Z (J ,A) then so is R'wy where R'wy <—> w e J ~1 ot p and (Vz£w)Rzy. This is because, roughly speaking, the hypothesis (that there are no such y>g) guarantees that there is a bound v < wet such that if Rzy is -3u Ruzy where R is £ CJ /A) , then for all z £ w, ^u Ruzy <—> (^u e S )Ruzy. of R
1
The E -definition
comes from existentially quantifying over such bounds v
and bounding the existential quantifier on u by S . But then
204 1
2
y = b(u) <—> (Vx€y) (xeuAx^B) A (Vxeu) (xeB —> xey) Now 2 is n
and, by the preceding, 1 is £ CJ /A). A Boolean
combination of n
and £
is £ (J , A ) . Note that this proof works
for any B which is I (J #A) . (2.5) COROLLARY:
If (J ,A) is 1-sound then for all x c j , all a — p positive n < a), x is I (J ,B) iff x is £ n + 1 (J *A1
i Proof:
<
n
By (.2.4), and by easy arguments as implicit in the proof
of (2.4), it suffices to prove that if x
c
x is I (J ,B) . So, suppose y •= (y ,.../Y
J
is I (J ,Al, then ) £ J , j) is L
such
that x = {zeJ : (J ,A) |=(|>[z,y]}. By soundness, for j < m there are i
< a>, £j < co.p such that y
Accordingly, letting 8 be the £
=
h a
A ^ J ' £ j ' P } CP = Pa,A^ *
formula in parameters j = h a,A C ± j ' E j ' p ) "A
*Cz,yo, .. .,yn__1)) , x = {zeJ :(Ja,A) |= 9 (z,i,f ,pl } . Letting k be such that 6 is the k t h £ -formula, x = {z: Ck,z,i,5) e B>. Thus, x is 2 (J ,B) in parameters k,i,£. op (2.6) PROPOSITION: • •—•
Let (J ,B) be amenable. There is at most one p
(J ,A) which is amenable, 1-sound and such that p = p ,B = A ot a, A 0^, / Suppose that (J ,A),(J ^ A 1 ) both have these properties. a a Let 8(x,i,y,z) express "x = h _.(i,y,z)" when interpreted in YJ,E amenable rudimentarily closed (X, E) . Let j be such that the th ~ Proof:
j — ^-formula is ^x6 (x, i,y,z) . Let M = { (i,y) : (j ,i,y) e B> . Let k be such that the k — E -formula is '(9(x,i,y,z) A 9(x',if,yf,z) A x = x 1 ) . Let R = {((i,y),(i',y')):(k,i,y,i',y') e B } . Let £ be such that the I—
I. -formula is ^x^x • (9 (x>i,y,z) A9 (X 1 ,i' ,yf ,z) Axex ') . Let
E = {((i,y),(i',y')):(£,i,y,if,y') e B>.
Let m be such that the
205 m—
I -formula is -3x (8 (x,i,y,z) AL[(X) ) (here U_ is the additional
unary predicate symbol interpreted by A,A 1 respectively). Let U = {(i,y):(m,i,y) € B } . Then, by the hypotheses, R is an equivalence relation on M and is a congruence for E and U; E is a well-founded, extensionalmod-R relation on M and (J ,£,A) = (M/R,E,U) = (J , ,e,A'), the isomorphisms being (i,y) I—> h
A (i/y#p),
and
(i,y) *—> h a ,,A (i ' y ' pI) (p - pJ^nP 1 = P j - / A f ) *
ThuS a
" a''
1
A = A .
(2.7) COROLLARY: p = p
/ P
=
L e t (J ,A) , (J-,A) be amenable and 1-sound. L e t
P- -> o = o
/ "^^
n - n - B = A
/ ^^
fY /\
R - A
rv 7\
Of 7\
ry 7\
Suppose m < GO, f: (J_A ) -> (J A) , f (p) = p and ot, 1 ot, f|j-: p
Proof:
(J-,B) p
•> (J ,B) . m p
Then f:
(J-,A) a
If m = o, there is nothing to prove, so suppose m > o.
Let M = (M/R,E,U) be the model constructed in (2.6) for (J , A ) , and let U be the analogous model constructed for (J_,A).
Then
h «// 1= n are £ (J ,B) , I (J_,B) by the same £ -formula. Accor1 M 'M op op o n+1 i n+1 r, ki t r n are L (J ,B) , I (J_,B) resM M n p n p
H
pectively by the same £
formula.
As M = (J ,A) , M = (J_,A) and
the isomorphisms are E (J ,A) in p, E^(J-,A) in p by the same £ -formula, and since f: (J-,A) •>1 (J ,A) , it is easy to see that f: (J_,A) •> . (J ,A) . a m+1 a (2.8) DEFINITION: • " • • •' " •' * •
Set p° = 3, p? = A Q = -0 and say that J_ is p
p
p
P
o-sound for all 3- For n < oo, having defined p*?, p^, A^J and the P P 3 notion of n-soundness of 3/ set a = p o , A = A o , and define P
p^?+1 = p 1 , p ^ + 1 = p 1 , A n 3 a,A -^3 a,A 3
P
= A 1 , and say that 3 is n+1ot,A
sound iff 3 is n-sound and (J ,A) is 1-sound.
Note that this is
coherent definition since, inductively using (2.3), (J ,A) is
206 amenable and that for n = o the above definitions coincide with those of (2.1). The next lemma is the basic fine structure lemma, and the Condensation Lemma plays a central role. (2.7).
The proof uses (2.3)-
The essential construction is that of STEPS 1,2 below,
the rest being essentially checking details. (2.9) LEMMA:
For all g, for all n < CD, setting a = p£, A = A^: p pa) For 1 < m < a) and x c j , x is Z (J ,A) iff x is — a ~m a Z (Jr.); if f ° r some n < ai.a, x —<= sri then x e Ja iff ~m+n p b) a is the least ordinal a 1 < g such that there is a € P(o)a')\JpQ which is ~n Z (jp ) , p£ is the <j -least p e JpQ p for which there is such an a which is Z (jo) in p. n p Further, there is a map g from a subset of wa onto J P
which is Z (J ) if n = o and is Z (J ) if n > o; a is p
*~j_
~n
p
minimal for the existence of such a g, when n > O. c) J o is n+1-sound. P
d) ("Downward Extension of Embeddings"): Let p = p g , B = A
. Suppose that m < ai, that (J-,B) is amenable,
that f: (J-,B) ->- (J ,B) . Then there are unique g,f such P m p ~ _ n+1 — n+1 — that: i) f 3 f, p = p- , B = A * , g is n+1-sound, —
p
p
ii) for k < n+1, let k1 = (n+1)-k, let p = p g , Pi, p
=
n+l
A
= 3 A
' n+l
= B, A O
Proof:
a 'K
= AQ ' K =
*' ?n+l = l' V l
= B].
O
= Ao ' Pv = PQ » Pv =
*• po
=
P5 '
[thus
=
THEN: for k < n+1, f (p. ) = p . , —_——
j^
j£
The proof is by induction on n for all g at once; a) , b)
are trivial for n = o.
STEP 1 is to prove c) for n assuming a) ,
207 b) for n < n and c) , d) for n < n.
STEP 2 is to prove d) for
n assuming a),b),c) for n < n, and d) for n < n.
STEP 3 is to
prove a) for n+1 assuming a)-d) for n < n and STEP 4 is to prove b) for n+1 assuming b),c),d) for n ^ n+1 and a) for n < n. STEP 1; n-sound. p = p^ p
By hypothesis if n > o and by definition if n = o, 3 is It remains to show that (J ,A) is 1-sound. (= p
_) .
Let
If p = a, there's nothing to prove so suppose
Ot, A
p < a.
Let p = p
_, and let X = h Ot , A
" (_u>xu)px{p}) , Thus Ot , A
U)p U {p} £ X, (X,AHX) -*> X (J /A). Let IT: (J-,h) -^-> (X,AnX) , and let TT (p) = p. ThusfJ^, A) is amenable, p ^ ot and J_ = h- -" CX
(o)Xo)px{p}) . Also p <
Ot
p and 7r|wp = id|wp.
such that a is Z (j f A) in p.
By a) for n, a ft J Q .
I.(J-,A) in p by the same E -formula, and a i J-. A = A = 0 and so a e JJ
= h " (a)Xcopx{p}) . CX
Ot , A
Let a e P(cjp)\J
be
Also, a is
If n = o,
. But then a = a since a i J , and so
By (2.3) this means that p <
p so that, in
J
Ot
fact, p = p; i.e. J is 1-sound. If n > o, let n = n+1. By d) for n, let f =>_ TT and 3 be such that f: J- -> . Jo, a = p-r, A = A-r and with the other properIs n+1 p p p ties of d ) . By a) for n and 3 it follows that a is Z _ (J-) 'm+l 3 but a i J-. Accordingly, a e J- . , and since a ft J n , 3 = 3P P+1 P But then a = p- = p o = a, A = A- = A O = A . Thus J = h " _ p P p p a a,A (u)Xa)px{p}) and so as before p ^ p which means p = p. Again this J means that (J ,A) is 1-sound. Let f: (J-,B) •* (J ,B) where p m p (J-,B) is amenable.
Let Y = range f.
and so setting X = h ^
Thus Y is closed for pairs
_ " (o)xyx{p}) , (x,AnX) -< . (J ,A) . Let CX, A
1 Ot
f*: (J-,A) -^-> (X,ADX) . The key technical point is: (*) : X is an €-end-extension of Y, i.e. if y e Y, x e Xny then x e Y.
208 Proof of_ (*): Let y £ Y, x £ Xny. Let x = h where z £ Y.
(i,z,p),
Then, letting j be such that the j — t
formula is
the definition of h _, x is the unique element of J such that a,A ^ a (j,x,i,z) £ B.
Let f(y) = y, f(z) = z.
Since f: (JV,B) -> (J ,B)
and since (J ,B) \= (-3x£y) (Cj ,x,i,z) £ B), which is a I
state-
ment, (J_,B) |= (^x£y) ((j,x,i,z) £ B) . Let x e J- be such that x £ y and (j,x,i,z) e B.
Let x' = f (x) .
Then x 1 £ y,
(j,x',i,z) £ B, so x' = x, i.e. x e Y. Now it's an easy consequence of (*) that f* ^ f, and of course f *: (J-,A) -> (J ,A) . Note that J- = h- -" (o)Xo)px{p}) # where f * (p) - p.
Thus p- - < p, p . - ^ p. Ot , A
Ot , A
'J
Further B - { (i,x) £ j _ : (J-,A) |-<j). (x,p) } . pa i seen a s f o l l o w s .
Let ( i , x )
£ B, x = f ( x ) .
Then (i,x). e B,
that is
(J a /A)
|= ( ) ) i ( x , p ) .
Since f*:
f
(J
l= + i ( X / P ) -
On t h e o t h e r hand, i f
* 2. f '
a'^)
with x £ J_,
(J~,A) •• (J ,A) and
t h e n for t h e same r e a s o n s a s above,
where x = f ( x ) .
But t h e n ( i , x )
This is easily
e B, so ( i , x )
(J-,A) |= (J>i (x,p) (J ,A)
|= §, (x,p) ,
£ B.
Suppose now t h a t ri < p , x e J - , § i s £•, and a c u^x) i s such a -*t h a t a = {y: CJ-fA) 1= <J> (y,x) } . Let x = f * (x) / n - f Cnl . Let -± __ __ _ _ x be h_ - (i,£,p) where i < GO, £ < a).p. Let j be such that the a, A j —
E. formula in i,f,p is: («3x) (x = h_ --(i,C,p) A d>(y,x)). 1 a ,A
Then a = {y: (j,i,CfY) € B } ; i.e. a is such that ({ (j, i,f) }xo).r|) n B = {(j,i,C)}xa.
Since (J_,B) is amenable, a e J_.
This proves:
(**):
p = pi - .
Once it's proved that p = p- - it will follow that B = A- -. I'll a, A a, A />^ prove this at the same time as I construct 3, f. The last step
209 will be to verify that f has the properties of d) and to verify its uniqueness.
The n+1-soundness of $ will then be clear.
If n = o, then a = $ , A = A = 0 . for n = o, J- is 1-sound.
Set $ = a, f = f*.
Accordingly, there is £ < oop, i < oo
such that if q = pi , then p = h-(i,£,5) .
Let q = f*(q), £ = f(?) ;
then p = h (i,£,q), which means that p <_ q. and so p = q = p..
By c)
But then p <
q,
Then f is as required.
Now suppose n > o; let n = n+1.
By d) for n, f*, let f,B
be unique such that f =>_ f*, ot = p-, A = A^ and for k ^ n, f(P v ) = P V ' K
f|j-
K
:
p
= n-k and p
(J- /A.) •* ,n (J p
K
K+±
/A ) where for such k, p,
= p§ , p, = p R , p
K = p-^- , p
= p
,A
= A^ ,
1
k = Ao . So in particular by c) for n, J;r is n+1-sound, i.e. k _ P i e (J-,A) is 1-sound. The proof that p = p- - is then essentially a a,A as above for n = o. Again, this means that p = p- ,
A
P
,1 n+1 B = A- - = A^ . 01, A
, n+1 e.mUt n + l x Thus f * (p^ ) = P D
p
p
p
B y ( 2 . 6 ) , a , A a r e u n i q u e s u c h t h a t 1p = p - - , B
= A--;
Ot , A
uniqueness of f* such that f* =>_ f, f * (p-
-) = p
Ot , A
the
Ot, A
/ Ot, A
f*: (J-,A) -> (J ,A) then follows easily by the 1-soundness of (J-,A) , (J ,A) . By (2.7) f *: (J-,A) -> . (J ,A) . If n = o this ot ot a m+1 a completes the proof of d ) .
If not, let n = n+1.
It follows from
d) for n, f* that, with the above notation, for k < n K
K
k
proof of d ) .
STEP 3:
Suppose l < m < a ) , p = p
, B = A
o
p
x £ J .
O
, a = p _ , A =
P
P
By (2.5) and the 1-soundness of (J ,A) , x is
£ (J ,B) iff x is Z _ (J ,A). If n = o, this is as required. ~m p ~m+l a If not, a) for n guarantees that x is Z _ (J ,A) iff x is ~m+l a
210
W n ( V ' i'e- iffxis W l ( V 'Iffl<wp'^S, and x e J
then by a) for n, x e Ja, and so by (2.3), x e J .
STEP 4:
The first two assertions of b) are immediate from the
definitions of p sound; but h
and p, and from a ) .
By c) for n, (J ,A) is
is £..(J ,A) and so h
is £-,
(Jft) , i.e.
a
E (J ). If n = o and = &, this is as required. If not, b) n+l pD for n guarantees there is f which is £ (J ) and maps a subset of wa onto J_. Let h(i,x) = h ^(i,x,p). Then f°h is I ,. (JD) and p otA ~n+i p maps a subset of cox J onto J . Composing with a I (J ) map a of oj.p onto orxj , foh°a is a map of a subset of oj.p onto J o . P p Since a is E (j ) , a is also L ( J J . That p is minimal for the existence of such a map follows from an easy generalization of the diagonal argument of (2.3)a); this generalization has for hypotheses the existence of g, a map from a subset of oo.ri onto J
which is I
(J ) where n < p / and the fact that p is minimal
with x e P(o).p)\JR which is J + T ( J D ) • This completes the proof of STEP 4 and of (2.9). (2.1O)
It seems in order to tie up a few loose ends and to make
a few remarks about the proof of (2.9) . A. It seems worthwhile to unravel the inductive proof of (2.9) and to give explicitly the definition of the g as guaranteed by b) and the f as guaranteed by d ) . Concerning g, let p , A, , p be as in d) for 3 and for K K k k < n+1, and let h = h (-#-*Pv)• Let a be a Z (J ) map of k p k ,A k K ~l p a subset of w.p onto ayxj . Then g can be taken to be h °...°h °a. Concerning f, let p , A , p , h be as above nO K K K K but for J3 (once J- is constructed!). Then f just makes the diagram commute, i.e. f°hn°...°ho = h n°...°ho °f. J^ore ^ accurately, let X h
k+i"(wxV-
Then
= range f and for k < n, let X, _ -=
'
letting
irs
\+i ^
V
/
Y
=
p
'
211 f |j= TT"1. p k+l B. 1.
The following is a good list of rather easy exercises.
Let n ^ 1, n < Po«
Prove that if f is a map from a subset
p
of o)T] into o)pD and f is E (J_) then f is not onto; that is, copo P ~n p p is a E (J)-cardinal. 2.
Let n > 1.
Then o)pD is £ -admissible (though it may not be p n even A -admissible) . P
3.
Note that it may be that p^ < 3 and JJ=* a).pfi is singular. P p P Also there may be n < p o and f, a map from a subset of wn P
cofinally into co$ which is I. (Jo) . ~l P 4. Prove that J D is E -uni form iz able (hint: let n > 2 and R be P n a I (J ) binary relation. Find R a Z.(J ,A). binary relation on J such that the uniform En-uniformization of R lifts to a (no p 1 k longer uniform, because of the p g for 1 < k < n-1) £n-uniformization of R, where p = p o , A = A o ) . p
(2.11)
p
In this paragraph I prove the basic "map-splitting"
lemma which will play an important role in the construction of morasses in L.
Strictly speaking, the only reason to consider
this lemma as part of the fine structure theory is the use of the J -hierarchy (which can be avoided by certain artifices) and the presence of "effective" notions like E beddings.
and E^elementary em-
On the other hand, the underlying intuitions most
certainly do come from the fine structure theory as the notation below suggests.
Typically, there are 3o/32f
E but not a E _ cardinal in Jn ,p. = ~n ~n+l 3. i LEMMA:
Let P o 'P 2 ' A o' A 2
be
such that
p ,P 2 are admissible, and (J V
0 / V l' V 2' f ' g 0' g l
be
SUCh
that
pn P
•
either A
n
such that v. is a
(i = 0,2). = A
= ^ or
,Ai) are amenable (i — 0,2). Let
212
i) v. i s a regular cardinal of J 1
f=CJ p ,A o )
ii)
g
THEN,
r\
+x
*o Jv2 '
t h e r e a r e p ,A_,f ,fn such t h a t
V^'V
i
CJp,A2)ff|jV-gi.g0.g0:Jv
1 1 0
1
is a regular cardinal of ^ V
Proof:
Ci = 0,2), P
(
CJ /A..} i s amenable, v., pi
1
Jp Q 'V + 0 , cof i n a l t J y V -
V ^ ' V ' g± = filJvi
(i
= °-11'
f
-
W
The proof proceeds by expressing (J ,A.) Ci — 0,2) as
directed unions of I -elementary substructures in the same way. These substructures collapse down to elements of J
Ci = 0,2) .
In this way the directed union systems become directed, commutative systems of models and maps which constitute E_(.J ,A.) predicates Q. c^ J
(i = 0,2) in the same E -defining formula.
The g -inverse image of Sl~ is a directed, commutative system of models and maps Q, <^ j
, by the I -elementarity of g.. .
Com-
pleting the diagram, the direct limit through Q. maps into (J
,An) and hence this direct limit is well-founded. Taking it 2 2 to be a transitive £-model yields (J ,A.} and the map into P
(J ,A9) is f r pQ ^ 1
f is just fn °f , and is also the completion 0 1 0
of the diagram between the direct limit through Q limit through Q . If f
and the direct
is not cofinal, since f. need only be
£ -elementary, cop^ can be redefined to be the point into which f
is cofinal.
If p ,p are successor ordinals (which means A = A 9 -= 0) , 0 2. 0 £ the above account is slightly inaccurate. Letting p. = y.+l (i = O,2), instead of constructing directly Pi' f o '^l' I construct y^ , and f.: J 1
*
Y
-> J
<>
, and then use the standard
Y
213 facts that f J
->• J and that there are canonical exten0 W Y2 sions, f ± 2. f., f . : J Y _ + 1 ->Q J y _ + i + 1 , f.CJ^l = J Y _ ^ . f. is Y
:J
0
Y
the extension of f. by rudimentary functions, i.e., for x £ J , f.(h(x,J )) = h(f.(x),J ) , where h is rudimentary. Y Y Y± i i _ _ 1 i+1 The E -elementarity of f. guarantees that this is unambiguous and that f. so defined extends f, and is E -elementary. The f. are obtained in essentially the same way as above. The J
(i = 0,2) are expressed in the same way as directed
unions of E -elementary substructures (n < w ) . The collapsed systems of models and £ -elementary maps form £ predicates Q. 1
J
c
J defined by the same £. formula over J (i — 0,2}. x p ~ vi ± is obtained as the direct limit through Q -the directed,
commutative system of models and E -elementary maps (n < CJ) which is the g -inverse image of ^ 2 *
Once again, I is obtained by
completing the diagram and F = f
°f|j . This said, I'll just Y 0 describe the directed union systems and the directed commutative systems of models and maps Q. Ci = 0,2) in two cases. CASE 1; "—"_""-_"-—_"•—
p. are limit ordinals (i -= 0,2) . i.
First, recall that h
is defined and lacks only the
definability over (M,B) when (M,B) is non-amenable and even then h is rudimentary in M and B. Hence, if a < p.,h in what follows, h denotes h _ ^ a a,A.nj i a Set t £ T. iff t = (ri/£/a} where n < p., a £ J
£J ;
is finite
and C < v. (i = 0,2) . For such t £ T ± let X t = h 1 " (u)Xu).£*{a}) . Thus, by the above, X
eJ
. For t,tf £ T , let t *3 t 1 iff
^ a 1 ; thus, if t ^ since p . is a limit ordinal, J p
i
t ? then Xt 5_ Xt, .
= LJ X^. t e r i fc
Clearly,
This is the directed
214 union system.
Also (T. , -< ) is £.. (J
,A.) by the same £..
formula, as is the function ((X ,A.nx ): t e T . ) . Now, for t e T., let a :(J *,A*) — > (X ,A.nx ) , let a* 1
t
"H+-
"C.
1
"C
i-
u.
0 (J A £€J J_ , while if (a) . Since n* £ n, if A,. = 0, (J_^.,A*)
be a
\ - ' n£' £>
P.'
A. ^ 0, p. is admissible so that a. and hence A* £ J 1
t
1
t
. In what
p .
follows, let 4Jl . = (J * , A * ) , and let h
= h _ . Then
|v/6l. j = h " (a)Xa).£;x{a*}) , and h
Since ? < v. and v. is a
t
1
n* t
t1
t
t
regular cardinal of J A* £ J
£ J
^vLx. t
p.
1
1
, this means that T)* < v. , and, if A. ^ 0,
, since the GCH condensation argument can then be carried
i out in J p
i If t ^
t 1 let a , = o^oo ..XJl -> yffi,. Also, a , is tt' t' t t O t' tt'
just the inverse of the transitive collapse of a" [X ] and —1 t' -1 tft,Cxt] is just h " (cjxco.^xfa (a)}) . Since J (v. is a regular cardinal of J
) ,a
is admissible
,eJ
Let fi± = ( ( X J t t ' ^ t t ^ ^ ^ 1 € T i / t : ^ t I ) # I t I S n O t h a r d t O see that Q, is L (J ,A.) in v. by the same Z_ formula. Let 1
-1 ft.. = g. Cft2].
Then Q
is a directed, commutative system of
models and E -elementary maps.
Let (((M,E,A) ,a 1 : t e f) be the
direct limit through ft . Thus, T = {t e T~'- ML . £ range g.} . Let f: (M,E,A) -> (J ,A_) be given by f°a^. = a °g. (t e T) . Thus 0 p£ 2. t t 1 (M,E,A) is well-founded and extensional. Also (M,E,A) |= "I am a J " (this is easily checked).
Thus, taking
(M,E) to be a transitive e-model, p ,A are defined by taking (J
,A, ) = (M,E,A), in which case f
range f £ range f .
is just f. Clearly
215 CASE 2:
p. are successor ordinals, say p. = Y-+1 (i = 0,2).
• -
i
i
Then f (J
) = J
and so f = f |j
:J
I
•> J
iff t = (n,£,a) where n < a ) , oo.£;
be the set of members of J
. Let t e T.
is finite.
For
which are the
< -least members of J satisfying a E v(J ) condition in paraJ y± n Y± meters from a).£ u a.
Thus X^ -5 t n
J and J = L-^ X^_. Y - 1 Y.i * .m t t£T.
t -3 t 1 iff n < n ' , S ^ E ' / a c a 1 ; X. -*?
Set
thus, if t -< t' then
X , . Now proceed as in CASE 1 (changing a few details)
to obtain y. , fo/f-, / a nd thence, as outlined above, to obtain
p =
i V
§3
1 and f
o' f r
MORASSES AND CONSTRUCTING THEM IN L. In this section I'll work gradually up to defining a morass
and constructing one in L.
First, I'll construct in L a simpler
object called a coarse morass which is then defined after the fact by abstraction from the properties of the object which was constructed.
A morass is then defined to be a coarse morass
which satisfies two further more subtle properties which Jensen aptly called continuity properties.
Finally, I (considerably)
modify the coarse morass construction to give a morass in L. As mentioned in the introduction, I assume V = L, but of course this is needed only to facilitate the presentation of the constructions and plays no real role (other than motivation) in the definition
of morass.
To fix ideas, I'll construct an w.
(coarse) morass, but the construction generalizes readily to arbitrarily uncountable regular K, giving K (.coarse) morasses. definitions are given in this full generality. general observations will help get under way.
The
A few simple and
216 (3.1) PROPOSITION: element of L
Suppose a < v and p e L
is definable in L
is such that every
in parameters from a u {p} .
Then a) the set of elementary substructures of L
containing p
and having transitive intersection with a is well-ordered by inclusion and the well-ordering coincides with the ordering of the intersections with a, b) the set of elementary substructures of L
containing p
and having transitive intersection with a is closed under unions of chains. Proof:
b) is evident.
For a) note that if p e X --? L
and
Lv
xna = 3 then X = {(t(£,p)) : £ < $, t is an L -term}. <3-2) REMARK:
Proposition (3.1) holds if "definable" is replaced
by "£,-definable" and "elementary substructures" is replaced by "s-elementary substructures" where s is a reasonably closed set of formulas containing all the I..-formulas.
The proof of a) in
this setting proceeds simply by restricting to E -terms t. Also, L
can be replaced by J
or by structures of the form (L ,A) ,
(J ,A) . x will be called non-projectible (because then x = p for all n ) if L
f= ZF~; if so, of course L
(3.3) PROPOSITION:
= J .
Suppose T is non-projectible or a limit of
no n-proj ectible ordinals, that c o < a < p < T , L
|= a is regular,
p e J , A e P(J ) n L . Then: P p T a) if X -df L , p,p, A £ X then (XnJ , XnA) -w> Ox
p
(J ,A) , a) p
b) there is an increasing continuous tower of length a consisting of elementary substructures of (J ,A) which contain p, have transitive intersection with a and lie in L . x Proof:
Suppose first that x is non-projectible.
a) is clear and
standard by relativation; b) is easy since using the closure
217 properties of L
and the fact that L
f= a is regular, such a
tower can be constructed working inside L . Now if x is a limit of non-projectible ordinals isolate everything inside some L , where T 1 < T is non-projectible. I'll now begin constructing the coarse morass. (3.4) DEFINITION:
v is u^-like iff v is non-projectible and
there is exactly one a which is an uncountable regular cardinal L of L ; this a is denoted by so that a = a>. . v is pretty iff J a v v v * 1 — v is a)o-like and there is p c a , p € L such that all elements of L
are L -definable in parameters from a V
V
u {p}; the < -least V
Li
such p is denoted by p . Let S 1 = {v: V is pretty}, S° = (a : V is pretty},
(3.5) DEFINITION:
S = {(a ,v): v is pretty}.
For a e S°, let S
= {v:a = a }.
(3.6) PROPOSITION: (MO) a) S is a set of pairs of priraitive-recursively (p.r.) closed ordinals such that if (a,v) e S, a < v < a^, b) if (a,v) , (a ' ,v') € 5 and a < a 1 then v < a 1 , c) 03, = max S Proof:
= sup S no) , u
= sup S
= sup S
Clear.
an elementary embedding and f(p
) = p
(3.8) PROPOSITION: a)
If f:L
-> L
is pretty then f|a
b)
There is at most one pretty f: L
c)
If v, / v 2 and f: L
•> L
=id|a
,f(a
)=a
-> L
is pretty then a
< a v
•
218 Proof:
For a) it's clear that f(a
) = a since a is the a), V V X l 2 l of L . The proof that fI a = id I a is by induction on 3 < a V V V V l l l l If 3 ^ 03 then clearly f (3) = 3, and if f (3) = 3 then f (3+D = 3+1. V
So let 3 > OJ be a limit ordinal and suppose that f|3 = id. | B. Since 3 < a
, there is, in L
f (g):f (a)) •*•
t
, g:a) •>
3.
By elementarity
f(-3), and for n < w f(g) (f(n)) = f(g) Cn) = f (g(-n» =
(since f|3 = id | 3 and g(n) < 3) g(n) . Hence f(g) = g:03 ->
f(3)
and so f(3) = 3. For b) , if f , f : L for C < a V
l
V
2
•> L
are pretty, then for L
, t(£,p
) is defined in L
V
V
2
L
-terms t,
and f ((t(£,p )) L v i ) = V 2 l
L V
2 ) = f 2 ((t(£,p v )) Z ) . For c ) , clearly a < a . If a V V V l 2 l isomorphic to the term models on a = a v
= a V 2 = a
V
l
the complete theories of (Lv ,p^ / ? ) ? < a /
V
then L ,L are V l V2 which are given by 2
(Lv
/P v '& £
Bv
the
elementarity of f, these theories coincide, hence so do the term models.
Since L
,L are both transitive and isomorphic to this l V2 term model, they 1 re equal, V
(3.9) DEFINITION: f: L
-> L
f: L
. If v
•> L V
l
V
For v ,v2 e S 1 , set y>1 T v 2 iff there is pretty T Vp let f
be the unique pretty
; conventionally set f V
2
1V2
(vn ) = v o . X
2
(3.10) PROPOSITION: (Ml) a) T is a tree on S b) (f
such that if v_ T v_ then a
< a
:v1 T v 2 ) is a commutative system of order-
preserving maps, f
| v_+l:v_+l -»• v.+l, f
(v ) = v 2 .
219 c) f
la
V
1V2
= idla
V
V
l
l
,f (a ) = a . V V V 1 2 Vl 2
d) setting a. = a f
: S 1 -> S 2
,S. = S nv .+1 (i = 1,2)., then i i and v. is minimal in S.. iff v~ is minimal in S 2
and if v, is the immediate successor of T, in S.. and T0 = f (T_) then v o is the immediate successor of T 0 in S 9 . 2 vxv2 1 2 I l Proof:
Everything but d) is now totally clear and d) is an
easy consequence of the elementarity of f (3.11) PROPOSITION:
a) (M2): If V
T v2, T
(3.lO)d)) and T 2 = f
(T.) then T.T T 2 and
f
(in fact, f
T T
12
|T,+1 1
= f
V2
IT.+I X
= f
V2
"V2
e $
Cas in
|L T
).
l
b) (M3): {a : V T T } is closed in a . c) (M4): If T is not maximal in 5
then
{a : V T T } is unbounded in a , d) CM5): If {a : V T T } is unbounded in a then T -= \J f VTT
Proof:
"v. VT
Clear from (.3.3).
I'll now define a coarse morass by abstraction from the preceding construction. (3.12) DEFINITION:
Let K > a) be regular.
A K-coarse morass is
a structure (S,S ,S ,T,f
) satisfying (M0)-(M5) except 1V2 Vl 2 that in (MO) " K " replaces "to " and " K + " replaces nco2". V
In fact, the very notion of coarse morass and the properties (M0)-(M5) were formulated in the attempt to capture in abstract form the salient properties of the structure just constructed.
220 This represents an important aspect of morasses: in addition to recapitulating certain important combinatorial properties, in addition to representing a tool for carrying out intricate inductive constructions/ the formulation of the notion of morasses may be thought of as a step beyond the combinatorial principles ^' V
, • towards isolating important structural features of
the constructible universe. Before going further I should elaborate a bit on the morass properties (M0)-(M5), and draw a few pictures to help the reader visualize morasses. FIGURE 1:
Arrows between points indicate the tree relation. The tree T and maps f V
express how the points in S 1V2
are
P Ul
"approximated" by a chain of height a)., of countable ordinals. Provided that the process of approximation has certain coherence properties, there f s hope of determining a structure of size to 2 ~ in the simplest case, the interval [w ,00 ) - as a direct limit through a tree of height u), consisting of countable structures.
Thus, there is the obvious connection with the
gap-two-two-cardinal problem of model theory (see [6], or [2]); in fact one of the major motivations for the formulation of the notion of morass was the attempt to solve this problem. There is little to say about the properties (MO),(Ml) which tell what sort of structure is being considered.
The property
221 (M2) is the first attempt to enforce certain coherence properties on the process of approximation, in that if a point is mapped to another by a tree map, then the former stands in the tree relation to the latter with the restriction of the tree map in question as witness. FIGURE 2:
V
1V2
The property (M3) says, in effect, that there are no gaps in the process of approximating a point x e S . The property (M4) can be thought of as being motivated from a practical point of view: in most inductive constructions, the limit stages are rather easy to handle.
Thus, it would be desirable to have as
many limit steps as possible in the process of approximation. A more systematic explanation runs like this: associated with each oj^-like point a are a certain set (S ) of oo^-like points. All but the largest o)^-like point
associated with a (if it
exists) can be expressed as a direct limit (of length a in the above construction) of structures of size smaller than a.
By
(M3) the conclusion of (M4) is equivalent to saying that T is a limit in T.
It is immediate from (M4) that the S
!
s are not
the levels of T, because if a is a successor point in the increasing enumeration of S , then S
has exactly one element.
The property (M5) merely guarantees that the process of approximation has certain natural properties, namely, a limit point in T is uniquely determined as the direct limit of the tree maps between its predecessors.
222 (3.13) DEFINITION: (S,S ,S ,T,f
A K-morass is a K-coarse morass
)
satisfying the properties (MO) ',
and (M6),(M7); the statements of (M6),(M7) follow, while (MO) 1 is obtained from (MO) by adding: d) for all a e S , S
is closed as a subset of sup S .
The properties (M6),(M7) are: (M6): Suppose v ,v2 e S ,a. = a
(i = 0,2). If v
is a
limit in S , if v Tv n and sup f v V. < v o , then v Tv. 1 2 o 1 a o 2 ^ v vn o o o 2 and f v = f v (note that (MO)' guarantees v. e $ and
Vl
°
v V
X
o2 °
a
2
in fact that vn e S* ) . 1 a 2 (M7): Suppose T is a limit in S. , T is the immediate T-predecessor of x and x = sup f-
! %'f.
Then, whenever a < a < a
and a € S , there is v e S_ n x such that setting v = f^ (v)_, there is no r\ with a
= a
and vTnTv.
n The properties (M6) and (M7) are the most subtle, difficult to prove, and difficult to motivate.
Like (M3) they are contin-
uity properties: (M3) expresses a sort of 'Vertical" continuity, while (M6) expresses a sort of "horizontal" continuity, and, as will be explained, (M7) expresses a sort of "diagonal" continuity. FIGURE 3:
DIAGRAM FOR (M6)
o
o
1
2
o
223 Before attempting to depict the situation described in (M7) it will be useful and instructive to present an alternative formulation equivalent to (M7): Suppose v ,vn £ S ,a. o £ I
V.
(i = 0,2), v
o
is a limit in
"v , a < a. < ou. Suppose that for all oV2 n e S n v , setting ru = f (n ) / there is r\. such that o a o 2 v vn o 1 o o 2 n Tri-jTru and let n* be the least such ru • Suppose that
S
a
,v^ = sup f
, v
o
V
°
a_ = sup{a .: n e S n v } (note that by (M2),(M3) this means 1 n* o a o 'o o that a., £ S° and in fact that if n e S n v then there is 1 o ao o n, £ S with n TruTrio) . Then there is vn e S such that l a o 1 2 1 a_ v Tv.Tv^ and v is the immediate T-predecessor of vn . o 1 2 o 1 This explains the above terminology: v
is the diagonal
limit of the n*. o FIGURE 4:
o
'o
"o
I °o
Thus (M7) again expresses the "saturatedness11 of the morass in that any stage in the process of approximation which arises naturally (e.g. v ) is already part of the morass.
A pragmatic
explanation of the first formulation of (M7) is that, again, limit steps in an inductive constructions are easier to handle. Though x is an immediate T-successor of T , it can be viewed as a diagonal limit (of the n* of the second formulation).
224 Now I'll construct a morass in L.
Once again, this con-
struction should be thought of as the motivating example behind the formulation of the notion of morass and the morass properties as attempts to capture certain important structural features of L encountered in the construction.
I'll construct an a)..-morass;
the generalization to arbitrary regular K > 0) is straightforward, as in the generalization to L[Aj when A c_ K. v e S 1 iff v is not a cardinal, v is a limit
(3.14) DEFINITION:
of non-projectible ordinals, and there is a unique uncountable cardinal, a
in L
(for K-morasses in L [ A ] when A c_ K, we
+
require v < K , V is not a cardinal, there is a> < a that L [A n a ] f= "a
is the largest cardinal and a
< v such is regular",
and v is a limit of ordinals T such that L [A n a J \= ZF ) . S,S ,S (a € S ) are defined as in (3.5). (3.15) DEFINITION:
If v e S , let $(v) be the least 3 ^ V such
that v is not a cardinal in
Jo±1. p+l Accordingly, letting 3 = 3(v), there is k < u> such that
there are p e J , r\ < v, a c ri, and f with f: a -*• p onto E (j ) in p. Such a k is positive. Jc p (3.16) DEFINITION:
v and f is
Let 3 = 3(v). n(v) is the least n for the
existence of p,n,a,f with r\ < v, a c j], f : a •> v and f is — onco Z ,, (Jo) in p. Let n = n(v). Set p(v) = p£ and let A(v) = A £ . n+x p p P (3.17) PROPOSITION: A = A(V)
Let V £ S , 3 = 3(v), n = n(v) , p = p (v) ,
Let y = Po + • THEN: p > V, CD.y < a v . p
Proof: Clearly, if n = 0, p = 3 ^ v. If n > 0, m < n, and m . _ . m m m m+1 . Pp > v, let 6 = p^ , B = A^ , q = Pg , n = Pg • Since J. = h. 0
" (o)xo).nx{q}) / there is a I. (J.) map g of a subset of
0, B
o).n onto v.
n ^ v.
^JL
0
This contradicts the definition of n(v) , unless
225 On the other hand, if f: a ->• ,_ v, a c ^ < v, and f is onto — E _ (J o ) , then f is E. (J ,A) . Also f ft J because J c j and ~n+l p ~1 p p p— p v is a regular cardinal of J . Thus, since v = oov, y < v. But J
= h " (o)Xo)yx{p }) . If ajy = v then, setting P rA p h = h (~/-/Pc ) / h°f maps a subset o f n onto J and n < coy: p, A p p this is impossible. If a < ojy < v, there is g € L , P
g: a -> _ , _ coy and so h°g maps a subset o f a onto J and v onto ^ JT v p a < ojy: again, this is impossible. Note that
(3.17) guarantees that J P n+1 p,A are a s above and p = p
= h P
" (coxa x{p}) where 'A
V
P
(3.18) DEFINITION:
p(v) is the < -least p e J j
J
= h P
P, A
"(coxa x{p}) V
P < p"+1). J
(thus, if
such that
p
ojy < a , then possibly V
Set Jl = (J ,A,p(v)).
p
V
QxGx is the formula: V ° r d i n a l y
(3.19) DEFINITION: (y < x A 0(x)) .
p
gordinalx
f is a Q-formula iff <|> is 0x6 (x) where 9 is E .
Q-elementary substructures and embeddings have the obvious meaning and are denoted by j{)X - ^
(3.20) DEFINITION:
Q^t ®'AsL "*"
Suppose v ,v e S .
dj-
Set v TV iff v
f \>2
but a) n ( v 1 ) =
2
b) p ( v x ) = v1 iff P ( v 2 ) = v 2 , and c) there is f: /SI
-> JJI such that f |a = id|a , V V V l X 2 l l ^ ) = a^ , f(p(v1)) = p(v 2 ) , f |L^ :L^ •+ L^ and if V
v_ < p(v,) then f(vn) = v o ; such a map is called pretty. (3.21) PROPOSITION:
pretty f : ^
•> JJiv
a) If v ,v
£ S 1 , there is at most one
, b) T is a tree; if V-,Tv2 then a
226 Proof:
Essentially as in (3.8)b),c) and (3.10)a)'b)
(3.22) DEFINITION:
(3.23)
If ^ T v ^
f
is the unique pretty
In practice, when trying to prove that vTv, it frequently
is relatively straightforward to find f,p,A,p such that (J_,A) is amenable, p > v, f:(J-,A) ->
(J , ,A (v)) , f(p) = p(v) ,
f|a- = id|a-,f(a-) = a v , ?|L_: L_ + if not then f (v) = v.
Q
L^, p = v iff p(v) = v, and
Note that this means that if v < p, then
v is a regular cardinal of J_. What then remains is to prove that n(v) = n(v), that p(v) = p, that p(v) = p and that A(v) = A.
This is usually done
Using (2.9)d), there are f*,$ such that, setting
as follows.
n = n(v) , 3 = 3(v) , then p = p^, A = A £ , f* D f ,f* : j - -*• p
p
—
p
j
n+1
p
and having the other properties of (2.9)d) . Note that if v < 6, then v i s a regular cardinal of J^. P
Now J- = h- -" (o)xa-x{p}) , since this carries down, via f P P/A v from (J
,A(v)).
of a- onto v. n(v) ^ n.
Thus there is g, a Z (J-,A) map from a subset
Such a g is then £ .i (Jo) •
Thus JS = 3(v) and
Suppose, towards a contradiction, that n(v) < n and
set m = n(v) . Then there is a, a E _ (J-J map from a subset of ~m+l p) m+1 a- onto v. If v < p, then, since m < n , v < p < p ^ , and so a v P is a /v E _ (J-) bounded subset of J _ . This means that a € Jr, m+l p m+l p
p
e
which is impossible since v is a regular cardinal of J-.
Hence
P
there is a E
(J-) map of a subset of a- onto *p = w.p, and,
since m+l < n, this means, by (2.9)b), that there is a E (J-) map of a subset of a- onto J-. v p
This contradicts (2.9)d) which
guaranteed the minimality of aj.p for this property. in this case too n(v) = n.
But then,
But p = p(v) , A = A(v) , as required,
227 and p (v) ^ T P .
Finally, suppose that q <
p.
Then
q = f(q) < p(v) . But then, if p e h- -" (u>xa-x{q}) , J P ,A V p(v) e h "(coxa x (q}) (in fact p(v) e h _" (a>xa^x{q}) ) . p ,A V P' "^ _ — contradicts the definition of p(v) , and so p(V) = p.
This
In what follows, the above schema will be used in different situations, and sometimes merely cited to complete proofs. (S,S°iS1,T,£
(3.24) LEMMA:
Proof: n € S
) m VTV
V
(MO),(M5) are clear.
is a coarse morass.
For (Ml)-(M4), first note that if
and a = a , then: (1) : If T € S L
n T)r then $(x)+l < n (this is clear since
= J , L
n
|= card x = a, but x is a cardinal of
nr
n
W' = pm
(2) : hence, for m < a>, setting p m
(J ,A ) e L ; in particular p m x] m (3): the function {JJ[ : x £ S
, A = A
P\X^
JSl e L , x r)
n n) is uniformly Z (L )
in a. This suffices for (M2),(M4), and (Ml) is easy, since S uniformly L (L ) in a, f 1
n
: L vxv2
-* L vx
Q
and f (a v2
) = a v±
v2
For (M3) , I'll refer to the schema of (3.23). a < a
be such that {a : vTx} is unbounded in a.
X = KJ {range f a
nx = a
T
: vTx and a
n n is
First, let
Let
< a } , and let Y = X n L .
n Y = a, and p(x) e X.
Now
Then
/FL |x is the union of a
tower of £ -elementary substructures of AjL
and L |Y is the
union of a tower of Q-elementary substructures of L , and so |x — ^ ^CL , L |Y -< L . In fact, it's easy to see that X = h^
" (o)xax{p(x) }) . Also, x < p(x) iff x e X.
f: (J-,A,p) — > P
_
JJL |x and let f 1 : LT
1
X
Now let
— > L | Y . It f s easy to X '
see that f => f' and that if x e X, then f(x) = x, and x is a
228 regular cardinal of J- and in fact is the cardinal successor in P J- of a which is, even without assuming T e X, the unique uncountable cardinal of L_.
Further, T is a limit of non-
projectible ordinals: this is because L |Y = L_ and K |Y is the union of a tower of Q-elementary substructures of L appropriate II^-properties hold.
in which the
This means that T e S
and
a = a-. To see that T T T is now easy using the argument of (3.23) . It remains only to verify that (M6),(M7) hold in order to conclude that (S,S ,S ,T,f
) i s a morass. This will be 1V2 Vl 2 done in the next two lemmas where the argument of (3.23) will be V
combined with (2.11). (3.35) LEMMA:
(M6) holds for (S,S°,S^Tjf
) V V
1 2 VV2
Proof:
First assume p (v ) > v . Let p. = p(v.), $. = $(v.),
—————
O
* O
1
1
1
1
A. = A(v.), p. = p(v.) (i = O,2) ; let n = n(v ) = n(v~), let f = f , let yg = f |L , and let y gn = idlL . Let HJ Oi , v v ' o v v ' v l ' v -' o 2 o 2 o 1
A-^f ,f v
be as guaranteed by (2.11).
is, in J
, the cardinal successor of a
By (2.9)d) applied to f fn ^ 1
Thus, among other things
there are 3_ ,f~, such that
f ,f. : J -> J , p = p n , A.. = A*? and with the other 1 , eL 1 61 1 1 3X n 62
properties of (2.9)d). successor of a
.
Thus, in J
, v, is the cardinal
Since f = f of , p
€ range f. , say
f 1 (p 1 ) = p 2 ; thus, also f (p ) = p . Again, apply (2.9)d), this time to f , to obtain f ^_ f such that f : J o -> JD and having the other properties of o p n+JL p_ (2.9)d) (that f is E , - and not just E -elementary follows o n+1 n from the fact that f is E -elementary since it's cofinal). Let g be a map of a subset of a
onto v o q e Jo , say by the E n-formula 9. o p n+l o
which is E
. (JR ) in o
229 Let g be the subset of J
which is defined over JD
and q (= f (q ) ) . By the E
by 9
-elementarity of f , g is a map
into v,, and f "g c g. Hence g is cofina
from a subset of a
into v . But v, is, in Jn , the cardinal successor of a aV 1 X 3 1 X 3 V 1 2 Hence, in the usual fashion, g gives rise to a £ (J ) map of a subset of a
onto v . Thus $-, = $(v..) and n(v ) < h.
If m < n, then p o > p_ and p. > V. . Thus, if x c j is ex i i i - vL E _ (J ) , x e J D , by (2.9)b). Thus, no such x can be a map ~m+l P, P, of a subset of a smaller ordinal onto v.. . This means that n(v ) > n, i.e. n(v-) = n, so that p_ = p(v 1 ), A Suppose p, ^ p(v,) . Let £ < a l
Pl,Al
^
= A(v ) .
, i < ca be such that
Let q 2 = f ^ p t v ^ ) .
Then p 2 =
h^^
(i,^,qo) . But then p 0 ^ q o so p < p(v ) . On the other hand, ^ Z J Z _L J 1 this means that in (J ,A^) the following E -formula holds: (•3q) (^?
) (-3i
But that means that in (J
(ifC/q) and q < p ). ^
,A ) the following E -formula holds: o
Hq) (3£
o
) (^i
(i#5q) and q < p ) . J °
This, however, contradicts the defining property of p Hence p- = p(v,) . But then f = f V
= p(v ) .
witnesses that v Tv^. and, oVl
as required, f |v = g = f |v ^ ' o1 o ^o v vn' o O 2
If p(v ) = v , then the proof proceeds essentially as above but with a few minor modifications which I now give. First, there's no need to invoke (2.11): set p. = v n , f = f , 1 1 o v vo o z fn = idlj . Let An = A o nj . Then A_ is also 1 ' P1 1 2 Q1 1 A^nJ\ = v*—J f (A n j j . 2
5
C
° °
?
Thus (J p
l
,A. ) i s amenable s i n c e
1
230 each f (A nj\ ) e J . o o £ p1
Also (J
p1
,A.) ««< (J ,AO) . 1 o p2 2
as before to obtain pw3-, >f , f.. as before.
Now proceed
If p, < 3 w then
p is a cardinal of J by (2.3)b), and p. is regular in J i 3X i ex since it is, in J o , the cardinal successor of a . The rest of V
h
2
the proof goes through exactly as before, but some care must be taken if p. = 3-. •
This is no obstacle to concluding, as before,
= 3(v..), n(v^) < n.
that 3
However, if p
= 3
another argument
is needed to conclude that n(v^) = n: if n(v ) < n, let m = n ( v n ) . Then p. = p. = 3 w and so there is a I ,.(Jo ) map a 1 1 3-, 1 ~m+l 3., of a subset of a onto 3n • Hence p o ^ a , which is impossible V V 1 3 2 1 2 +1 m+1
since m+1 < n and so p
=
3n •
The proof that p n = p(v-, ) is as
before. (3.26)
The following will be useful in the proof of (M7).
PROPOSITION:
Suppose that T T T and T = sup f- " T . Then
wp (T) = sup f- "ojp (x) . Proof:
If not, let wp ' = sup f- "copd) . Then, setting TT = A ( T ) D J ,, (J ,,A') is amenable, since A 1 = LJ A(x)nS
A1 ^
J
f- (A(x)nSr) and each f- (A(x)nSr) e J ..
=
Also,
)
(J ,,Af) -J
(J
.,A(T)).
Let a = idlj ,.
Let n = n(T) = n(T) .
p p1
o p (T) ' p' By (2.9)d), let a,3' be such that a D O,O: JO. -* Jo. ,, n n — P n p {T) = p o l , A 1 = Ao with the other properties of (2.9)d). Also, P P
by (2.9)d) applied to f-
viewed as a E -elementary map between
(Jp(-,,A(?) and ( JpIf A'" let f = f-^ ?: J ^ ^ Accordingly as in (3.25), there is a Z of a
onto T .
because 3
1
(J 1
^
+ 1
Jg,.
) map of a subset
This means that 3 ( T ) = 3 / but this is absurd,
< 3 ( T ) since
p _ , = p ' < p = p / % . 3(T) 3
231 (3.27) LEMMA: Proof:
(M7) holds for ( S j S ^ S ^ T j f
)v Tv.
I'll prove the second formulation of (M7). Recalling the
notation there, I'll first prove that there is v Tv. with a
= a . This will be by cases depending on whether or not
p(v ) = v , the case where this is true being the easy case. Then I'll prove that this v is an immediate T-successor of v . Adopting the notation of the second formulation of (M7), for n € S n v / let X = f . "L .. Thus L |x -< L , o a o n n*n 9 n* v 'n o v O
O
O
Z
O
Z
O
Z
and a o n X = a .. Also n < n -, = > X ex . Thus 2 n n* o,o o,l n ^ — n , o o ofo o,l
setting X =
<J
X
n eS nv o a o o
n
;
o
L
V
|x ^ 2
L °
V
, and since 2
an = sup{a . : n £ v }, a n n X = a., , and a n € X. Also for i n* o o z i z o n e S n v , r\~ e X, since v i s a l i m i t i n S and i f n < n' / o o t o 2 o a oo o t h e n ri 2 e sup{n2:
x
•• o € S
n
o "v = o 2 i n v . so L | x
T h u s , s i n c e v« = s u p f n v } , X i s
cofinal
o |x — * L .
in fact L
-<- L
2
and
2
Q V
V
2
Let g.: L
— > L |x. Then it is easily seen that V l 2 g 1 (a ) = a^, g. la^. = id|a , that a is the unique uncountable 1
V
cardinal of L
and that v, is a limit of non-projectible 1
i ordinals; i.e. v. e S , a f
v v_ o 2
"L v o
= a
. Also, it's clear that - 1
i
c x , a n d so l e t g = yg n of L :L ->• L . A l s o , l e t — ^o l v v ' v V Q \) o 2 o o l
n = n(vo ) = n ( vzo ) . First suppose p(v ) > v , p(v 2 ) > v 2 - Recall that P ± = P ( v ± ) , A ± = A ( v ± ) , p ± = p(v ± ) (i = 0,2). By (2.11) let f ,f. ,p^,A J
be such that p_ > v., v^ is a regular cardinal of
(and hence v.. is in J
the cardinal successor of <x^) ,
232 (J
,A ) i s amenable, f. pi-
f
o
:
=> g.
1
(J
p
o
In f a c t ,
(i = 0 , 1 ) ,
1
,A ) -> , . . -, ( J ,A ) , f : ( J ,A., ) -> (J ,A ) . o o,cofinal p. 1 1 p_ 1 o p n Z? L i z
f
=
i s c o f i n a l i n wp 2 : t h i s i s because f-.°f
and by (3.26) f
is cofinal in a)p9.
o z in fact Q-elementary between (J Px
o 2 But then f, is E. and
,An) and (J -L
, A Q ) ; also P2
^
P 2 € range f , since p 2 € range f , say p 2 = f-i(P-i)• applying (3.23), v T If p(vQ) = and let A_ = l
is amenable.
VQ,
V
f
Now,
.
p(v2) = v 2 , let p x = v x , let f± = g ± (i = 0,1)
C J f (A n L ) ; thus, as before, (J ,A1) _, o o n pi n eS n v o 1 o a o o Also, let p
be such that f (p ) = p 2 .
Again by
(3.23), v x Tv 2 . Since T is a tree, this means that v Tv_ . It remains only o 1 to see that v is the immediate T-predecessor of v,. Suppose that a < a 1 < a_ and a 1 e S . Since a 1 < a_ there is o 1 1 n € S n v such that a 1 < a .; accordingly, there can be no o a o n* o o f n' € S f. such that n Tn Tn*. But this also means that there can o a o o o be no v 1 e S , such that v Tv'Tv., because, if there were such a v',then setting f (n ) = n ', by (M2) n Tn • . Also V V O O O O f
o
vo v i(n o }l = fv'v K)l
and S
°'
again by (M2)
' n o Tf v v
(
V-
o 1 Since T is a tree, this means ri'Tn* (since r)*Tf (ri )) which o o o v v. o o 1 is a contradiction. (3.28)
The morass just constructed has the additional property,
assuming V = L, that H W
2
= {^J \J5{ \. The following definition V veS 1
which will be used in the next section, is an attempt to formulate an abstract version of this property.
233 Suppose (S,S°,S1,T,f
DEFINITION:
)
Suppose V is a function with domain S
is a K-morass. such that
ii) for v € S iii)
n T , l/(v) = 1/(T) n T U = <~J l/(v) i s s u c h t h a t (U I I {V}:OJ ; L < v < u)2) veS K + -,
+
enumerates LK J Then (S,S ,S ,T,f ) is universal with respect to V ' v 1 v 2 y v 1 Tv 2 ^ iff whenever vTx, f (S,S°,S ,T,f
|v: (v,e,(/(v)) -*
)
(T,£,1/(T)).
is universal iff it is universal with
V V
1 2 VV2
respect to some V satisfying i ) , ii), iii). In the above construction, set (£,£) e t^(v) iff £, C < v and ? € the ^—- (in < ) subset of v. j
§4.
COMBINATORIAL APPLICATIONS. In this section I'll give some examples of combinatorial
applications of morasses.
Most of these were found very early,
and for some it was seen, after the fact, the the full morass structure wasn't really needed.
Notably absent is the appli-
cation to the gap-two-two-cardinals theorem, long the principal application of morasses, but whose length and difficulty ruled out including it here.
Very recently Shelah and I, [8], and
Velleman Clo], independently proved that the existence of morasses is equivalent to Martin's Axiom type principles, which in turn provide new and systematic methods for applying morasses.
For other applications of morasses see [3] and [7].
(4.1) Recall the principle D
: there is a sequence
(C : a limit, a < a^) such that
234 i) C
is closed unbounded in a,
ii) if cf a < a) then o.t(C ) < u> , then C o = C
iii) if 3 is a limit of C ot
p
n p\
ot
The following is easily seen to be an equivalent formulation. There is a sequence (C : a limit, a < ajn) such that: ot 1
i ) C
<£
is a closed subset of the set of limit ordinals < a
which is unbounded if cf a < w, i i 1 ) : as in ii), iii1) if 6 € C then C o = C n 3. ot p ot Finally, if S £ (ID ,(0 is closed unbounded in o>2/ then D
is
the principle: there is a sequence (C : a limit, ot € S) such that i ) C is a closed subset of Lim n a n S which is unbounded S ' ot if cf a > u>, ii) , and iii 1 ). (4.2) LEMMA: •
If (SjS^S^Tjf
)
is an u^-morass then
holds.
Proof:
Let v e S . a).
Set T e C iff T < v, and there is vTv v
such that x = sup f- "x.
The easy verifications are left to the
reader. (4.3)
Let K > co be regular.
Recall the principle
0
: There
is a sequence (W : a < K ) such that: i) W
c_ p( a ) , card W
< K,
ii) whenever Y £ K, there is a closed unbounded C £ K such that for o t e C , Y n a , (4.4) LEMMA:
C n a e W .
Let K be an infinite cardinal.
universal K -morass then
<>
+
holds.
If there is a
235 Proof:
Let (S,S ,S ,T,f
)
be a K -morass universal V
V2 V 2 with respect to I/.
For a £ S° n K
set a e K
iff there is
v € S , T < v such that a = (3 < a: (x,3) e l/(v)}, and let K^ = {a n 3: a e K , 3 £ a}. Let w generated by K +
a € K \ S ° , let
1
be the Boolean algebra
u {T : v e S } u a, where T w
= {a-: vTv}. For
= (j).
Now let Y c^ K , let (v,x) be such that v e S
+/
x < v,
+
Y = (3 < K : (x,3) € l/(v)}. Set a e C iff a e T and letting v be that element of S such that vTv, x £ range f- . Let a be a vv o Suppose a e C, and, let vf e S
the least element of C. that v'Tv. that f
Then C n a = T t\a 1
(T )
Further, if T' is such
= x, then Y n a = {3 < a: 1
f , : (v'/C^fv )) ->
(T',3)
€ l/(v')}, because
(v,€,(/(v)) and f , la = id I a.
1
{3 < a: (T ,3) e l/(v')} e K (4.5)
€W.
be such
But
c w . a — a
Let K > a) be regular.
Recall Silver's principle (W) :
there is a family F £ P(K) , card F = K such that: (*): for a < K, setting F|a = {xna: X e F}, card F|a and there is a sequence (W : a < K) such that W e H , W ot a K a and: (**): whenever s c F and card s < K, there is a < K — o that for all 3 £ a , si 3 = {xn3: Xes} e W o . o p (4.6) LEMMA: If K is an infinite cardinal and there is a
< K, c PP(a), — such
univer-
sal K -morass, then (W) , holds.
V
V
rv
Clearly card F = K , and for a < K , card F|a < K + . If a e S ° , set a £ K iff a c s and there is v £ S , x < v such that either a — a (1) a = (n: (x,n) € l/(v)}, or (2) there is some vTv, x < v
such that
a = {f- v (n) :(T,n) € l/(v) ,n e s 1 } .
236 Let K Let Ka = K
1 2 = {{T 1 : n e a}:a e K }; let K = {{T'u{n}:n £ a}:aeK }. a t| I a a ri a
^
°
= {{T'}:n £
generated by
= {{T'u{n}}:n e S 1 na} / and let n
K3Q , let a K *s ^
*—^
So}.
Let W
be the Boolean algebra
K . a
If B £ K + \ S ° , let 3* = the least element of S°\B, and let W
= {{xn3: X£A}:A e W p
}.
Then clearly W
€ H +, W Ot
p
K 1
£ PP(a).
01
For (**) , suppose s c^ F, card s < K; say s = {T : v € s }.
Let
(v. : i < ^) be the increasing enumeration of s . The proof goes contains all the {T 1 } for n 1 n e S and all the {T'u{n}} for n e S n3, suppose, without loss P 11
by induction on £.
Since each W
of generality that £ is a limit ordinal and that (**) all the (s)^ = {T
1
holds for
: i < £}, for £ < £, with p\ playing the role
of the a of (**) . o Let X = sup v., let v* > X and T * be such that (V*,T*) is lexicographically minimal such that s
= { V : ( T * , V ) e l/(v*)}.
Choose v*Tv* minimal such that x*, X and each v. range £-
, and a-* > sup
B.
I'll prove that a-^ witnesses that (**)
holds for s.
Let
X be such that f-^ ^(X) = X, and let x* be such that ^ ^ ^ l ] £ Ka ' s i n c e V -1 * , s = { V : ( T * , V ) € I/(v*)}. Hence {T± : i < O = s a-, e W v. v* a-. 1
V*
By the definition of W (y e K \S°) it clearly suffices to prove that siB € W o for B € S°nK + , B > a-., p v* So, let B > ot-^, B e S°nK + . X T X ' T X , then f-
"s
A A1
s|3 -
^
A
."s|
V
If there is A' e S
with
€ K o and so again sIB e W_, since p
p
237 Suppose then that there are X ,X2 such that XTX.TXJTX, X 2 is an immediate T-successor of X
and a. 1
1 9
v.,v. (i<5) be such that f 1
1
tively (i<£) . Thus f
9
(v.) = v., f. ,(v.) = v. respec1 1 A_A i 1 2 (vl) = v.. For i < £ let n. be the 1
1
X
there is 6 < £ such that a 1
n
By (M7) , sup a i<£ n i > 3.
= a. . A 2
Thus,
e
n 3 = (T1 n ou )u{v,} = T 1
V. 1
1 2 {T1 u{v }: 6 < i < O X v1
1
1
9
least n such that vTTn T vf.
For 6 < i < £, T
Let
A A
A A~ 1
< 3 < a. . 2
V. 1
A
1
i
u{v.}. V.
1
But
1
1
e K
c wo. A l - P
But s13 =
I
{T1 ngr i < e}u{Tf u3: 9 < i < ?}, and, by the induction hypoV V i i 0 theses {T1 n3: i < 6} = (s) 13 e W n . Since W o is closed under v_^ P P finite unions, si 3 e W n . §5.
GETTING MORASSES BY FORCING. My approach to adding a (K,1) morass is to add a somewhat
simpler object which is then "thinned" to obtain
a morass.
This is essentially as in Jensen's original treatment, see [9]. However my treatment here is based on a new set of conditions introduced by Velleman, see [10], who has kindly permitted me to present them here and in [8]. The basic difference is that Jensen's conditions involved first adding a D -sequence and then picking out a subtree of a tree with maps along it which is canonically associated with the D
sequence, whereas Velleman1s
conditions permit great latitude in simultaneously assembling the tree and the maps. First, I'll define the generic object. (5.1)
a and 3 are the same kind of ordinal if they're both zero,
or both non-zero limits, or both successors.
f: a -> 3 is nice
iff f is order-preserving, a and 3 are the same kind of ordinals,
238 for all y < a, y and f (y) are the same kind of ordinals and if Y+l < a then f(y+1) = f(y)+1 and, finally, if a = a+1, then 3 = f(a)+l. DEFINITION:
Let K ^ co be regular.
A (K,1) preinorass is a triple
(T, ~* ,f ) such that xy x*4y i)
(T, -f
) is a tree, { K } * K + £ T C K + 1 * K + , (dom T) nk = K,
and for a < K, 0 < y
= {T:(a,r) e T} < K is a limit,
ii) for x € T, set x = (£(x),0(x)) (warning: il(.x) need not coincide with the level of x in (T, ^
) ) ; then (f
a commutative system of nice maps f
: x -J y) is
: 0 (x) -> 0 (y) ,
iii) if x -$ y, x < 0(x) , w = (£(x) ,T) , z = U(y) ,f (x))., xy then w ^ z and f = f T. wz xy' iv 1 ) {Z(x): x —£ y} is either empty or a coinitial segment of dom T n ZCy), and is non-empty if &(y) is limit, v 1 ) if l(y) is limit then 0(y) =
<^J range f , xy x^ y
vi 1 ) if 0(x) is limit, x ~f y and X = sup range f
< 0(y)
then setting z = U(y),X) f x - ^ z, f = f xz xy The next lemma justifies the introduction of premorasses by showing that they can be "thinned" to yield morasses. LEMMA (Jensen, essentially): If there is a (K,1) premorass there is a (K,1)-morass. Proof:
To fix ideas, take K = a>. .
•
is straightforward.
As usual, the generalization
j_
First, define a normal sequence ($ : 5 < U K )
of countable ordinals by induction: 3 the least 3 > B £ such that y S° = (3 ? : 5 < 0J1>.
= 0, $, = sup $ r , $ r
< 3 for all n < 3.
Set
is
239 If x e T, set v(x) = Jt(x)+0(x), a(x) = l(x) . let f
: v(x)+l •> v(y)+l be such that f
5 < 0(x), f f
If x *< y,
|a(x) = id|a(x), for
(a(x)+5) = a(y)+f xy (?) f f xy (v(x)) = v(y) .
is nice.
Then
Set v(x) -< v(y) iff x -< y.
If x e T, a(x) e S°, v = v(x) let D, v = {0}x{(g f Y Qp): 3 < a(x)} U {l}x (S°nv(x) ) U _ {2}*{(v(y), v(z),5,£): y < D
z, v(z) < v, f
(?) = S h
Thus
£ 3 x~v codes up the segment of the premorass internal to v.
Further, if v < v(w) and a(w) e S°, then D
= D , x n 3 x^ v. v v(w) = (L [D ] is admissible.
v is D-admissible iff 0%
Set v(x) € S 1 iff x € T, a(x) e S°, v(x) is a limit of D-admissibles, and ^
f= a(x) = ^
Note that all elements of L C D ]
are
. S={ (a(x) ,v(x) ) :v(x) eS }, E (<^ )-definable in
parameters from a . For V,T e S , set v —•£ f
^_ f
, f :
^t
-+ /0l .
in which case, call it f CLAIM:
x iff v ^
x and there is
Such a map is unique if it exists,
+
(S,S°,S , — ^ + , f
)
+ . is a morass.
Proof of CLAIM (and thus of lemma): (Ml) is satisfied because of the Q-elementarity of f
.
(M2 is satisfied since by the proper
ties of the premorass v » < v, and because f- ($1,-.) = {^Ql ) this means f_ | L - [ D - ] : 0 ^ - - ^ W 01 V .(M3) follows from the properties of the premorass and easy model-theoretic arguments. (M4) uses the fact that if n € S /$l
|= a
argument
and r\ > x then since x is regular, a familiar Lowenheim-Skolem-type
for
Ol
can be carried out within /fl^
elementary tower of height a intersections with a
of submodules of (^
to
g i v e a^i ,D ) whose
form a closed-unbounded subset of S
n a .
240 Thus, among other things, a trivial in —=? .
is a limit in S , and so x is non-
The conclusion now follows readily.
(M5) is trivially verified.
(M6) follows from the corres-
ponding property vi) of the premorass, the fact that f_
:
AK - -»• JJ\ , since £Tl ^ -< X
TT
O
A
til-
A
O
, and the fact that f-
T
XT
is cofinal in A. For (M7), first, a can be taken, without loss of generality with a- < a < a . there is T 1 e S
a
Then, by the property iv 1 ) of the premorass,
such that x *< T 1 ^
T.
For v e S
v = f- (v) and let v 1 be that element of S v1 " ^
v -^
v.
Let g =
^J
f , V
a-
n x, let
such that
. Then, by arguments
V
X
similar to those for (M6) , g: C7Z , +
CL
. But then x 1 -^-
x
and g = f , . This completes the proof of the CLAIM and the LEMMA. I turn now to presenting Velleman's conditions for adding a premorass. (5-3)
DEFINITION:
p = (t, - < , f
)
e P iff
i) t c^ (K+1) x K , card t < K, t ^ 0 ==*> ((K,O) e t and dom t n K is a successor ordinal), (t, ^ a e dom t n K, O < y
P
) is a tree, and for
= {x:(a,x) e t} < k is a limit,
ii):ii),iii),iv'),vi) of (5.1) hold, replacing T by t (note that for iii) this means that if £(y) - K, then (K,f
(x)) e t
and for vi) if £(y) = K, then (K,X) e t ) , (iii): v 1 ) of (5.1) holds with the additional requirement that A(y) < K. tp =
For p € P, p = (tP, - < P , f x y ^ x - < P y '
^—l ({a}x Y P )u{K}xs P . ae(dom t^)DK
and
Then, for p,q e P, p < q iff
241 (iv) (t q , » < q ) is a subtree of (t P , * < P ) and if x ^ then f
P
= f
q
q
; for 6 e (dom t ) n K, y
q
q
y
= y? and if
x < y q / x e t q then x ~ < P ( < $ , T ) iff x — < q (6,x) . P
= (P, <) .
The following lemma summarizes the important properties of P . The reader can either carry out the verifications (straightforward, except possibly for c)) or consult [8] where P
is presented in greater detail.
(5.4) LEMMA: a) P
is K-complete (i.e. every decreasing sequence
of length < K has a (not necessarily greatest) lower bound), b) if p £ P, a < K, ( 3 < K , y < $ there is q < p such that a e dom t , $ £ S q and y e ^—' K ' _q y =
range f , where y xy
(K,3)/
c) P
has the K
-C.C.
This immediately gives P (5.5) LEMMA:
In V
there is a (K,1) premorass.
(5.6) REMARKS: a) If A £ Q\
(or, in the general case, if A c K ) ,
the entire construction of (5.2) can be relativized to A.
The
approach is to start from the models 4ft , ^ = ' Anv(x)], e,
D v
/ X ) ' Anv(x)) with Jl(x) = a) 1 .
Attention
is restricted to transitivizations of Q-elementary substructures of the <7l, v(x) having the form ( L v ( y ) [ D y ( y ) , A ] , e, D v ( y ) f A ) with y **< x and the inverse of the transitivizing map extending f In particular if 2 case 2 - K and 2
= % and 2 ~ ^ 7 ^ or i n t h e ^ e n e r a l = K ) and A is chosen such that Anco codes
242 up H and AnCcD- ,u)n) codes up H , then the morass which results a). 1 2 ^ a) ' from the thinning of (5.2) relativized to A is universal. b) Roughly speaking, the idea of (5.4)c) is as follows. P can be partitioned into K equivalence classes, the content of the equivalence relation being, roughly, that p and q are equivalent just in case p,q have the same underlying structure and differ only in that S P , S q do not coincide (though they will have the same order type - this will be part of the meaning of having the same underlying structure). power K , there is Y c^ X of power K equivalence class.
Then, if X c p has
coming from the same
But then there is Z c y of power K
such
that {S : p e z} forms a A-system, i.e., there is a such that whenever p,q e Z, p ^ q implies S
n S K
= a which is an initial K
segment of S P , s q , and either all members of SP\a are less than K
K
all members of S \a, or conversely.
"K
The definition of the
equivalence relation will be such as to guarantee that two equivalent conditions whose S 's have the A-property (i.e. the property which defines a A-system) are compatible.
Accordingly,
the elements of Z are pairwise compatible, and so X could not have been an antichain.
243 REFERENCES Cl]
DEVLIN, K.J., ON HEREDITARILY SEPARABLE HAUSDORFF SPACES IN THE CONSTRUCTIBLE UNIVERSE, FUND.MATH. Vol.82 (1974) N— 1, pp.1-10.
[2]
DEVLIN, K.J., ASPECTS OF CONSTRUCTIBILITY, LECTURE NOTES IN MATH. Vol.354 (Springer, Berlin, 1973).
[3]
DEVLIN, K.J., to appear.
[4]
DEVLIN, K.J., and JENSEN, R.B., MARGINALIA TO A THEOREM OF SILVER, IN LECTURE NOTES IN MATH. 499, (Springer, Berlin, 1975).
[5]
DODD, A.J., and JENSEN, R.B., THE CORE MODEL, ANNALS MATH. LOGIC 20^ (1981) 43-75.
[6]
JENSEN, R.B., THE FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY, ANNALS MATH. LOGIC 4_ (1972) 229-308.
[7]
REBHOLZ, J., SOME CONSEQUENCES OF THE MORASS AND DIAMOND, ANNALS MATH. LOGIC 7_ (1975) 361-386.
[8]
SHELAH, S., and STANLEY, L., S-FORCING AND MORASSES I. A "BLACK BOX" THEOREM FOR MORASSES, WITH APPLICATIONS: SUPERSOUSLIN TREES AND GENERALIZED MARTIN'S AXIOM, to appear.
[9]
STANLEY, L., CHARACTERIZING WEAK COMPACTNESS, to appear.
[10]
VELLEMAN, D., Ph.D. THESIS, UNIVERSITY OF WISCONSIN, 1980.
LIST OF PARTICIPANTS
Abraham, U., (formerly Avraham, V.) Department of Mathematics, The Hebrew University, Jerusalem, Israel. Ambos, K., Mathematisches Instituts, Theresienstr., D-8000 Munchen, BRD. Andersen, J., Crewe & Alsager College, Alsager, Stoke-onTrent. Apter, A., Department of Mathematics, M.I.T., Cambridge, MA 02x39, US. Arens, Y., Department of Mathematics, Evans Hall, U.C.B., Berkeley, CA 94720, U.S. Baumgartner, J., Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, U.S. Blass, A., Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, U.S. Brown, M., Peterhouse, Cambridge. Carlson, T., Department of Mathematics, University of Colorado, Boulder, Colorado, U.S. Carr, D.M., Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1. Dahlhaus, S., Osthofener Weg 17, 1000 BLN-W 38, W. Germany. Dodd, A. New College, Oxford. Donder, H-D., Seminar fur Logik, 53000 Bonn, Beringstr. 6, W. Germany. Drake, F., School of Mathematics, The University, Leeds LS2 9JT. Franek, F., 66 Spadina, Apart. 904, Toronto, Canada. Erdos, P., Mathematical Institute, Hungarian Academy of Science, Realtanoda u 13-15, Budapest V, Hungary. Falaki, M., Wolfson College, Oxford. Farrington, P., School of Mathematics, The University of Leeds LS2 9JT. Ferro, R., Via Gabelli 57, 35100 Padua, Italy. Friedman, S., Department of Mathematics, U.C.B. Berkeley, CA 94720, U.S. Gardiner, A., Department of Pure Mathematics, University of Birmingham, PO Box 363, Birmingham B15, 2TT. Gilow, C , Department of Mathematics, Cornell University, Ithaca, New York, U.S. Gold, A., University of Windsor, Ontario, Canada N9B 3(4. Griffor, E., Department of Mathematics 2-247, M.I.T., Cambridge, MA 02139, U.S. Groszek, M. , Department of Mathematics, Harvard, Cambridge MA 02138, U.S. Guaspari, D., St. John f s College, Annapolis, Maryland, U.S. 21404. Hajnal, A., Mathematical Institute, Hungarian Academy of Science, Realtanoda u 13-15, Budapest V, Hungary. Halpern, J.D., Department of Mathematics, University of Alabama, Birmingham AL5243. Harrington, L., Department of Mathematics, U.C.B., Berkeley, CA 94720, U.S.
246
Huber-Dyson, V University of Calgary, Department of Philosophy, Calgary, Alberta T2N 14N, Canada. Hyland, M., King's College, Cambridge. Isbell, J., Department of Mathematics, SUNY Buffalo, U.S. Izouvaros, A., Department of Mathematics, University of Thessalonika, Greece. Jenkins, J., Peterhouse, Cambridge, CB2 1QU. Johnstone, P., Department of Pure Mathematics, 16 Mill Lane, Cambridge. Jones, M., New Hall, Cambridge. Kanamori, A., Department of Mathematics, Harvard, Cambridge, MA 02138, U.S. Kastanas, I., Department of Mathematics, Caltech, Pasadena, CA 9110*, U.S. Kechris, A., Department of Mathematics, Institute of Technology, Pasadena,CA 9110b*, U.S. Kimmel, K., Forschungsinstitut fiir Anthropotechnik, 5309 Mechenheim, Buschstrasse, BRD. Koepke, P., Department of Mathematics, U.C.B., Berkeley, CA 94720, U.S. Koppelberg, B., Mathematical Institute, FU Berlin, KoniginLuise-Str. 24, 1U00 Berlin (West) 33. Legrand, M. , Penn State University, University Park, Penn. 16801, U.S. Levinski, J-P., Universite Paris VII, U.E.R. de Mathematiques, 2 Place Jussieu, 75221 Paris, France. Libert, D. Department of Mathematics, U.C.B. Berkeley, CA 94720, U.S. Lin, C , Department of Mathematics, 2-270, M.I.T., Cambridge, MA 02139, U.S. Lindstrom, I., Department of Philosophy, Stanford University, Stanford, California 94305, U.S. Linton, F., Department of Mathematics, Wesleyan University, Middletown CT 06457, U.S. Maass, W., Mathematisches Institut der Universitat, Theresienstr. 39, D-8000 Munchen 2, W. Germany. Mackenzie, K., Max Rayne House, 109 Camden Road, London NWl. Magidor, M., Department of Mathematics, The Hebrew University, Jerusalem, Israel. Martin, D.A., Department of Mathematics, UCLA, Los Angeles, U.S. Mathias, A.R.D., Peterhouse, Cambridge. Mignone, R., Department of Mathematics, Penn State University, Univesity Park, PA 16802, U.S. Miller, A.W., Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, U.S. Milner, E., Department of Mathematics & Statistics, Faculty of Science, University of Calgary, 2920 24 Ave N.N., Calgary, Canada. Mitchell, R. , The School of Mathematics, The University, Leeds LS2 9JT. Mitchell, W., 420 E 70th St., Apartment 3A, New York City, NY 10021, U.S. Moerdijk, I., Roeterstraat 15, Amsterdam, Netherlands.
247
Mosbach, M., Berthold-Schwarz-Str. 25, D-6700 Ludwigshafen/RH. 14, BRD. Moss, J., 1 Colosseum Terrace, Albany St., London NWl. Normann, D., Institute of Mathematics, Boks 1053, University of Oslo, Blindern, Oslo 3, Norway. Normann, S., Institute of Mathematics & Statistics, Boks 35, Agricultural University of Norway, Ass-NLH, Norway. Pearce, J., Department of Mathematics, UCB., Berkeley, CA 94720, U.S. Pelletier, D., Department of Mathematics, York University, Toronto M3J 1P3, Canada. Philp, B., Pure Mathematics Department, University of Birmingham, Edgbaston, Birmingham 15. Prikry, K., Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455. Quinsey, J., Mathematical Institute, 24/29 St. Giles, Oxford. Rado, R., 14 Glebe Road, Reading RG2 7AG. Rothacker, E., Steinkuhlstr. 22, D-463 Bochum 1, W. Germany. Labib Sami, R., Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt. Shelah, S., Department of Mathematics, The Hebrew University, Jerusalem Israel. Shore, R., Department of Mathematics, Cornell University, Ithaca, NY 14bb3, U.S. Silver, J., Department of Mathematics, UCB., Berkeley, CA 94720, U.S. Singh, D., Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq. Slaman, T., Department of Mathematics, Harvard, Cambridge, MA 0213b, U.S. Smith, B., Department of Pure Mathematics, University of Cambridge. Smorynski, C , Mathematisch Insitut der Rijksuniversiteit te Utrecht, Budapestlaan 6, Utrecht, Netherlands. Stanley, L., Universite de Clermont, Complexe Scientifique des Ceseaux, Mathematiques Pures, B.P. 45-63170, Aubiere, France. Stekeler, P., Rheingutstr. 12, D775, Konstanz, W. Germany. Stoltenberg-Hansen, H., Institute of Mathematics, Boks 1053, University of Oslo, Blindern, Oslo 3, Nowary. Tall, F., Department of Mathematics, University of Toronto, Ontario, Canada. Tavares, J., Instituto Superior de Engenharia de Coimbra, Quinta da Nova, Coimbra, Portugal. Taylor, R., Department of Philosophy, Columbia University, New York City, U.S. Thiele, F., Breisgauer Str. 30, D1000 Berlin 38, W. Germany. Thomason, A., Peterhouse, Cambridge. Watson, S., Department of Mathematics, University of Toronto, Ontario, Canada. Welch, P., Mathematical Institute, 24/29 St. Giles, Oxford. Wolfsdorf, K., Grossbeeren Str., 78, D-l Berlin 61, Germany. Wong, H., Room L, Department of Mathematics, Bedford College, Regent's Park, London. Zieqler, M., Dietrich-Schafer-Weg 38, 1000 Berlin (West) 41.
