New $\Sigma ^1_3$ facts
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- by Sy D. Friedman PDF
- Proc. Amer. Math. Soc. 127 (1999), 3707-3709 Request permission
Abstract:
We use “iterated square sequences” to show that there is an $L$- definable partition $n:L{\text -}Singulars \to \omega$ such that if $M$ is an inner model not containing $0^\#$:
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For some $k, M \models \{\alpha |n(\alpha )\leq k\}$ is stationary.
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For each $k$ there is a generic extension of $M$ in which $0^\#$ does not exist and $\{\alpha |n(\alpha )\leq k\}$ is non-stationary.
This result is then applied to show that if $M$ is an inner model without $0^\#$, then some $\Sigma ^1_3$ sentence not true in $M$ can be forced over $M$.
References
- René David, A very absolute $\Pi ^{1}_{2}$ real singleton, Ann. Math. Logic 23 (1982), no. 2-3, 101–120 (1983). MR 701122, DOI 10.1016/0003-4843(82)90001-8
- Sy D. Friedman, David’s Trick, to appear, Proceedings of the European Summer Meeting of the ASL, Leeds, England, 1998.
- Sy D. Friedman, Fine Structure and Class Forcing, book, rough draft.
Additional Information
- Sy D. Friedman
- Affiliation: Department of Mathematics Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 191285
- Email: sdf@math.mit.edu
- Received by editor(s): November 25, 1997
- Received by editor(s) in revised form: February 13, 1998
- Published electronically: May 13, 1999
- Additional Notes: The author’s research was supported by NSF Contract #9625997-DMS
- Communicated by: Carl G. Jockusch, Jr.
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3707-3709
- MSC (1991): Primary 03E45, 03E55, 03E15, 03D60
- DOI: https://doi.org/10.1090/S0002-9939-99-04914-X
- MathSciNet review: 1610964