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Reflecting stationary sets and successors of singular cardinals

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Abstract

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n-supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares.

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References

  • [BSh360] Baldwin, J.T., Shelah, S.: Smoothness in the primal framework. Ann. Pure Appl. Logic (to appear)

  • [BD] Ben David, S.: A Laver-type indestructability for accessible cardinals. In: Drake, F.R., Truss, J.K. (eds.). Logic Colloquium 86, pp. 9–19. North-Holland, 1988

  • [GS] Gitik, M., Shelah, S.: On a certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Logic28, 35–42 (1989)

    Google Scholar 

  • [JSh387] Jech, T., Shelah, S.: Full reflection for the ℵ n 's. J. Symb. Logic55, 822–830 (1990)

    Google Scholar 

  • [L] Laver, R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Isr. J. Math.29, 385–388 (1978)

    Google Scholar 

  • [MgSh204] Magidor, M., Shelah, S.: When does almost free imply free? J. A.M.S. (accepted)

  • [MkSh367] Mekler, A.H., Shelah, S.: The consistency strength of “Every stationary set reflects”. Isr. J. Math.67, 353–366 (1989)

    Google Scholar 

  • [Sh-b] Shelah, S.: Proper Forcing. (Lect. Notes Math. vol. 940, 496 + xxix pp. Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  • [Sh68] Shelah, S.: Jonsson algebras in successor cardinals. Isr. J. Math.30, 57–64 (1978)

    Google Scholar 

  • [Sh88a] Shelah, S.: Classification theory for non-elementary classes. II. Abstract elementary classes. Appendix: On stationary sets. In: Baldwin, J.T. (ed.). Classification theory. Proceedings, Chicago 1985. (Lect. Notes Math., vol. 1292, pp. 483–497) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  • [Sh100] Shelah, S.: Independence results. J. Symb. Logic45, 563–573 (1980)

    Google Scholar 

  • [Sh108] Shelah, S.: On successors of singular cardinals. In: Boffa, M., van Dalen, D., McAloon, K. (eds.). Logic Colloquium 78. Amsterdam 1979, pp. 357–380

  • [Sh111] Shelah, S.: On power of singular cardinals. Notre Dame J. Formal Logic27, 263–299 (1986)

    Google Scholar 

  • [Sh237e] Shelah, S.: Remarks on squares. In: Around classification theory of models. (Lect. Notes. Math., vol. 1182, pp. 276–279) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  • [Sh282] Shelah, S.: Successors of singulars, cofinalities of reduced products of cardinals and productivity of chain conditions. Isr. J. Math.62, 213–256 (1988)

    Google Scholar 

  • [Sh300] Shelah, S.: Universal classes, Chaps. I–IV. In: Baldwin, J.T. (ed.). Classification theory. Proceedings, Chicago 1985. (Lect. Notes Math., vol. 1292, pp. 264–418) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  • [Sh355] Shelah, S.: ℵ ω+1's has a Jonsson algebra. (Preprint) to appear in: Cardinal Arithmetic. Oxford University Press

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Shelah, S. Reflecting stationary sets and successors of singular cardinals. Arch Math Logic 31, 25–53 (1991). https://doi.org/10.1007/BF01370693

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