The calculus of partition sequences, changing cofinalities, and a question of Woodin
HTML articles powered by AMS MathViewer
- by Arthur W. Apter, James M. Henle and Stephen C. Jackson PDF
- Trans. Amer. Math. Soc. 352 (2000), 969-1003 Request permission
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
- Arthur W. Apter, AD and patterns of singular cardinals below $\Theta$, J. Symbolic Logic 61 (1996), no. 1, 225–235. MR 1380685, DOI 10.2307/2275606
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Moti Gitik, Regular cardinals in models of $\textrm {ZF}$, Trans. Amer. Math. Soc. 290 (1985), no. 1, 41–68. MR 787954, DOI 10.1090/S0002-9947-1985-0787954-5
- James M. Henle, $\gamma$-Ramsey and $\gamma$-ineffable cardinals, Israel J. Math. 30 (1978), no. 1-2, 85–98. MR 508255, DOI 10.1007/BF02760831
- J. M. Henle, Magidor-like and Radin-like forcing, Ann. Pure Appl. Logic 25 (1983), no. 1, 59–72. MR 722169, DOI 10.1016/0168-0072(83)90054-4
- J. M. Henle, Researches into the world of $\kappa \rightarrow (\kappa )^{\kappa }$, Ann. Math. Logic 17 (1979), no. 1-2, 151–169. MR 552419, DOI 10.1016/0003-4843(79)90024-X
- J. M. Henle, Some consequences of an infinite-exponent partition relation, J. Symbolic Logic 42 (1977), no. 4, 523–526. MR 491196, DOI 10.2307/2271873
- S. Jackson, “A computation of $\mathbfit {\delta } 15$,” Mem. Amer. Math. Soc., to appear.
- Steve Jackson, $\textrm {AD}$ and the projective ordinals, Cabal Seminar 81–85, Lecture Notes in Math., vol. 1333, Springer, Berlin, 1988, pp. 117–220. MR 960900, DOI 10.1007/BFb0084974
- Steve Jackson, AD and the very fine structure of $L(\textbf {R})$, Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 1, 77–81. MR 980305, DOI 10.1090/S0273-0979-1989-15766-2
- S. Jackson, “Structural consequences of AD,” to appear in the Handbook of Set Theory, M. Foreman, A. Kanamori, M. Magidor eds.
- S. Jackson, “The weak square property,” to appear in the Journal of Symbolic Logic.
- S. Jackson and F. Khafizov, “Descriptions and cardinals below $\mathbfit {\delta } 15$,” preprint.
- 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.
- Alexander S. Kechris, $\textrm {AD}$ and projective ordinals, Cabal Seminar 76–77 (Proc. Caltech-UCLA Logic Sem., 1976–77) Lecture Notes in Math., vol. 689, Springer, Berlin, 1978, pp. 91–132. MR 526915
- Alexander S. Kechris, The axiom of determinacy implies dependent choices in $L(\textbf {R})$, J. Symbolic Logic 49 (1984), no. 1, 161–173. MR 736611, DOI 10.2307/2274099
- Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin, The axiom of determinacy, strong partition properties and nonsingular measures, Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79) Lecture Notes in Math., vol. 839, Springer, Berlin-New York, 1981, pp. 75–99. MR 611168
- Alexander S. Kechris, Robert M. Solovay, and John R. Steel, The axiom of determinacy and the prewellordering property, Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79) Lecture Notes in Math., vol. 839, Springer, Berlin-New York, 1981, pp. 101–125. MR 611169
- A. Kechris and W. H. Woodin, “ Generic codes for uncountable ordinals, partition properties, and elementary embeddings,” circulated manuscript, 1980.
- E. M. Kleinberg, $\textrm {AD}\vdash$ “the $\aleph _{n}$ are Jonsson cardinals and $\aleph _{\omega }$ is a Rowbottom cardinal”, Ann. Math. Logic 12 (1977), no. 3, 229–248. MR 469769, DOI 10.1016/S0003-4843(77)80002-8
- Eugene M. Kleinberg, Infinitary combinatorics and the axiom of determinateness, Lecture Notes in Mathematics, Vol. 612, Springer-Verlag, Berlin-New York, 1977. MR 0479903
- E. M. Kleinberg, Strong partition properties for infinite cardinals, J. Symbolic Logic 35 (1970), 410–428. MR 309734, DOI 10.2307/2270698
- Yiannis N. Moschovakis, Ordinal games and playful models, Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79) Lecture Notes in Math., vol. 839, Springer, Berlin-New York, 1981, pp. 169–201. MR 611173
- Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
- John R. Steel, Closure properties of pointclasses, Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79) Lecture Notes in Math., vol. 839, Springer, Berlin-New York, 1981, pp. 147–163. MR 611171
- J. Steel, “Scales in $L(\mathbb {R})$,” Cabal Seminar 79–81, Lecture Notes in Mathematics, 1019, 107–156, Springer–Verlag, 1983.
- John R. Steel, $\textrm {HOD}^{L(\textbf {R})}$ is a core model below $\Theta$, Bull. Symbolic Logic 1 (1995), no. 1, 75–84. MR 1324625, DOI 10.2307/420947
- 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.
Additional Information
- Arthur W. Apter
- Affiliation: Department of Mathematics, Baruch College, New York, New York 10010
- MR Author ID: 26680
- 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
- MR Author ID: 255886
- ORCID: 0000-0002-2399-0129
- Email: jackson@jove.acs.unt.edu
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1695015