Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 8
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Foreman, Matthew 2013. Calculating quotient algebras of generic embeddings. Israel Journal of Mathematics, Vol. 193, Issue. 1, p. 309.

    Larson, Paul 2005. Saturation, Suslin trees and meager sets. Archive for Mathematical Logic, Vol. 44, Issue. 5, p. 581.

    Takahashi, Juji 1989. Models of Set Theory in Which Every Normal Precipitous Ideal is Uniformly Normed. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, Vol. 35, Issue. 6, p. 537.

    Johnson, C. A. 1987. Distributive and related ideals in generic extensions. Nagoya Mathematical Journal, Vol. 106, p. 91.

    Johnson, C. A. 1986. Precipitous Ideals on Singular Cardinals. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, Vol. 32, Issue. 25-30, p. 461.

    Laver, Richard 1984. Precipitousness in forcing extensions. Israel Journal of Mathematics, Vol. 48, Issue. 2-3, p. 97.

    Jech, Thomas J. and Mitchell, William J. 1983. Some examples of precipitous ideals. Annals of Pure and Applied Logic, Vol. 24, Issue. 2, p. 131.

    Kakuda, Yuzuru 1981. Logic Symposia Hakone 1979, 1980.


On a condition for Cohen extensions which preserve precipitous ideals

  • Yuzuru Kakuda (a1)
  • DOI:
  • Published online: 01 March 2014

T. Jech and K. Prikry introduced the concept of precipitous ideals as a counterpart of measurable cardinals for small cardinals (Jech and Prikry [1]). To get a precipitous ideal on ℵ1( W. Mitchell used the Cohen extension by the standard Levy collapse that makes κ = ℵ1, where κ is a measurable cardinal in the ground model (Jech, Magidor, Mitchell and Prikry [2]). His proof essentially used the fact that , where is the notion of forcing of Levy collapse and j is the elementary embedding obtained by a normal ultrafilter on κ.

On the other hand, we know that has the κ-chain condition. In this paper, we show that the κ-chain condition for notions of forcing plays an essential role for preserving normal precipitous ideals. Namely,

Theorem 1. Let κ be a regular uncountable cardinal and I a normal ideal on κ. Let be a notion of forcing with the κ-chain condition. Then, I is precipitous iff ⊩“the ideal on κ generated by I is precipitous”.

As an application of Theorem 1, we have the following theorem.

Theorem 2. Let κ be a regular uncountable cardinal, and Γ be a κ-saturated normal ideal on κ. Then, {a < κ; the ideal of thin sets on a is precipitous} has either Γ-measure one or Γ-measure zero, and the ideal of thin sets on κ is precipitous iff {α< κ; the ideal of thin sets on α is precipitous) has Γ-measure one.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]T. Jech and K. Prikry , Ideal of sets and the power set operation, Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593595.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *