Journal of Symbolic Logic

Cardinal-preserving extensions

Sy D. Friedman

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A classic result of Baumgartner-Harrington-Kleinberg implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that ω2L is countable: { X ∈ L | X⊆ ω1L and X has a CUB subset in a cardinal-preserving extension of L} is constructible, as it equals the set of constructible subsets of ω1L which in L are stationary. Is there a similar such result for subsets of ω2L? Building on work of M. Stanley, we show that there is not. We shall also consider a number of related problems, examining the extent to which they are “solvable” in the above sense, as well as defining a notion of reduction between them.

Article information

J. Symbolic Logic, Volume 68, Issue 4 (2003), 1163-1170.

First available in Project Euclid: 31 October 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E35: Consistency and independence results 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals

Descriptive set theory large cardinals innermodels


Friedman, Sy D. Cardinal-preserving extensions. J. Symbolic Logic 68 (2003), no. 4, 1163--1170. doi:10.2178/jsl/1067620178.

Export citation


  • J. Baumgartner, L. Harrington, and E. Kleinberg Adding a closed unbounded set, Journal of Symbolic Logic, vol. 41 (1976), pp. 481--482.
  • K. Devlin and R. Jensen Marginalia to a theorem of Silver, Lecture Notes in Mathematics, vol. 499, Springer-Verlag,1975.
  • S. Friedman The $\Pi^1_2$-singleton conjecture, Journal of the American Mathematical Society, vol. 3 (1990), pp. 771--791.
  • L. Harrington and A. Kechris $\Pi^1_2$-singletons and $0^\#$, Fundamenta Mathematicae, vol. 95 (1977), no. 3, pp. 167--171.
  • R. Jensen The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229--308.
  • D. A. Martin and R. M. Solovay A basis theorem for $\Sigma^1_3$ sets of reals, Annals of Mathematics, vol. 89 (1969), no. 2, pp. 138--160.
  • M. Stanley Forcing closed unbounded subsets of $\omega_2$, to appear.