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Measurable cardinals and a combinatorial principle of Jensen1
Published online by Cambridge University Press: 12 March 2014
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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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- Copyright © Association for Symbolic Logic 1973
Footnotes
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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