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The weak square property

Published online by Cambridge University Press:  12 March 2014

Steve Jackson
Steve Jackson*
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA, E-mail: jackson@unt.edu
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Abstract

We formulate and prove a combinatorial property assuming AD + V = L(ℝ). As a consequence, we show that every regular κ which is either a Suslin cardinal or the successor of a Suslin cardinal is -supercompact. In particular, all the projective ordinals are -supercompact.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 640 - 657
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Becker, Howard, Determinacy implies that ℵ2 is supercompact, Israel Journal of Mathematics, vol. 40 (1981), no. 3, pp. 229–234.CrossRefGoogle Scholar
[2]Becker, Howard, A property equivalent to the existence of scales, Transactions of the American Mathematical Society, vol. 287 (1985), no. 2, pp. 591–612.CrossRefGoogle Scholar
[3]Becker, Howard and Jackson, Steve, Supercompactness within the projective hierarchy, this Journal, vol. 66 (2001), p. fix, (this issue).Google Scholar
[4]Becker, Howard S. and Kechris, Alexander S., Sets of ordinals constructible from trees and the third Victoria Delfino problem, Axiomatic set theory (Boulder, Colorado, 1983), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 13–29.CrossRefGoogle Scholar
[5]Harrington, L. and Kechris, A. S., On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154.CrossRefGoogle Scholar
[6]Jackson, Steve, Structural consequences of AD, Handbook of Set Theory, to appear.Google Scholar
[7]Kechris, Alexander S. and Woodin, W., Generic codes for uncountable ordinals, partition properties and elementary embeddings, circulated manuscript.Google Scholar
[8]Martin, Donald A. and Steel, John R., The extent of scales in L(ℝ), Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 86–96.CrossRefGoogle Scholar
[9]Moschovakis, Yiannis N., Descriptive set theory, Studies in logic, vol. 100, North–Holland, 1980.Google Scholar
[10]Moschovakis, Yiannis N., Ordinal games and playful models, Cabal seminar 77–79, Lecture Notes in Mathematics, vol. 839, Springer, Berlin, 1981, pp. 169–201.CrossRefGoogle Scholar
[11]Steel, John R., Closure properties of pointclasses, Cabal seminar 77–79, Lecture Notes in Mathematics, vol. 839, Springer, Berlin, 1981, pp. 147–163.CrossRefGoogle Scholar
[12]Steel, John R., Scales in L(ℝ), Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 107–156.CrossRefGoogle Scholar
[13]Woodin, W. Hugh, AD and the uniqueness of the supercompact measures on , Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 67–71.CrossRefGoogle Scholar