The calculus of partition sequences,

changing cofinalities, and

a question of Woodin

Authors:
Arthur W. Apter, James M. Henle and Stephen C. Jackson

Journal:
Trans. Amer. Math. Soc. **352** (2000), 969-1003

MSC (1991):
Primary 03E15, 03E35, 03E60

DOI:
https://doi.org/10.1090/S0002-9947-99-02554-4

Published electronically:
October 15, 1999

MathSciNet review:
1695015

Full-text PDF

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 ``ZFDC is the only regular, uncountable cardinal .'' This gives a partial answer to a question of Woodin.

**[A]**A. Apter, ``AD and patterns of singular cardinals below ,''*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, ``-Ramsey and -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 ,''*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 ,''*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 ,''*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 ,'' 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 ,''*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, `` the are Jonsson cardinals and 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 ,'' Cabal Seminar 79-81, Lecture Notes in Mathematics, 1019, 107-156, Springer-Verlag, 1983. CMP**16:07****[St3]**J. Steel, ``HOD is a core model below ,''*Bulletin of Symbolic Logic*1 (1995) 75-84. MR**97a:03059****[W]**W. H. Woodin, ``AD and the uniqueness of the supercompact measures on ,'' Cabal Seminar 79-81, Lecture Notes in Mathematics, 1019, 67-71, Springer-Verlag, 1983. CMP**16:07**

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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:
https://doi.org/10.1090/S0002-9947-99-02554-4

Received by editor(s):
September 30, 1997

Published electronically:
October 15, 1999

Additional Notes:
The first author’s research was partially supported by PSC–CUNY grant 665337

Article copyright:
© Copyright 1999
American Mathematical Society