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Saturated ideals

  • Kenneth Kunen (a1)

In this paper, we give consistency proofs for the existence of a κ-saturated ideal on an inaccessible κ, and for the existence of an ω2-saturated ideal on ω1. We also include an historical survey outlining other known results on saturated ideals.

§1. Forcing. We assume that the reader is familiar with the usual techniques in forcing and Boolean-valued models (see Jech [3] or Rosser [11]), so we shall just specify here some of the less standard notation.

“cBa” abbreviates “complete Boolean algebra”.

If P (= ‹P, <›) is a notion of forcing, is the associated cBa. We write VP for .

Notions like κ-closed, κ-complete, etc. always mean < κ. Thus, P is κ-closed iff every decreasing chain of length less than κ has a lower bound, and a Boolean algebra is κ-complete iff sups of subsets of of cardinality less than κ always exist.

If is a cBa, x̌ is the object in representing x in V. In many cases, especially with ordinals, the ̌ is dropped. V̌ is the Boolean-valued class representing V — i.e.,

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[4] T. J. Jech , Two remarks on elementary embeddings of the universe, Pacific Journal of Mathematics, vol. 39 (1971), pp. 395400.

[5] R. B. Jensen , The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.

[6] K. Kunen , Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.

[8] K. Kunen and J. B. Paris , Boolean extensions and measurable cardinals, Annals of Mathematical Logic, vol. 2 (1971), pp. 359378.

[12] E. C. Smith Jr. and A. Tarski , Higher degrees of distributivity and completeness in Boolean algebras, Transactions of the American Mathematical Society, vol. 84 (1957), pp. 230257.

[14] R. M. Solovay and S. Tennenbaum , Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201245.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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