Tehnical Report CS0598 - 1989 - CS Technion
TECHNION - Israel Institute of Technology
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
Computer Science Department
THE TWO-CARDINALS TRANSFER PROPERTY AND RESURRECTION OF SUPERCOMPACTNESS by
S. Ben-David and S. Shelah
Technical Report #598 December 1989
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
1
THE TWO-CARDINALS T~ANSFER PROPERTY AND RESURRECTION OFSUPERCOMPACTNESS
Shai Ben-David Department of Computer Science Technion-Israel"Institute of Technology Haifa, Israel
Saharon Shelah Department of Mathematics The Hebrew University Jerusalem, Israel
ABSTRACT
We show that the transfer property ~ for singular A, does not imply (even) the existence of a non-reflecting stationary subset of 1..+. The result assumes the consistency of ZFC with the existence of inifinitely many super compact cardinals. We employ a technique of "resurrection of supercompactness". Our forcing extention destroys the supercompactness of some cardinals, to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extention.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
2
1. INTRODUCTION
Th~ results presented in this paper extend .our previous work on the relative strength' ot combinatorial proPerties of successors.of singular cardinals. In a seminal paper [I-72]Jensen has presented a collection of combinatorial properties that hold in the constructable universe L. From the point of view of applications of set theory to other branches of mathematics, these p1'Qperties are 'all you have to show about L'. Ever since that paper these properties were applied to a wide spectrum of questions to provide consistency results inside set theory as well as in other branches of mathematics ([S-74l,'[E-SOl, [F-S3l, to mention just a few). It seems natural to ask to what degree can these propertieS replace the axiom'V
=L?
Is there any
combinatorial principle that implies all these properties? What is the relative strength of these properties? What are the implication relations among them?
The picture seems to be basically settled for limit cardinals and for successors of regular cardinals, [Mi-72l. [G-76l. Our investigations has focused on successors of singular cardinals. Essentially we have been able to prove, assuming the consistency of the existence of large cardinals, that all the nontrivial implications among these properties are not provable in ZFC (see [BS-S5l. [BS-S6l. [BM-S6]). Here we examine the strength of the model theoretic two-cardinals-transfer-property
< It 10 lt o> -+ . Jensen [J72l has shown that it is implied by 0... A quite straight forward argument can show that it implies the weaker Ok principle. We show that < It 1.ltO> -+ does not imply the existence of a non-reflecting stationary subset of A+ for any singular A (as long 'as ZFC is consistent with the existence of infinitely many supercompact cardinals). It follows that the implications from 0.. to < It l,lt 0> -+ is strict and that O~ (or. equivalently. the existence of a speci31 A+ - Aronszajn tree) does not imply the existence of a nonreflecting stationary subset of A.+. We use a novel technique which we call "resurrection of super compactness". We start with a model V in which A is a limit of super cdmpact cardinals and therefore all stationary subsets of A+ reflect We extend it through forcing to a model V[al in which the two cardinal transfer property holds.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
3
Now we have to argue why we still have reflection of all stationary subsets of A+ (although our forcing has inevitably. distroyed the supercompactness of a final segment of cardinals below A). Instead of applying the commonly used combinatorial analysis to our forcing partial order, we demonstrate the reflection property by showing that we could "resurrect" tlie supercompactness of any cardinal p below A by a further forcing extensionQ p that preserves reflection of appropriate subsets of A+•
2. PROOF OUTLINE
The proof is based on a translation of the transfer property to a combinatorial principle SA,. We show
can be forced using a 'mild' forcing notion. The mildness of the forcing notion guarantees that over certain models where every stationary subset of A+ reflects, such forcing extension would not distroy the reflection. The natural candidate for exhibiting reflection of all stationary subsets of A+ is a model in which Ais
a limit of supercompact cardinals. Let V be such a model, standard compactness arguments show that O~
fails in V. It follows that if we extend V to a model V[G] of -+ the super compactness of a final segment of the cardinals below Awill be deStroyed. We wish to show that our extension was mild enough to retain some of the supercompactness consequences - namely the reflection of all stationary sub-
To exhibit this we use a technique we call "resurrection of supercompactness". We further extend V[G] to a model V[G][Q]. We show that V[G][Q] the supercompactness of certain cardinals is resurrected.
Consequently, in V[G][Q] we do have the desired reflection principle. All that is left to do is make sure that reflection of some stationary S ~ A+ in V[G][Q] can only occur if S was already a reflecting stationary subset of A+ in V[G]. More precisely, for every super compact cardinal p below A we exhibit the existence of a forcing notion Q p such that: (i) Q p pre~ves statinarity of subsets of S~+ = (a -7 . (Actually
both properties are equivalent to a seemingly stron~er transfer property.) We refer the reader to [S-87] for the full theorem and its proof. To gain a feeling for the content of the new
S). principle let us demonstrate its strength by proving the following corollary of the above
theorem directly. Corollary 1: Proof:
For a strong limit singular cardinal A.. S). implies O~.
Let (C~: a < 1..+. i < cofl.) be as). sequence. Define A a to be u
i ..) be the ~ generic S). sequence. A condition r
E
R is a closed boun~:d subset of A+ such that
-<XE r
implies that for some
i < cof>.., C~ is not bounded in a. R is ordereed by end extensions. R is designed to introduce a closed
unbounded subset in A+, along which, for each a some C~ contains an unbounded subset of a. The model in which our theorem is realized is V [P*Rl so we will sWdy the properties of P*R (rather than those of R).
Let us work in the ground model V. P*R can be represented as the set of all pairs (p,r) such that p E ~, and p It- "r e RIt. Note that as ~ does not introduce any new sets of size S A, each member of R is
in V. It is easy to see that p It- r e R" iff r ~ dom (p)+ I, for every a e r there is some C~ in p (a) that is It
unbounded in a, and of course r is closed (as a subset of dom (p)+ 1). Lemma 8: {(p,r): (p,r)e P*R and super) = dom(p») is dense in P·R. Proof:
Given any (p,r)e P·R define a one-level-extension q of p as in the proof of Lemma 3. As
~(=dom(p)+l) =dom(q), is a member of each
Cb we may define r'= r U
{~} to get (q,r') in P·R above
o
(p,r).
From now on let us assume that all the members of P·R have this property (the second coordinate is a closed cofinal subset of the domain of the first). Lemma 9:
aE
dom (p')
Proof:
For any (p,r) E P·R and any a < A+ there is some (p',r') ~ (p,r) such that
= suPer').
This is an easy consequence of Lemmas 8 and 6.
Lemma 10: P·R is J.l-strategically-closed for any regular J.l < A. Proof: The strategy for player I will be an adaptation of the strategy presented in the proof of Lemma 5.
Let «pjrj): i < p) be the sequence played so far and define (pp,r p) -the next condition picked by player 1.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
9
We start with the successor. stages. For p =2 we pick anyone level extention of (Plrl). For p successor bigger than 2, we modify the definition of the C~l. from Lemma 5. by defining for i < i o, define C~l
=0
f6r i o S i < j let C~l
define r p =r p-l
U
= {~} U
C~ and for j S i'C~l = {y}
U
C~. As 'Ye C~l for i '2 j we may
(y}o Now we are left with the limit stages:Let 'Y be .u dom (P;)(= .u super;)}. We re~t ' ~ implies the existence of a non-reflecting stationary subset of A.+. Proof: Just note that the conclusion of Theorem 15 is all we need to make our proof of the main theorem ~~~
0
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
13
REFERENCES [BS-85]
Ben David, S., and Shelah, S., "Souslin trees and successors of singular cardinals", Annals ofPure and Applied Logic, 1985.
[BM-86] Ben David, S., and Magidor, M. "The weak
nal afSymbolic Logic, 1986.
oi. is really weaker than the full Q", Jour~ sub
omega +! "
[BS-86]
Ben David, S., and Shelah, S., "Non-Special Aronszajn trees on Israel Journal ofMathematics, 53, 93-96,1986.
[BD-86]
Ben David, S., "A Laver-type indi'structability for accessible cardinals". Logic Colloquium '86, F.R. Drake, J.K. Truss (eds.), North-Holland, pp. 9-19, 1988.
(B-80]
EkIof, P., "Set theoretic metods in homological algebra and abelian groups". Les Presses de 1'Universite de Montreal, 1980.
[F-831
Fleissner, W.G., "If all Moore spaces are metrizable, then there is an inner model with a measurable cardinal". Trans. of the American Mathematical Society, 273, pp. 365-373, 1983.
[0-76]
Gregory, J., "Higher Souslin trees and the generalized continuum hypothesis" , Journal ofSymbolic Logic 41, pp. 663-671, 1976.
[J-n]
Jensen, R.B., "The fine structure of the constructable hierarchy". Annals ofMathematical Logic, 4, pp. 229-308, 1972.
[L-78]
Laver, R., "Making the supercompactness of lC indistructable under lC-directed closed forcing". IsraeLJournal ofMathematics, 29, pp. 385-388, 1978.
[M-82]
Magidor, M., "Reflecting stationary sets". Journal of Symbolic Logic, 47, pp. 755-771, 1982.
[Mi-72j8 Mitchell, W., "Aronszajn trees and the independence of the transfer property". Annals ofMathematical Logic, 5, pp. 21-46, 1972. [S-74]
Shelah, S., "Infinite abelian groups, Whitehead problem and some constructions". Israel Journal of Mathemdtics, 18, pp. 243-256, 1974.
[S-75]
She1ah, S., "A compactness theorem for singular cardinals for free algebras. Whithead problem and transversals". Israel Journal ofMathematics, 21, pp. 329-349, 1975.
[S-78]
She1ah, S., "On the successors of singular cardinals". Logic Colloquium '78. M. Baffa, D. Van Dalen and K. McAloon (eds.), North-Holland, Amsterdam, 1979.
[S-87]
Shelah, S., "Gap 1 two cardinal principle and omitting type theorems for L(Q)" Israel JournalofMathematitJS, 65,1989, pp. 133-152.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
Computer Science Department
THE TWO-CARDINALS TRANSFER PROPERTY AND RESURRECTION OF SUPERCOMPACTNESS by
S. Ben-David and S. Shelah
Technical Report #598 December 1989
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
1
THE TWO-CARDINALS T~ANSFER PROPERTY AND RESURRECTION OFSUPERCOMPACTNESS
Shai Ben-David Department of Computer Science Technion-Israel"Institute of Technology Haifa, Israel
Saharon Shelah Department of Mathematics The Hebrew University Jerusalem, Israel
ABSTRACT
We show that the transfer property ~ for singular A, does not imply (even) the existence of a non-reflecting stationary subset of 1..+. The result assumes the consistency of ZFC with the existence of inifinitely many super compact cardinals. We employ a technique of "resurrection of supercompactness". Our forcing extention destroys the supercompactness of some cardinals, to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extention.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
2
1. INTRODUCTION
Th~ results presented in this paper extend .our previous work on the relative strength' ot combinatorial proPerties of successors.of singular cardinals. In a seminal paper [I-72]Jensen has presented a collection of combinatorial properties that hold in the constructable universe L. From the point of view of applications of set theory to other branches of mathematics, these p1'Qperties are 'all you have to show about L'. Ever since that paper these properties were applied to a wide spectrum of questions to provide consistency results inside set theory as well as in other branches of mathematics ([S-74l,'[E-SOl, [F-S3l, to mention just a few). It seems natural to ask to what degree can these propertieS replace the axiom'V
=L?
Is there any
combinatorial principle that implies all these properties? What is the relative strength of these properties? What are the implication relations among them?
The picture seems to be basically settled for limit cardinals and for successors of regular cardinals, [Mi-72l. [G-76l. Our investigations has focused on successors of singular cardinals. Essentially we have been able to prove, assuming the consistency of the existence of large cardinals, that all the nontrivial implications among these properties are not provable in ZFC (see [BS-S5l. [BS-S6l. [BM-S6]). Here we examine the strength of the model theoretic two-cardinals-transfer-property
< It 10 lt o> -+ . Jensen [J72l has shown that it is implied by 0... A quite straight forward argument can show that it implies the weaker Ok principle. We show that < It 1.ltO> -+ does not imply the existence of a non-reflecting stationary subset of A+ for any singular A (as long 'as ZFC is consistent with the existence of infinitely many supercompact cardinals). It follows that the implications from 0.. to < It l,lt 0> -+ is strict and that O~ (or. equivalently. the existence of a speci31 A+ - Aronszajn tree) does not imply the existence of a nonreflecting stationary subset of A.+. We use a novel technique which we call "resurrection of super compactness". We start with a model V in which A is a limit of super cdmpact cardinals and therefore all stationary subsets of A+ reflect We extend it through forcing to a model V[al in which the two cardinal transfer property holds.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
3
Now we have to argue why we still have reflection of all stationary subsets of A+ (although our forcing has inevitably. distroyed the supercompactness of a final segment of cardinals below A). Instead of applying the commonly used combinatorial analysis to our forcing partial order, we demonstrate the reflection property by showing that we could "resurrect" tlie supercompactness of any cardinal p below A by a further forcing extensionQ p that preserves reflection of appropriate subsets of A+•
2. PROOF OUTLINE
The proof is based on a translation of the transfer property to a combinatorial principle SA,. We show
can be forced using a 'mild' forcing notion. The mildness of the forcing notion guarantees that over certain models where every stationary subset of A+ reflects, such forcing extension would not distroy the reflection. The natural candidate for exhibiting reflection of all stationary subsets of A+ is a model in which Ais
a limit of supercompact cardinals. Let V be such a model, standard compactness arguments show that O~
fails in V. It follows that if we extend V to a model V[G] of -+ the super compactness of a final segment of the cardinals below Awill be deStroyed. We wish to show that our extension was mild enough to retain some of the supercompactness consequences - namely the reflection of all stationary sub-
To exhibit this we use a technique we call "resurrection of supercompactness". We further extend V[G] to a model V[G][Q]. We show that V[G][Q] the supercompactness of certain cardinals is resurrected.
Consequently, in V[G][Q] we do have the desired reflection principle. All that is left to do is make sure that reflection of some stationary S ~ A+ in V[G][Q] can only occur if S was already a reflecting stationary subset of A+ in V[G]. More precisely, for every super compact cardinal p below A we exhibit the existence of a forcing notion Q p such that: (i) Q p pre~ves statinarity of subsets of S~+ = (a -7 . (Actually
both properties are equivalent to a seemingly stron~er transfer property.) We refer the reader to [S-87] for the full theorem and its proof. To gain a feeling for the content of the new
S). principle let us demonstrate its strength by proving the following corollary of the above
theorem directly. Corollary 1: Proof:
For a strong limit singular cardinal A.. S). implies O~.
Let (C~: a < 1..+. i < cofl.) be as). sequence. Define A a to be u
i ..) be the ~ generic S). sequence. A condition r
E
R is a closed boun~:d subset of A+ such that
-<XE r
implies that for some
i < cof>.., C~ is not bounded in a. R is ordereed by end extensions. R is designed to introduce a closed
unbounded subset in A+, along which, for each a some C~ contains an unbounded subset of a. The model in which our theorem is realized is V [P*Rl so we will sWdy the properties of P*R (rather than those of R).
Let us work in the ground model V. P*R can be represented as the set of all pairs (p,r) such that p E ~, and p It- "r e RIt. Note that as ~ does not introduce any new sets of size S A, each member of R is
in V. It is easy to see that p It- r e R" iff r ~ dom (p)+ I, for every a e r there is some C~ in p (a) that is It
unbounded in a, and of course r is closed (as a subset of dom (p)+ 1). Lemma 8: {(p,r): (p,r)e P*R and super) = dom(p») is dense in P·R. Proof:
Given any (p,r)e P·R define a one-level-extension q of p as in the proof of Lemma 3. As
~(=dom(p)+l) =dom(q), is a member of each
Cb we may define r'= r U
{~} to get (q,r') in P·R above
o
(p,r).
From now on let us assume that all the members of P·R have this property (the second coordinate is a closed cofinal subset of the domain of the first). Lemma 9:
aE
dom (p')
Proof:
For any (p,r) E P·R and any a < A+ there is some (p',r') ~ (p,r) such that
= suPer').
This is an easy consequence of Lemmas 8 and 6.
Lemma 10: P·R is J.l-strategically-closed for any regular J.l < A. Proof: The strategy for player I will be an adaptation of the strategy presented in the proof of Lemma 5.
Let «pjrj): i < p) be the sequence played so far and define (pp,r p) -the next condition picked by player 1.
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
9
We start with the successor. stages. For p =2 we pick anyone level extention of (Plrl). For p successor bigger than 2, we modify the definition of the C~l. from Lemma 5. by defining for i < i o, define C~l
=0
f6r i o S i < j let C~l
define r p =r p-l
U
= {~} U
C~ and for j S i'C~l = {y}
U
C~. As 'Ye C~l for i '2 j we may
(y}o Now we are left with the limit stages:Let 'Y be .u dom (P;)(= .u super;)}. We re~t ' ~ implies the existence of a non-reflecting stationary subset of A.+. Proof: Just note that the conclusion of Theorem 15 is all we need to make our proof of the main theorem ~~~
0
Technion - Computer Science Department - Tehnical Report CS0598 - 1989
13
REFERENCES [BS-85]
Ben David, S., and Shelah, S., "Souslin trees and successors of singular cardinals", Annals ofPure and Applied Logic, 1985.
[BM-86] Ben David, S., and Magidor, M. "The weak
nal afSymbolic Logic, 1986.
oi. is really weaker than the full Q", Jour~ sub
omega +! "
[BS-86]
Ben David, S., and Shelah, S., "Non-Special Aronszajn trees on Israel Journal ofMathematics, 53, 93-96,1986.
[BD-86]
Ben David, S., "A Laver-type indi'structability for accessible cardinals". Logic Colloquium '86, F.R. Drake, J.K. Truss (eds.), North-Holland, pp. 9-19, 1988.
(B-80]
EkIof, P., "Set theoretic metods in homological algebra and abelian groups". Les Presses de 1'Universite de Montreal, 1980.
[F-831
Fleissner, W.G., "If all Moore spaces are metrizable, then there is an inner model with a measurable cardinal". Trans. of the American Mathematical Society, 273, pp. 365-373, 1983.
[0-76]
Gregory, J., "Higher Souslin trees and the generalized continuum hypothesis" , Journal ofSymbolic Logic 41, pp. 663-671, 1976.
[J-n]
Jensen, R.B., "The fine structure of the constructable hierarchy". Annals ofMathematical Logic, 4, pp. 229-308, 1972.
[L-78]
Laver, R., "Making the supercompactness of lC indistructable under lC-directed closed forcing". IsraeLJournal ofMathematics, 29, pp. 385-388, 1978.
[M-82]
Magidor, M., "Reflecting stationary sets". Journal of Symbolic Logic, 47, pp. 755-771, 1982.
[Mi-72j8 Mitchell, W., "Aronszajn trees and the independence of the transfer property". Annals ofMathematical Logic, 5, pp. 21-46, 1972. [S-74]
Shelah, S., "Infinite abelian groups, Whitehead problem and some constructions". Israel Journal of Mathemdtics, 18, pp. 243-256, 1974.
[S-75]
She1ah, S., "A compactness theorem for singular cardinals for free algebras. Whithead problem and transversals". Israel Journal ofMathematics, 21, pp. 329-349, 1975.
[S-78]
She1ah, S., "On the successors of singular cardinals". Logic Colloquium '78. M. Baffa, D. Van Dalen and K. McAloon (eds.), North-Holland, Amsterdam, 1979.
[S-87]
Shelah, S., "Gap 1 two cardinal principle and omitting type theorems for L(Q)" Israel JournalofMathematitJS, 65,1989, pp. 133-152.
