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A Laver-Type Indestructability for Accessible Cardinals
Logic Colloquium '86
F.R.Drake and J.K. Truss (Editors)
9
0 Elsevier Science Publishers B.V. (North-Holland), 1988
A LAVER-TYPE INDESTRUCTABILITY FOR ACCESSIBLE CARDINALS Shai Ben David, Department of Mathematics, Hebrew University of Jerusalem. INTRODUCTION In [L78] Laver introduced a forcing notion super-compactness of a super compact x
PH
that makes the
indestructible under x-directed-
closed forcing extensions. to an accessible cardinal and
We wish to collapse a large cardinal H
still have indestructibility for some of its compactness properties. If x
A
is a super compact and
stationary subset S of
A
is a cardinal above it then every
reflects, provided all its elements have co-
x.
finalities less than
As an application
We get an indestructibility theorem for this property. we get a model where for every long as [a E S:cof(a.)'
every stationary subset S C A
# A} is stationary in
super compact cardinal, p notion P
A
regular ( H
A.
reflects as
x is a
We prove that if
x 5 A then there is a forcing
and
so that: p
(i) P is p-closed (so in particular
and cardinals below it are not
collapsed); (ii) In
(iii) P (iv) if
V[PI x = p+;
does not collapse cardinals or change cofinalities above x;
P
is generic for
order then in V[Pl[k] For
A
regular
P
P
then V[P]
1
"If R
is a x-closed partial
every stationary subset of
S
reflects".
x to A+.
is just the usual Levy collapse of
For
is the Levy collapse folowed by a A+-strategically closed
singular
A,P forcing of size A+. 1.
A VERY WEAK BOX PRINCIPLE Assume
(A+)<'
=
Let (A :a. < )'1
A+.
+ a
sequence of subsets of ]Aa] 5 A
IJ
and
P
Aa = PcA(i+).
c
),+:]XI
For
be an increasing and continuous such that for each a,
< A}
6 < A+
a limit ordinal we say that
a
Ai.
Let
S*(A+) =
i<6 the definition of
and every initial segment of
I6 < A+:&
* +
S (1 )
* +)
s (A
uniquely defined.
is a member of
is not approachable}.
depends upon the sequence
such sequences agree on a closed unbounded subset of class of
c6
(Aa:CI
A+,
Note that although
< A'),
as any two
the equivalence
modulo the ideal of non-stationary subsets of A +
is
S.Ben David
10
Let us denote by 0;
the property
'I
S
* (A+
".
A+
is not stationary in
(The superscript s stands for Shelah who introduced this property in [ 5 7 8 ] . ) Another way of formulating 0; (b) the sequence
3 (Aa:Q <)'A
is:
such that:
A,
(a) for each a , IAaI 2
A>
(Aa:a <
(c) for each limit 6 <
A+
is 5-increasing and continuous, there is a closed
c6,
6,
cofinal in
otp(c ) = cof(6) and for every 6 E c6, c6n6 E A6 (by continuity A 6 * 6 =a
Oh
above and (c'): For each limit
6 , otp(c ) = cof 6 and for every 6
6 < A+ there is a closed c6, cofinal in 6 Ec6, c6r16 E A 6'
OBSERVATION ( 1 ) If
(K,,s0)
+
(A+,A)
One way to see it is:
holds.
then 0;
the transfer property implies the existence of a
A+-special Aronszajn tree, which is equivalent to the principle 0;
and
0;
implies 0;. ( 2 ) p<'
'
implies 0'.
=
directly.
THEOREM 1 . O Shelah
This follows from ( 1
A,
For a singular
( [ 5781 ) :
supercompact cardinal H
implies that there is no
0;
A.
cof(h) < H. <
such that
COROLLARY The failure of 0;
but can also be easily proven
for a singular
is consistent with
ZFC
if
the existence of a supercompact cardinal is. REMARK
In
[ S 7 8 ] a model of
*
model f o r 7 0
ZFC for - 0
is constructed, and in [BS86] a YJJ
Iyw
is constructed.
Both models are forcing extensions of a
ground model with a supercompact cardinal. The relevance of 0;
to our topic is through Shelah's theorem 20 of [ S 7 8 ] .
This theorem implies that: SA+ and <' V[+I 14
+
P
is a '-closed
If VkO;,
then if
forcing notion,
s is stationary in A+
h
S
is a stationary subset of
V-generic for P, then
$1.
The next theorem shows that any model can be extended by a 'mild' forcing to where 0; THEOREM 1 . 1
holds (for any given For every cardinal h
A). such that '2 2 h
for all
< 1,
is a A-strongly-strategically-closed forcing not'ion PA such that and a generic set for PA introduces a 0; sequence. For a definition of paragraph of section 2 .
'' A-strongly-strategically-closed
"
there
lPAl = 2'
see the first
A Laver-Type Indestructability for Accessible Cardinals REMARK
11
The cardinal arithmetic assumption can be weakened at the cost of
complicating the definition of 0; along the lines of Definitions 1 , 2 in [5781. proof. -
A condition p
A3
p = (A :a < 1.1 <
;
in
is just an initial segment of a 0:
PA
is a set of 5 A
each A
is 5-increasing, continuous, and for each limit 6 5 p exists a closed cofinal subset c6 proper initial segment of p
q
c6
if Dom(p) 5 Dom(q)
and
q
(=
Dom(p))
of order type cof(6)
belongs to
sequence,
subsets of a , the sequence
A6.
there
such that every
The order is end-extensions:
(Dom(p)) = p.
CLAIM For every p
Proof. c6
E PA and
p
p < A+
there is an extension of
p,q
such that
E Dom(q). Let p = (A :a < 11)
and
a
6
closed cofinal in &:p 5
11 5 p .
For each limit p 5 6 5 p
of order-type cof(6).
6 5 p}.
Let q
be
Now, for
6
11
(Aa:a < p)^ (A6:p 5
6
pick some 5 p
let
5 p).
CLAIM PA is A-strongly-strategically closed.
Proof.
Having a chain of conditions of cardinality
condition extending all of them should extend
(pi:i < p ) , any pi; on the other hand, if
u
iEp U pi is a condition in PA. so to i
extends
U pi then
guarantee that the union of this chain is a condition. obstacle to
U p. being a member of PA
i
The only possible
is that maybe for 6
such that otp(c6) = cof(6)
Let us defineplayer 1's strategy H, by
H (pi:i 5
a)
-
=
=
iLlllDom(pi),
and for each
a!i[
"i] AADom(pa)
A is {a:a c {Dompi:i a and pi was picked by player I}) -Dom(pa) where a is the closure of a in the ordinal topology. Note that by the where
assumption p <
A
a,
I A ~ ~ ~ ( ~ L~ A.) I
Now at any limit stage p , (Dom(pi):pi was picked by player I) is cofinal in 6 = U Dom(p. ) , so i
3
21-I 5
Every initial segment of such
a
is a member of some
for some stage i < P that was played by player ADom ( p i shown that H is a winning strategy for player I.
I.
We have thus
S. Ben David
12
The definition of
PA and the previous claim imply that a generic set for sequence: this completes the proof of Theorem 1.1.
PA introduces a 0;
2. ON !.I-STRATEGICALLY-CLOSEDFORCING NOTIONS
A partial order
DEFINITION
is p-strongly-strategically-closed if player
has a winning strategy in the following game.
I
Each player in his turn
chooses a condition p E p extending all the conditions that have been chosen at previous stages, thus constructing an increasing chain in p (Pi:i < a ). Player I1 gets to play all even and limit stages and Player I the odd stages. Player I wins if for any a 5 p
.
(P. :i < a)
there is condition in
Clearly, any p-closed p
a p-strongly-strategically-closed p so
it is
extending
is a !.I-strategically-closedpartial order
p
distributive (forcing via such
(p,m)
p
is !.I-strongly-strategically-closedand adds no p-sequences to the
ground model). In Theorem 2.5 we indicate why being p-strongly-strategically-closed is strictly weaker than being p-closed. We will now generalize some theorems due to Magidor and Shelah (respectively) concerning closed partial orders, to strongly-strategically-closed partial orders. THEOREM 2.1
(Magidor [M82], Lemma 3 )
notion
regular) and
(H
p
Let p
be a X-closed separative forcing
a cardinal above the cardinalities of p
and the
set of its dense subsets. (a) In V[col(X,
(b) For every 6, of
V[6]
p
there is a
generic filter over
is a x-closed forcing extension of g-generic over
V,
V,6,
such that
v[6].
there is a x-closed forcing extension
making it a model of the form V[col(x,
and such that
6 is
the g-generic object defined in this model by part (a). OBSERVATION 2.2 "
P
The conclusions of Theorem 2.1 still hold when the assumption
is x-closed"
is replaced by
"
p is x-strongly-strategically-closed" .
This was independently noticed by Matt Foreman. Proof.
One just repeats Magidor's proof but in the construction of the
sequence
(pa:CY < 8)
V[col(H,
uses Player's I strategy.
define a V-generic subset-for (pas < x)
an increasing sequence one-to-one mapping of
x
onto
generic set of thecollapse. (pa:"
< 0)
,
if
F(P)
{D
p
More specifically, in by constructing inductively
of elements of p.
c P:D
Let p
is dense in
Let
F
be the
introduced by the
be the minimal upper bound of
is a dense subset of
p
we define pD
to be some
A Laver-Type Indestructability for Accessible Cardinals canonical extension of
in
p
F(p)
is the strategy for Player I; if
pp = H( (Pa:@ < p ) * @ d )
and
to be any common extension of p and q x-closed any sequence of length < x
of
pick
if such exists, and p
p
of elements of
H
P
P
otherwise,
Note that as the collapse to
x
is
is a member of the
ground mode1,so it belongs to the domain of the strategy H. x-strongly-strategically-closed and
where H
of p
F(p) is some member q
and proceed as in the previous case.
13
As
p
is
is used in successor stages (the choice
can be regarded as being made by Player 11) an upper bound to
pp
ba:a < p) Let
exists for all
p <
x.
be the p-generic filter generated by
G
check that V[col(x,
+,:a < M).
It remains to
is a x-closed extension of V[G].
This is done by
repeating the argument of Magidor's proof.
p
REMARK We do not know if the theorem still holds when the assumption on is replaced by:
Shelah IS781 (Theorems 20, 21, 22) Supposeforaregular p
THEOREM 2 . 3 above it,
V[61,
S
p is x-strategically-closed.
S
5 gcf(X+).
Then there is a p-closed forcing
is not stationary if and only if
REMARK The assumption
S
THEOREM 2 . 4
X for all
is not stationary.
For p , ~ , s 5 'A
S.
It follows (for every
S
g A+)
from
p < p.
A+
as above if
(S\S*(A+))~S<~is stationary
p,
then for any p-strategically-closed forcing
S
is stationary in V[i].
Note that we don't assume strong-strategical-closure. demand on
and
such that in
5 gcf(X+) relates to the cardinal arithmetics of
the cofinalities of the elements of the assumption 2 ' 5
[s\S*(X')lnSx+
p
This relaxes the
p by allowing games in which Player I has to play the limit
stages in order to guarantee a winning strategy. Proof.
The proof is the proof of Shelah's theorem.
submodel N < H
of size , Isuch that its A+
Once forms an elementary
is some 6 E
(A++) constructs ih this model a cofinal sequence of conditions of of Player I is used (instead of the p-closedness of
p
S
and
p. The strategy
in Shelah's proof) to
guarantee that there is some p E p above all the members of the sequence. is stationary allows to pick 8 outside The assumption " (S\S*(A')) 'I
S*(X')
so there is some cofinal sequence in
segments are members of THEOREM 2 . 5
6
such that all its initial
N.
If there exists a super compact cardinal, then the assertion
S. Ben David
14
P
'I
is A-strongly-strategically-closed
A-closed
1.1 <
More precisely, there are cardinals
(i)
is strictly weaker than
"
p
p
is p-closed.
(iii) The cardinality of
K
is
Let
A and a partial order p
so that:
x).
Let
one of them extends the other.
PA
+W
V[D]
and
A
(the omegath
indestructible under
K
be the partial order of Theorem 1 . 1 .
PA,
if
p
V[6.J.
If
is a cardinal such that
D 5 PA
If
and q
It follows that if
is a super compact cardinal in
holds in
= K
is A-strongly-strategically-closed.
PA
then it is X-directed closed as in then H
A
Apply Laver's method to make
x-directed closed forcings. [PA[= A+;
A+.
be a super-compact cardina1;set
K
cardinal above
0:
is
is A-strongly-strategically-closed.
(ii) No dense subset of
Proof.
P
'I
'I.
is H-closed
are compatible, then
PA D
is V-generic for
is dense in
cof(A) < K < A.
PA
then
This
contradicts Theorem 1 . 0 . This theorem aswers a question raised by Forman.
3. THE MAIN RESULT The next theorem is the key to the results of this paper.
It is a
generalization to K-strongly-strategically-closed partial orders, of a claim that appears implicitly in the proof of Magidor's Theorem 2 o'f [M82]. Let
THEOREM 3 . 1
below n.
n be a supercompact cardinal and
p
If
V[col(p,
p
some regular cardinal
is a x-directed-strongly-strategically-closed
V[cOl(p,
satisfies:
A s 5 S C K , if s\S*(A) stationary in
'I.
COROLLARY 3.2
For
"
For every
is stationary in
K,p,
p
A,
h 1 K s.t. cof(A) 1 K then for some a <
as above, V[cOl(P,
(a) For any cardinal A > p, 0;
forcing in
P,
then the model obtained by further forcing via
and every
A,
Sna
satisfies:
implies that every stationary subset of
sA+ reflects. CP
(b) For any
p > p
if 1Jcp = p
then every stationary subset of
S"
CP
reflects.
LEMMA 3.3
Proof. As
The theorem holds for K-directed-closed P's.
Pick a cardinal A * K
above K
and
2"'.
is a supercompact cardinal (in the ground model V)
elementary embedding
j:V+ M
is
to some transitive model M
there is an
of
ZFC s.t.
A Laver-Type Indestmctability for Accessible Cardinals N
j(x) > A * and ~ A 5 M,
is its critical point,
follows that
P, col(p,
M.
Extend
conclude that for some P-generic extension of
j to an embedding of
-G
M[j(col(p,
v[col(p,
po
is in
j[P]
be a
T
P
j(x)-directed-closed.
= {j(a):a <
A}
p
and
=
Sup(B)
of
p
M[j(colp,
pol/-j(p,
'j(T)ny = j"E'
)J
'.
As
in this model
A
of
p
disjoint from
is a closed unbounded subset of X
M[j(colp,
E
is not stationary in A .
M[col( p,
E
was stationary in
is x-closed,
P
E
A.
j"E.
there is a
It is not hard to check
disjoint from
E,
therefore
On the other hand, in Now we get the desired contra-
is a x-closed extension of
diction by recallingthat j(colp,
G
has cofinality p
therefore it follows that already in M[j(col(p,
closed cofinal subset H
In
I f this is not the case, let
I .
is not stationary in
col(p,
Therefore,
E G and
'j"E
j-'(€i)
is
then in M[col(p,
and
M[j(COl(p,
is j(x)-closed in
po
p
such that
in
E 6)
of size < A * < j(x)
name for a stationary
pok j(p) 'j"E is stationary in
(as j(H)>A)
j[P]
-
is
j(P)
that
-
so G ={j(p):p
M
be a generic set for
j[P]
is a x-closed forcing
E j[P] above all members of G (in V[j(col(p,
let
,B
M, jlP
is realised as a stationary subset E
CLAIM
is a
= jot), V= M[co~(~,
I.(!
i , M[j(col(p,
is a directed subset of
is x-directed-closed
there is some
T
M[COl(p,
so
M[COl(p,
By the closure properties of
and as
It
M.
M[j(col(p,
Applying Observation 2 . 2 to our case
in
E
to M[j(COl(p,
j(col(p,
implies A
and all their subsets and the names in their
forcing languages are members of V[COl(p,
5 A*
15
is forced to be a stationary subset of (s?,\s;)
its
stationariness cannot be destroyed by such forcing extensions (Thm. 2 . 3 ) . M [ j ( c o l ( p , c ~ ) ) l * [ j ( P ) ] I ~ " c o f j2( ~l(X) )
of this model and has cardinality p and
jot) >
A,
so
j"(h)
is bounded in
j"E is a stationary initial segment of implies therefore that in
"
(as cof(X) 2 K ) j"(A)
there, as in this model
j(A)). j(E).
V[col(p,
is a member
j(x)
is
The elementariness of E
p+
It follows that in this model j
has a stationary initial
segment.
Proof of the theorem forcing in
V'
model, where
Let
Q
be a x-directed-strongly-strategically-closed
Let p be COl(H,<6) in the sense of this 6 is some cardinal above 2IQ1. P is x-directed closed, so =
V[col(p,
S. Ben David
16
by the previous lemma in
By Lemma 3 . 3 x-closed.
x
S 5 S,,
and
Sx
and
V'[hl.
S\S*(A)
As
-
for some P
such that in
is stationary, then
is H-closed and in
It follows that in
The same a
and
S
S
is
is still stationary in
for some a < A Sna
V'[$]
V'[Q]
is a stationary
cof(A) 1 H
V"Q1
is
S\S*(A)
a.
is stationary in
It follows from Theorem 2 . 3 that if in V"Q1
subset of
a.
A, Sna
is V'[Q][P]
V'[h]
V'[Ql[Fl = V'[h]. in
cof X 2 x
if
then for some a <
A,
stationary in
V'[fi],
is therefore a reflection point for
S
the same holds
is stationary in
in
V"Q1.
The following theorem is a straightforward corollary to Theorem 3 . 1 . THEOREM 3 . 4
If
ZFC
cardinal then so is
is consistent with the existence of a supercompact ZFC with "For every H-directed-strongly-strategically-
closed forcing notion cof(X) 2 H
g
after forcing via
and every stationary subset
is stationary then
S
g
for every ordinal
S of
S
n
reflects", this for every
A
s.t.
if {aEs:cof(a)+ # cofll successor of a regular
cardinal.
4 . APPLICATIONS
If
THEOREM 4 . 1
V
V[&l
" ZFC + a-many supercompact cardinals ", 6 such that cof(6) = a there is a forcing extension
is a model of
then for every ordinal
adding no a-sequences to
V,
in which every stationary subset of
K6+ 1
reflects.
Proof.
Let
(H.:i < a )
be an increasing sequence of supercompact cardinals.
lim N ~ ,and let (xi:i < a ) a i
Let
H
=
partial orders Pi; Po in
V[hil
and
ei:i < p) for g.
)I.
is colt h,,
Pi+, = Pi*Qi+l.
followed by
col( Xp
0'
'"0.
be an increasing sequence of regular By induction on Qi+,
i < a
define
is col(Xi,
In limit stages P p is the inverse limit of
,
P
).
Let
g
= Pa
and let b be generic
Standard arguments about the Levy collapse show that in
V[i]
is a cardinal, lim xi = 8, and X z is a cardinal, so it is i
each
b+,-
Hi+,-directed-closed forcing notion (for any that any
A -sequence of V[hl -
Pi, P,-generic).
is an element of
V
is a
It follows
and in particular
p adds
no &sequences. Let
R
be the forcing notion described in.section1 for adding a 0'
#6 sequence to V[h]. -
Recall that
R
preserves cardinalities and cofinalities.
A Laver-Type Indestmctability for Accessible Cardinals
v[~I[1 ~ I"Every stationary subset of K , + ~ reflects".
CLAIM 4 . 2
Proof of the claim S . = {p
E
S 5
Let
K,+,
S:cof(p) < A , }
The function f:S + a
be stationary.
X.1
f ( p ) = min{i:cof(p) <
defined by Let
17
is pressing down so for some
is stationary.
Let u s show that
i < a
Si reflects.
Qi denote the V[il
Levy collapse colOti, <%+I). i V[g]*[Qi] = VIPil*[P1l*[Qi], and P *Qi isa K.-closed forcing notion (as an By Theorem 2 . 4 , as R
iteration of two x.-closed notions). strategically-closed in V[Pi], closed
Q.
is x.-strongly-
can be regarded as R*Q*
for some
K.-
Q*.
In V[P]*[R],S*(K6+1)= p'mod the non stationary ideal, so therefore without
xs+l1.
loss of generality Si 5 K,+,\S*( that
Si
Applying Theorem 2 . 4 , we conclude
is still a stationary subset of
(k,+ljJ[pR1 is
an ordinal of cofinality xl.
[pR1 &+' , 1
reflects".
some p <
sinp i s stationary in p .
is stationary in
p
REMARK 4 . 3 in V[p]
as V[pR]
(Xi:i < a)
and
=
such that for some Qi:
(a) Qi
Sup 1 i
every stationary subset of If
ZFC
s:ii
A+
is a forcing notion
.
%Xi '
reflects.
is consistent with the existence of arbitrarily many
X,
existence of a non-reflecting stationary subset of
Oi
Oi
a forcing notion BX
without collapsing cardinals or changing cofinalities and
such that for each regular the demands (a) and (b) for
Assume
does not imply the
X.
In [B,S861 we constructed for every singular X
that introduces
THEOREM 4 . 5
V[pRlk"Sinp
p is a Levy collapse of superthen if R
supercompact cardinals then, for every singular
Proof.
reflects
In particular, for
is X.-closed ;
then in V[pR] COROLLARY 4 . 4
(though
V[pQi].
does not destroy stationarity of subsets of
(b) R*Qi
cof(h,+l tv[pRl
For the same p ,
is a submodel of
What we actually proved was that if
compact cardinals
V[_p][Qi]
Col(Ki,<&+l)
every stationary subset of
and so "every stationary subset of
'I
in
makes the old K,+,
no longer a cardinal).
corollary (b) in V[p][Qi]
(%+l) VIP1[R1 -
ZFC
1-(
<
X there is some forcing notion Qp satisfying
p,X
of Remark 4 . 3 .
is consistent with the existence of a proper class
of supercompact cardinals.
Then
ZFC
is consistent with "For every cardinal
every stationary subset of { a < X:cof(a)+ # A}
reflects".
S. Ben David
18
REMARK
(a) The result is the strongest that can be expected in the sense
4.6
X
that if
= p',
p
a regular cardinal, then
{a < X:cof(a)
= p} is a non-
A.
reflecting stationary subset of
( b ) The assumption is a 'natural' one in the sense that reflection of
Xt
stationary subsets of
implies strong failure of KJx
(even 0
Xp - a - fails for all p < Xi.
sequence defined for ordinals of cofinalities 2 p
In [B,S86] it is shown that 0 is consistent with the existence of superXlJ compact cardinals K s.t. cof(X) < K < p ) . It follows that a model of ZFC with full reflection as above has an inner model with proper class of large cardinals.
Proof.
Let
be a model of
V
(xCY:aE On,a successor).
ZFC with unboundedly-many supercompactcardinals
Without loss of generality every limit of super-
compact cardinals is a singular cardinal. regular limit and replace V
( ~ ~ E: Ona) notion (1)
is
H(6)
by
of
(If not, let
is increasing and continuous.
p so that: For each CY p
= P *Pa
where
is
Pa
6 be the first such
Let us also assume that
V.)
We define a proper class forcing
K
~
-+ C.C. ~
(and a set) and
Pa
K -directed-strongly-strategically-closed.
( 2 ) If
p
is a successor of a regular cardinal in
V[i],
then in
V
p
is one of the supercompacts in our sequence. 1 3 ) If in
V"1
p
is p'
is regular, then for some a E On
p
and
Pa+, = Pa*co~(~t<~CYtl) *re c~l(p.
tion.
p is by induction on a.
The definition of
where
= Pa *Col(K
pa+ 1 sense of
v[pa].
For a limit a
? ~ = v ' ~ = s KU p is the PA p
P
a+1
Col(Xt,
(P :B < a). It is quite B p does satisfy our list of demands. (One
is the inverse limit of
straightforward to check that this has to use the inaccessibility of the closure property of
Y
= P *R *Col(X+,
forcing notion defined in section 1 for intro-
VIPal [RCYl.
At limit a's, P
K+
For successor a
Col(K , < H ~ + ~ isthe ) Levy collapse in the
Pa
are still cardinals in
REMARK
K
~
for a < y
+to~ get the
for P "a+1 - c.c* to verify that for limit y K
Y
and and
V"].)
4.7 'Actually we get a somewhat stronger result in
v"].
Not only
do we have the full reflection but for every p-directed-stronqly-strategicallyclosed
R
in
V[i][fi]
we still have relfection for the stationary subsets of
ordinals of cofinality less than
p.
A Laver-Type Indestructability for Accessible Cardinals Added in Proof. questions:
The proofs of Theorems 4 . 1 and 4.5 raise the following
Could we get the models of reflection by just Levy collapsing the
supercompact cardinals of the ground model? 0:
19
Do we really have to force the
sequences to get f u l l reflection of stationary sets? Shela has given a partial answer to these questions by proving that one can
force over the ground model
V
the supercompact cardinals of V
v*
to get an intermediate extension V' are still supercompact in
V'
so
that
and the model
obtained by iteratively Levy collapsing the supercompact cardinals of V'
(in the manner described in the proofs of Theorems 4 . 1 and 4 . 5 ) has the desired reflection properties.
S. Ben-David, S. Shelah, Non-special Aronszajn trees on Journal of Mathematics 53, ( 1 9 8 6 ) 93-96.
,fi
Israel
R. Laver, Making the supercompactness of x indestructible under x-directed closed forcings, Israel Journal of Mathematics 29, ( 1 9 7 8 ) 385-388.
M. Magidor, Reflecting stationary sets, Journal of Symbolic Logic 47, 4,
( 1 9 8 2 ) 755-771.
S. Shelah, On successors of singular cardinals, in M. Boffa, D. Van Dalen, K. McAloon (eds) Logic Colloquium '78, North Holland ( 1 9 7 9 )
357-380.
F.R.Drake and J.K. Truss (Editors)
9
0 Elsevier Science Publishers B.V. (North-Holland), 1988
A LAVER-TYPE INDESTRUCTABILITY FOR ACCESSIBLE CARDINALS Shai Ben David, Department of Mathematics, Hebrew University of Jerusalem. INTRODUCTION In [L78] Laver introduced a forcing notion super-compactness of a super compact x
PH
that makes the
indestructible under x-directed-
closed forcing extensions. to an accessible cardinal and
We wish to collapse a large cardinal H
still have indestructibility for some of its compactness properties. If x
A
is a super compact and
stationary subset S of
A
is a cardinal above it then every
reflects, provided all its elements have co-
x.
finalities less than
As an application
We get an indestructibility theorem for this property. we get a model where for every long as [a E S:cof(a.)'
every stationary subset S C A
# A} is stationary in
super compact cardinal, p notion P
A
regular ( H
A.
reflects as
x is a
We prove that if
x 5 A then there is a forcing
and
so that: p
(i) P is p-closed (so in particular
and cardinals below it are not
collapsed); (ii) In
(iii) P (iv) if
V[PI x = p+;
does not collapse cardinals or change cofinalities above x;
P
is generic for
order then in V[Pl[k] For
A
regular
P
P
then V[P]
1
"If R
is a x-closed partial
every stationary subset of
S
reflects".
x to A+.
is just the usual Levy collapse of
For
is the Levy collapse folowed by a A+-strategically closed
singular
A,P forcing of size A+. 1.
A VERY WEAK BOX PRINCIPLE Assume
(A+)<'
=
Let (A :a. < )'1
A+.
+ a
sequence of subsets of ]Aa] 5 A
IJ
and
P
Aa = PcA(i+).
c
),+:]XI
For
be an increasing and continuous such that for each a,
< A}
6 < A+
a limit ordinal we say that
a
Ai.
Let
S*(A+) =
i<6 the definition of
and every initial segment of
I6 < A+:&
* +
S (1 )
* +)
s (A
uniquely defined.
is a member of
is not approachable}.
depends upon the sequence
such sequences agree on a closed unbounded subset of class of
c6
(Aa:CI
A+,
Note that although
< A'),
as any two
the equivalence
modulo the ideal of non-stationary subsets of A +
is
S.Ben David
10
Let us denote by 0;
the property
'I
S
* (A+
".
A+
is not stationary in
(The superscript s stands for Shelah who introduced this property in [ 5 7 8 ] . ) Another way of formulating 0; (b) the sequence
3 (Aa:Q <)'A
is:
such that:
A,
(a) for each a , IAaI 2
A>
(Aa:a <
(c) for each limit 6 <
A+
is 5-increasing and continuous, there is a closed
c6,
6,
cofinal in
otp(c ) = cof(6) and for every 6 E c6, c6n6 E A6 (by continuity A 6 * 6 =a
Oh
above and (c'): For each limit
6 , otp(c ) = cof 6 and for every 6
6 < A+ there is a closed c6, cofinal in 6 Ec6, c6r16 E A 6'
OBSERVATION ( 1 ) If
(K,,s0)
+
(A+,A)
One way to see it is:
holds.
then 0;
the transfer property implies the existence of a
A+-special Aronszajn tree, which is equivalent to the principle 0;
and
0;
implies 0;. ( 2 ) p<'
'
implies 0'.
=
directly.
THEOREM 1 . O Shelah
This follows from ( 1
A,
For a singular
( [ 5781 ) :
supercompact cardinal H
implies that there is no
0;
A.
cof(h) < H. <
such that
COROLLARY The failure of 0;
but can also be easily proven
for a singular
is consistent with
ZFC
if
the existence of a supercompact cardinal is. REMARK
In
[ S 7 8 ] a model of
*
model f o r 7 0
ZFC for - 0
is constructed, and in [BS86] a YJJ
Iyw
is constructed.
Both models are forcing extensions of a
ground model with a supercompact cardinal. The relevance of 0;
to our topic is through Shelah's theorem 20 of [ S 7 8 ] .
This theorem implies that: SA+ and <' V[+I 14
+
P
is a '-closed
If VkO;,
then if
forcing notion,
s is stationary in A+
h
S
is a stationary subset of
V-generic for P, then
$1.
The next theorem shows that any model can be extended by a 'mild' forcing to where 0; THEOREM 1 . 1
holds (for any given For every cardinal h
A). such that '2 2 h
for all
< 1,
is a A-strongly-strategically-closed forcing not'ion PA such that and a generic set for PA introduces a 0; sequence. For a definition of paragraph of section 2 .
'' A-strongly-strategically-closed
"
there
lPAl = 2'
see the first
A Laver-Type Indestructability for Accessible Cardinals REMARK
11
The cardinal arithmetic assumption can be weakened at the cost of
complicating the definition of 0; along the lines of Definitions 1 , 2 in [5781. proof. -
A condition p
A3
p = (A :a < 1.1 <
;
in
is just an initial segment of a 0:
PA
is a set of 5 A
each A
is 5-increasing, continuous, and for each limit 6 5 p exists a closed cofinal subset c6 proper initial segment of p
q
c6
if Dom(p) 5 Dom(q)
and
q
(=
Dom(p))
of order type cof(6)
belongs to
sequence,
subsets of a , the sequence
A6.
there
such that every
The order is end-extensions:
(Dom(p)) = p.
CLAIM For every p
Proof. c6
E PA and
p
p < A+
there is an extension of
p,q
such that
E Dom(q). Let p = (A :a < 11)
and
a
6
closed cofinal in &:p 5
11 5 p .
For each limit p 5 6 5 p
of order-type cof(6).
6 5 p}.
Let q
be
Now, for
6
11
(Aa:a < p)^ (A6:p 5
6
pick some 5 p
let
5 p).
CLAIM PA is A-strongly-strategically closed.
Proof.
Having a chain of conditions of cardinality
condition extending all of them should extend
(pi:i < p ) , any pi; on the other hand, if
u
iEp U pi is a condition in PA. so to i
extends
U pi then
guarantee that the union of this chain is a condition. obstacle to
U p. being a member of PA
i
The only possible
is that maybe for 6
such that otp(c6) = cof(6)
Let us defineplayer 1's strategy H, by
H (pi:i 5
a)
-
=
=
iLlllDom(pi),
and for each
a!i[
"i] AADom(pa)
A is {a:a c {Dompi:i a and pi was picked by player I}) -Dom(pa) where a is the closure of a in the ordinal topology. Note that by the where
assumption p <
A
a,
I A ~ ~ ~ ( ~ L~ A.) I
Now at any limit stage p , (Dom(pi):pi was picked by player I) is cofinal in 6 = U Dom(p. ) , so i
3
21-I 5
Every initial segment of such
a
is a member of some
for some stage i < P that was played by player ADom ( p i shown that H is a winning strategy for player I.
I.
We have thus
S. Ben David
12
The definition of
PA and the previous claim imply that a generic set for sequence: this completes the proof of Theorem 1.1.
PA introduces a 0;
2. ON !.I-STRATEGICALLY-CLOSEDFORCING NOTIONS
A partial order
DEFINITION
is p-strongly-strategically-closed if player
has a winning strategy in the following game.
I
Each player in his turn
chooses a condition p E p extending all the conditions that have been chosen at previous stages, thus constructing an increasing chain in p (Pi:i < a ). Player I1 gets to play all even and limit stages and Player I the odd stages. Player I wins if for any a 5 p
.
(P. :i < a)
there is condition in
Clearly, any p-closed p
a p-strongly-strategically-closed p so
it is
extending
is a !.I-strategically-closedpartial order
p
distributive (forcing via such
(p,m)
p
is !.I-strongly-strategically-closedand adds no p-sequences to the
ground model). In Theorem 2.5 we indicate why being p-strongly-strategically-closed is strictly weaker than being p-closed. We will now generalize some theorems due to Magidor and Shelah (respectively) concerning closed partial orders, to strongly-strategically-closed partial orders. THEOREM 2.1
(Magidor [M82], Lemma 3 )
notion
regular) and
(H
p
Let p
be a X-closed separative forcing
a cardinal above the cardinalities of p
and the
set of its dense subsets. (a) In V[col(X,
(b) For every 6, of
V[6]
p
there is a
generic filter over
is a x-closed forcing extension of g-generic over
V,
V,6,
such that
v[6].
there is a x-closed forcing extension
making it a model of the form V[col(x,
and such that
6 is
the g-generic object defined in this model by part (a). OBSERVATION 2.2 "
P
The conclusions of Theorem 2.1 still hold when the assumption
is x-closed"
is replaced by
"
p is x-strongly-strategically-closed" .
This was independently noticed by Matt Foreman. Proof.
One just repeats Magidor's proof but in the construction of the
sequence
(pa:CY < 8)
V[col(H,
uses Player's I strategy.
define a V-generic subset-for (pas < x)
an increasing sequence one-to-one mapping of
x
onto
generic set of thecollapse. (pa:"
< 0)
,
if
F(P)
{D
p
More specifically, in by constructing inductively
of elements of p.
c P:D
Let p
is dense in
Let
F
be the
introduced by the
be the minimal upper bound of
is a dense subset of
p
we define pD
to be some
A Laver-Type Indestructability for Accessible Cardinals canonical extension of
in
p
F(p)
is the strategy for Player I; if
pp = H( (Pa:@ < p ) * @ d )
and
to be any common extension of p and q x-closed any sequence of length < x
of
pick
if such exists, and p
p
of elements of
H
P
P
otherwise,
Note that as the collapse to
x
is
is a member of the
ground mode1,so it belongs to the domain of the strategy H. x-strongly-strategically-closed and
where H
of p
F(p) is some member q
and proceed as in the previous case.
13
As
p
is
is used in successor stages (the choice
can be regarded as being made by Player 11) an upper bound to
pp
ba:a < p) Let
exists for all
p <
x.
be the p-generic filter generated by
G
check that V[col(x,
+,:a < M).
It remains to
is a x-closed extension of V[G].
This is done by
repeating the argument of Magidor's proof.
p
REMARK We do not know if the theorem still holds when the assumption on is replaced by:
Shelah IS781 (Theorems 20, 21, 22) Supposeforaregular p
THEOREM 2 . 3 above it,
V[61,
S
p is x-strategically-closed.
S
5 gcf(X+).
Then there is a p-closed forcing
is not stationary if and only if
REMARK The assumption
S
THEOREM 2 . 4
X for all
is not stationary.
For p , ~ , s 5 'A
S.
It follows (for every
S
g A+)
from
p < p.
A+
as above if
(S\S*(A+))~S<~is stationary
p,
then for any p-strategically-closed forcing
S
is stationary in V[i].
Note that we don't assume strong-strategical-closure. demand on
and
such that in
5 gcf(X+) relates to the cardinal arithmetics of
the cofinalities of the elements of the assumption 2 ' 5
[s\S*(X')lnSx+
p
This relaxes the
p by allowing games in which Player I has to play the limit
stages in order to guarantee a winning strategy. Proof.
The proof is the proof of Shelah's theorem.
submodel N < H
of size , Isuch that its A+
Once forms an elementary
is some 6 E
(A++) constructs ih this model a cofinal sequence of conditions of of Player I is used (instead of the p-closedness of
p
S
and
p. The strategy
in Shelah's proof) to
guarantee that there is some p E p above all the members of the sequence. is stationary allows to pick 8 outside The assumption " (S\S*(A')) 'I
S*(X')
so there is some cofinal sequence in
segments are members of THEOREM 2 . 5
6
such that all its initial
N.
If there exists a super compact cardinal, then the assertion
S. Ben David
14
P
'I
is A-strongly-strategically-closed
A-closed
1.1 <
More precisely, there are cardinals
(i)
is strictly weaker than
"
p
p
is p-closed.
(iii) The cardinality of
K
is
Let
A and a partial order p
so that:
x).
Let
one of them extends the other.
PA
+W
V[D]
and
A
(the omegath
indestructible under
K
be the partial order of Theorem 1 . 1 .
PA,
if
p
V[6.J.
If
is a cardinal such that
D 5 PA
If
and q
It follows that if
is a super compact cardinal in
holds in
= K
is A-strongly-strategically-closed.
PA
then it is X-directed closed as in then H
A
Apply Laver's method to make
x-directed closed forcings. [PA[= A+;
A+.
be a super-compact cardina1;set
K
cardinal above
0:
is
is A-strongly-strategically-closed.
(ii) No dense subset of
Proof.
P
'I
'I.
is H-closed
are compatible, then
PA D
is V-generic for
is dense in
cof(A) < K < A.
PA
then
This
contradicts Theorem 1 . 0 . This theorem aswers a question raised by Forman.
3. THE MAIN RESULT The next theorem is the key to the results of this paper.
It is a
generalization to K-strongly-strategically-closed partial orders, of a claim that appears implicitly in the proof of Magidor's Theorem 2 o'f [M82]. Let
THEOREM 3 . 1
below n.
n be a supercompact cardinal and
p
If
V[col(p,
p
some regular cardinal
is a x-directed-strongly-strategically-closed
V[cOl(p,
satisfies:
A s 5 S C K , if s\S*(A) stationary in
'I.
COROLLARY 3.2
For
"
For every
is stationary in
K,p,
p
A,
h 1 K s.t. cof(A) 1 K then for some a <
as above, V[cOl(P,
(a) For any cardinal A > p, 0;
forcing in
P,
then the model obtained by further forcing via
and every
A,
Sna
satisfies:
implies that every stationary subset of
sA+ reflects. CP
(b) For any
p > p
if 1Jcp = p
then every stationary subset of
S"
CP
reflects.
LEMMA 3.3
Proof. As
The theorem holds for K-directed-closed P's.
Pick a cardinal A * K
above K
and
2"'.
is a supercompact cardinal (in the ground model V)
elementary embedding
j:V+ M
is
to some transitive model M
there is an
of
ZFC s.t.
A Laver-Type Indestmctability for Accessible Cardinals N
j(x) > A * and ~ A 5 M,
is its critical point,
follows that
P, col(p,
M.
Extend
conclude that for some P-generic extension of
j to an embedding of
-G
M[j(col(p,
v[col(p,
po
is in
j[P]
be a
T
P
j(x)-directed-closed.
= {j(a):a <
A}
p
and
=
Sup(B)
of
p
M[j(colp,
pol/-j(p,
'j(T)ny = j"E'
)J
'.
As
in this model
A
of
p
disjoint from
is a closed unbounded subset of X
M[j(colp,
E
is not stationary in A .
M[col( p,
E
was stationary in
is x-closed,
P
E
A.
j"E.
there is a
It is not hard to check
disjoint from
E,
therefore
On the other hand, in Now we get the desired contra-
is a x-closed extension of
diction by recallingthat j(colp,
G
has cofinality p
therefore it follows that already in M[j(col(p,
closed cofinal subset H
In
I f this is not the case, let
I .
is not stationary in
col(p,
Therefore,
E G and
'j"E
j-'(€i)
is
then in M[col(p,
and
M[j(COl(p,
is j(x)-closed in
po
p
such that
in
E 6)
of size < A * < j(x)
name for a stationary
pok j(p) 'j"E is stationary in
(as j(H)>A)
j[P]
-
is
j(P)
that
-
so G ={j(p):p
M
be a generic set for
j[P]
is a x-closed forcing
E j[P] above all members of G (in V[j(col(p,
let
,B
M, jlP
is realised as a stationary subset E
CLAIM
is a
= jot), V= M[co~(~,
I.(!
i , M[j(col(p,
is a directed subset of
is x-directed-closed
there is some
T
M[COl(p,
so
M[COl(p,
By the closure properties of
and as
It
M.
M[j(col(p,
Applying Observation 2 . 2 to our case
in
E
to M[j(COl(p,
j(col(p,
implies A
and all their subsets and the names in their
forcing languages are members of V[COl(p,
5 A*
15
is forced to be a stationary subset of (s?,\s;)
its
stationariness cannot be destroyed by such forcing extensions (Thm. 2 . 3 ) . M [ j ( c o l ( p , c ~ ) ) l * [ j ( P ) ] I ~ " c o f j2( ~l(X) )
of this model and has cardinality p and
jot) >
A,
so
j"(h)
is bounded in
j"E is a stationary initial segment of implies therefore that in
"
(as cof(X) 2 K ) j"(A)
there, as in this model
j(A)). j(E).
V[col(p,
is a member
j(x)
is
The elementariness of E
p+
It follows that in this model j
has a stationary initial
segment.
Proof of the theorem forcing in
V'
model, where
Let
Q
be a x-directed-strongly-strategically-closed
Let p be COl(H,<6) in the sense of this 6 is some cardinal above 2IQ1. P is x-directed closed, so =
V[col(p,
S. Ben David
16
by the previous lemma in
By Lemma 3 . 3 x-closed.
x
S 5 S,,
and
Sx
and
V'[hl.
S\S*(A)
As
-
for some P
such that in
is stationary, then
is H-closed and in
It follows that in
The same a
and
S
S
is
is still stationary in
for some a < A Sna
V'[$]
V'[Q]
is a stationary
cof(A) 1 H
V"Q1
is
S\S*(A)
a.
is stationary in
It follows from Theorem 2 . 3 that if in V"Q1
subset of
a.
A, Sna
is V'[Q][P]
V'[h]
V'[Ql[Fl = V'[h]. in
cof X 2 x
if
then for some a <
A,
stationary in
V'[fi],
is therefore a reflection point for
S
the same holds
is stationary in
in
V"Q1.
The following theorem is a straightforward corollary to Theorem 3 . 1 . THEOREM 3 . 4
If
ZFC
cardinal then so is
is consistent with the existence of a supercompact ZFC with "For every H-directed-strongly-strategically-
closed forcing notion cof(X) 2 H
g
after forcing via
and every stationary subset
is stationary then
S
g
for every ordinal
S of
S
n
reflects", this for every
A
s.t.
if {aEs:cof(a)+ # cofll successor of a regular
cardinal.
4 . APPLICATIONS
If
THEOREM 4 . 1
V
V[&l
" ZFC + a-many supercompact cardinals ", 6 such that cof(6) = a there is a forcing extension
is a model of
then for every ordinal
adding no a-sequences to
V,
in which every stationary subset of
K6+ 1
reflects.
Proof.
Let
(H.:i < a )
be an increasing sequence of supercompact cardinals.
lim N ~ ,and let (xi:i < a ) a i
Let
H
=
partial orders Pi; Po in
V[hil
and
ei:i < p) for g.
)I.
is colt h,,
Pi+, = Pi*Qi+l.
followed by
col( Xp
0'
'"0.
be an increasing sequence of regular By induction on Qi+,
i < a
define
is col(Xi,
In limit stages P p is the inverse limit of
,
P
).
Let
g
= Pa
and let b be generic
Standard arguments about the Levy collapse show that in
V[i]
is a cardinal, lim xi = 8, and X z is a cardinal, so it is i
each
b+,-
Hi+,-directed-closed forcing notion (for any that any
A -sequence of V[hl -
Pi, P,-generic).
is an element of
V
is a
It follows
and in particular
p adds
no &sequences. Let
R
be the forcing notion described in.section1 for adding a 0'
#6 sequence to V[h]. -
Recall that
R
preserves cardinalities and cofinalities.
A Laver-Type Indestmctability for Accessible Cardinals
v[~I[1 ~ I"Every stationary subset of K , + ~ reflects".
CLAIM 4 . 2
Proof of the claim S . = {p
E
S 5
Let
K,+,
S:cof(p) < A , }
The function f:S + a
be stationary.
X.1
f ( p ) = min{i:cof(p) <
defined by Let
17
is pressing down so for some
is stationary.
Let u s show that
i < a
Si reflects.
Qi denote the V[il
Levy collapse colOti, <%+I). i V[g]*[Qi] = VIPil*[P1l*[Qi], and P *Qi isa K.-closed forcing notion (as an By Theorem 2 . 4 , as R
iteration of two x.-closed notions). strategically-closed in V[Pi], closed
Q.
is x.-strongly-
can be regarded as R*Q*
for some
K.-
Q*.
In V[P]*[R],S*(K6+1)= p'mod the non stationary ideal, so therefore without
xs+l1.
loss of generality Si 5 K,+,\S*( that
Si
Applying Theorem 2 . 4 , we conclude
is still a stationary subset of
(k,+ljJ[pR1 is
an ordinal of cofinality xl.
[pR1 &+' , 1
reflects".
some p <
sinp i s stationary in p .
is stationary in
p
REMARK 4 . 3 in V[p]
as V[pR]
(Xi:i < a)
and
=
such that for some Qi:
(a) Qi
Sup 1 i
every stationary subset of If
ZFC
s:ii
A+
is a forcing notion
.
%Xi '
reflects.
is consistent with the existence of arbitrarily many
X,
existence of a non-reflecting stationary subset of
Oi
Oi
a forcing notion BX
without collapsing cardinals or changing cofinalities and
such that for each regular the demands (a) and (b) for
Assume
does not imply the
X.
In [B,S861 we constructed for every singular X
that introduces
THEOREM 4 . 5
V[pRlk"Sinp
p is a Levy collapse of superthen if R
supercompact cardinals then, for every singular
Proof.
reflects
In particular, for
is X.-closed ;
then in V[pR] COROLLARY 4 . 4
(though
V[pQi].
does not destroy stationarity of subsets of
(b) R*Qi
cof(h,+l tv[pRl
For the same p ,
is a submodel of
What we actually proved was that if
compact cardinals
V[_p][Qi]
Col(Ki,<&+l)
every stationary subset of
and so "every stationary subset of
'I
in
makes the old K,+,
no longer a cardinal).
corollary (b) in V[p][Qi]
(%+l) VIP1[R1 -
ZFC
1-(
<
X there is some forcing notion Qp satisfying
p,X
of Remark 4 . 3 .
is consistent with the existence of a proper class
of supercompact cardinals.
Then
ZFC
is consistent with "For every cardinal
every stationary subset of { a < X:cof(a)+ # A}
reflects".
S. Ben David
18
REMARK
(a) The result is the strongest that can be expected in the sense
4.6
X
that if
= p',
p
a regular cardinal, then
{a < X:cof(a)
= p} is a non-
A.
reflecting stationary subset of
( b ) The assumption is a 'natural' one in the sense that reflection of
Xt
stationary subsets of
implies strong failure of KJx
(even 0
Xp - a - fails for all p < Xi.
sequence defined for ordinals of cofinalities 2 p
In [B,S86] it is shown that 0 is consistent with the existence of superXlJ compact cardinals K s.t. cof(X) < K < p ) . It follows that a model of ZFC with full reflection as above has an inner model with proper class of large cardinals.
Proof.
Let
be a model of
V
(xCY:aE On,a successor).
ZFC with unboundedly-many supercompactcardinals
Without loss of generality every limit of super-
compact cardinals is a singular cardinal. regular limit and replace V
( ~ ~ E: Ona) notion (1)
is
H(6)
by
of
(If not, let
is increasing and continuous.
p so that: For each CY p
= P *Pa
where
is
Pa
6 be the first such
Let us also assume that
V.)
We define a proper class forcing
K
~
-+ C.C. ~
(and a set) and
Pa
K -directed-strongly-strategically-closed.
( 2 ) If
p
is a successor of a regular cardinal in
V[i],
then in
V
p
is one of the supercompacts in our sequence. 1 3 ) If in
V"1
p
is p'
is regular, then for some a E On
p
and
Pa+, = Pa*co~(~t<~CYtl) *re c~l(p.
tion.
p is by induction on a.
The definition of
where
= Pa *Col(K
pa+ 1 sense of
v[pa].
For a limit a
? ~ = v ' ~ = s KU p is the PA p
P
a+1
Col(Xt,
(P :B < a). It is quite B p does satisfy our list of demands. (One
is the inverse limit of
straightforward to check that this has to use the inaccessibility of the closure property of
Y
= P *R *Col(X+,
forcing notion defined in section 1 for intro-
VIPal [RCYl.
At limit a's, P
K+
For successor a
Col(K , < H ~ + ~ isthe ) Levy collapse in the
Pa
are still cardinals in
REMARK
K
~
for a < y
+to~ get the
for P "a+1 - c.c* to verify that for limit y K
Y
and and
V"].)
4.7 'Actually we get a somewhat stronger result in
v"].
Not only
do we have the full reflection but for every p-directed-stronqly-strategicallyclosed
R
in
V[i][fi]
we still have relfection for the stationary subsets of
ordinals of cofinality less than
p.
A Laver-Type Indestructability for Accessible Cardinals Added in Proof. questions:
The proofs of Theorems 4 . 1 and 4.5 raise the following
Could we get the models of reflection by just Levy collapsing the
supercompact cardinals of the ground model? 0:
19
Do we really have to force the
sequences to get f u l l reflection of stationary sets? Shela has given a partial answer to these questions by proving that one can
force over the ground model
V
the supercompact cardinals of V
v*
to get an intermediate extension V' are still supercompact in
V'
so
that
and the model
obtained by iteratively Levy collapsing the supercompact cardinals of V'
(in the manner described in the proofs of Theorems 4 . 1 and 4 . 5 ) has the desired reflection properties.
S. Ben-David, S. Shelah, Non-special Aronszajn trees on Journal of Mathematics 53, ( 1 9 8 6 ) 93-96.
,fi
Israel
R. Laver, Making the supercompactness of x indestructible under x-directed closed forcings, Israel Journal of Mathematics 29, ( 1 9 7 8 ) 385-388.
M. Magidor, Reflecting stationary sets, Journal of Symbolic Logic 47, 4,
( 1 9 8 2 ) 755-771.
S. Shelah, On successors of singular cardinals, in M. Boffa, D. Van Dalen, K. McAloon (eds) Logic Colloquium '78, North Holland ( 1 9 7 9 )
357-380.