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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

New $\Sigma^1_3$ facts


Author: Sy D. Friedman
Journal: Proc. Amer. Math. Soc. 127 (1999), 3707-3709
MSC (1991): Primary 03E45, 03E55, 03E15, 03D60
Published electronically: May 13, 1999
MathSciNet review: 1610964
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Abstract | References | Similar Articles | Additional Information

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 $0^\#$:

(a)
For some $k, M \models \{\alpha|n(\alpha)\leq k\}$ is stationary.
(b)
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 $M$.


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Additional Information

Sy D. Friedman
Affiliation: Department of Mathematics Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: sdf@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04914-X
PII: S 0002-9939(99)04914-X
Keywords: Class forcing, absoluteness, partitions
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.
Article copyright: © Copyright 1999 American Mathematical Society