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Measurable cardinals and a combinatorial principle of Jensen1

  • Keith J. Devlin (a1)
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At the end of his paper [2], Silver shows how methods developed by Jensen and Solovay in order to prove results about the constructible universe L may be adapted to prove corresponding results for the universe L[μ], where μ is a normal measure on some uncountable cardinal ρ. In this paper we pursue this in greater depth. Jensen proved, in fact, much stronger results than those considered in [2], and we shall show that all of these carry over from L to L[μ], More precisely, we show that if V = L[μ] is assumed, then for any regular uncountable cardinal κ and any uncountable λ < κ, + (κ, λ) holds, and that + (κ, κ) holds just in the case κ is not ineffable. This result was proved to hold in L by Jensen, who first formulated the principles + (κ, λ).

Our proof differs in detail from Jensen's, and at one point (in choosing the set B of + ) differs fundamentally from his argument. However, the fact remains that our argument is modelled closely upon Jensen's, and it should be made clear that in many parts it is a straightforward adaption of his proof to the L[μ] situation. It is regrettable that, at the time of our writing this, Jensen's proof still only exists in the rough, handwritten form of [1]; so we shall give our argument in some detail, even those parts which are merely “translations” of Jensen's original arguments.

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1

We are grateful to the referee for pointing out several misprints in the original manuscript, and for suggesting one or two improvements in the exposition.

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[1] Jensen, R. B., Some combinatorial properties of L and V (handwritten notes).
[2] Silver, J. H., Measurable cardinals and Δ3 1 well-orderings, Annals of Mathematics (2), vol. 94 (1971), pp. 414446.
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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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