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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by Sy D. Friedman PDF
Proc. Amer. Math. Soc. 127 (1999), 3707-3709 Request permission


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 $0^\#$:

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

  2. 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 $M$.

  • 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.
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Additional Information
  • Sy D. Friedman
  • Affiliation: Department of Mathematics Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 191285
  • Email:
  • 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:
  • MathSciNet review: 1610964