Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

The calculus of partition sequences, changing cofinalities, and a question of Woodin

Author(s): Arthur W. Apter; James M. Henle; Stephen C. Jackson
Journal: Trans. Amer. Math. Soc. 352 (2000), 969-1003.
MSC (1991): Primary 03E15, 03E35, 03E60
Posted: October 15, 1999
MathSciNet review: 1695015
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory ``ZF$+$DC$+\omega _1$ is the only regular, uncountable cardinal $\le\omega _{\omega _1+1}$.'' This gives a partial answer to a question of Woodin.

References:

[A]
A. Apter, ``AD and patterns of singular cardinals below $\Theta $,'' Journal of Symbolic Logic 61 (1996) 225-235. MR 97d:03071

[ER]
P. Erd\H{o}s and R. Rado, ``A partition calculus in set theory,'' Bull. Amer. Math. Soc. 62 (1956) 427-489. MR 18:458a

[G]
M. Gitik, ``Regular cardinals in models of ZF,'' Trans. Amer. Math. Soc. 290 (1985) 41-68. MR 86i:03065

[H1]
J. Henle, ``$\gamma $-Ramsey and $\gamma $-ineffable cardinals,'' Israel Journal of Mathematics 30 (1978) 85-98. MR 80a:04004

[H2]
J. Henle, ``Magidor-like and Radin-like forcing,'' Annals of Pure and Applied Logic 25 (1983) 59-72. MR 85e:03119

[H3]
J. Henle, ``Researches into the world of $\kappa \rightarrow(\kappa )^\kappa $,'' Annals of Mathematical Logic 17 (1979) 151-169. MR 81c:03047
[H4]
J. Henle, ``Some consequences of an infinite-exponent partition relation,'' Journal of Symbolic Logic 42 (1977) 523-526. MR 58:10461

[J1]
S. Jackson, ``A computation of ${\boldsymbol{\delta}}^{1}_{5}$,'' Mem. Amer. Math. Soc., to appear. CMP 98:08

[J2]
S. Jackson, ``AD and the projective ordinals,'' Cabal Seminar 81-85, Lecture Notes in Mathematics, 1333, 117-220, Springer-Verlag, 1988. MR 89k:03055

[J3]
S. Jackson, ``AD and the very fine structure of $L({\mathbb{R}})$,'' Bull. Amer. Math. Soc. 21 (1989) 77-81. MR 90a:03075

[J4]
S. Jackson, ``Structural consequences of AD,'' to appear in the Handbook of Set Theory, M. Foreman, A. Kanamori, M. Magidor eds.

[J5]
S. Jackson, ``The weak square property,'' to appear in the Journal of Symbolic Logic.

[JKh]
S. Jackson and F. Khafizov, ``Descriptions and cardinals below ${\boldsymbol{\delta}}^{1}_{5}$,'' preprint.

[JMa]
S. Jackson and D. A. Martin, ``Pointclasses and well-ordered unions,'' Cabal Seminar 79-81, Lecture Notes in Mathematics, 1019, 56-66, Springer-Verlag, 1983. CMP 16:07

[Ke1]
A. Kechris, ``AD and projective ordinals,'' Cabal Seminar 76-77, Lecture Notes in Mathematics, 689, 91-132, Springer-Verlag, 1978. MR 80j:03069

[Ke2]
A. Kechris, ``The axiom of determinacy implies dependent choices in $L({\mathbb{R}})$,'' Journal of Symbolic Logic 49 (1984) 255-267. MR 86f:03092

[KeKlMW]
A. Kechris, E. Kleinberg, Y. Moschovakis, and W. H. Woodin, ``The axiom of determinacy, strong partition properties, and nonsingular measures,'' Cabal Seminar 77-79, Lecture Notes in Mathematics, 839, 75-100, Springer-Verlag, 1981. MR 83f:03047

[KeSoSt]
A. Kechris, R. Solovay, and J. Steel, ``The axiom of determinacy and the pre-well-ordering property,'' Cabal Seminar 77-79, Lecture Notes in Mathematics, 839, 101-125, Springer-Verlag, 1981. MR 83f:03042

[KeW]
A. Kechris and W. H. Woodin, `` Generic codes for uncountable ordinals, partition properties, and elementary embeddings,'' circulated manuscript, 1980.

[Kl1]
E. Kleinberg, ``${AD}\vdash$ the $\aleph _n$ are Jonsson cardinals and $\aleph _\omega $ is a Rowbottom cardinal,'' Annals of Mathematical Logic 12 (1977) 229-248. MR 57:9550

[Kl2]
E. Kleinberg, Infinitary Combinatorics and the Axiom of Determinateness, Lecture Notes in Mathematics, 612, Springer-Verlag, 1977. MR 58:109

[Kl3]
E. Kleinberg, ``Strong partition properties for infinite cardinals,'' Journal of Symbolic Logic 35 (1970) 410-428. MR 46:8839

[Mo1]
Y. N. Moschovakis, ``Ordinal games and playful models,'' Cabal Seminar 77-79, Lecture Notes in Mathematics, 839, 169-201, Springer-Verlag, 1981. MR 84h:03115

[Mo2]
Y. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980. MR 82e:03002

[St1]
J. Steel, ``Closure properties of pointclasses,'' Cabal Seminar 77-79, Lecture Notes in Mathematics, 839, 147-163, Springer-Verlag, 1981. MR 84b:03066

[St2]
J. Steel, ``Scales in $L(\mathbb{R})$,'' Cabal Seminar 79-81, Lecture Notes in Mathematics, 1019, 107-156, Springer-Verlag, 1983. CMP 16:07

[St3]
J. Steel, ``HOD$^{L({\mathbb{R}})}$ is a core model below $\Theta $,'' Bulletin of Symbolic Logic 1 (1995) 75-84. MR 97a:03059

[W]
W. H. Woodin, ``AD and the uniqueness of the supercompact measures on $\mathcal{P}_{\omega _1}(\lambda)$,'' Cabal Seminar 79-81, Lecture Notes in Mathematics, 1019, 67-71, Springer-Verlag, 1983. CMP 16:07

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03E15, 03E35, 03E60

Retrieve articles in all Journals with MSC (1991): 03E15, 03E35, 03E60

Additional Information:

Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College, New York, New York 10010
Email: awabb@cunyvm.cuny.edu

James M. Henle
Affiliation: Department of Mathematics, Smith College, Northampton, Massachusetts 01060-2165
Email: jhenle@smith.edu

Stephen C. Jackson
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: jackson@jove.acs.unt.edu

DOI: 10.1090/S0002-9947-99-02554-4
PII: S 0002-9947(99)02554-4
Received by editor(s): September 30, 1997
Posted: October 15, 1999
Additional Notes: The first author's research was partially supported by PSC--CUNY grant 665337
Copyright of article: Copyright 1999, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia