Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

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 $M$.


References [Enhancements On Off] (What's this?)

  • [82] René David, A very absolute Π¹₂ real singleton, Ann. Math. Logic 23 (1982), no. 2-3, 101–120 (1983). MR 701122, 10.1016/0003-4843(82)90001-8
  • [98] Sy D. Friedman, David's Trick, to appear, Proceedings of the European Summer Meeting of the ASL, Leeds, England, 1998.
  • [99] Sy D. Friedman, Fine Structure and Class Forcing, book, rough draft.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03E45, 03E55, 03E15, 03D60

Retrieve articles in all journals with MSC (1991): 03E45, 03E55, 03E15, 03D60


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-99-04914-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