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Coding over a measurable cardinal

Published online by Cambridge University Press:  12 March 2014

Sy D. Friedman
Sy D. Friedman*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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Extract

The purpose of this paper is to extend the coding method (see Beller, Jensen and Welch [82]) into the context of large cardinals.

Theorem. Suppose μ is a normal measure on κ in V and 〈 V, A〉 ⊨ ZFC. Then there is a 〈V, A〉-definable forcing for producing a real R such that:

(a) V[R] ⊨ ZFC and A is V[R]-definable with parameter R.

(b) V[R] = L[μ*, R], where μ* is a normal measure on κ in V[R] extending μ.

(c) V ⊨ GCH → is cardinal and cofinality preserving.

Corollary. It is consistent that μ is a normal measure, R ⊆ ω is not set-generic over L[μ] and 0+ ∉ L[μ, R].

Some other corollaries will be discussed in §4 of the paper.

The main difficulty in L[μ]-coding lies in the problem of “stationary restraint”.

As in all coding constructions, conditions will be of the form belonging to an initial segment of the cardinals, where p(γ) is a condition for almost disjoint coding into a subset of γ +. In addition for limit cardinals γ in Domain(p), 〈p γ′∣γ′ < γ〉 serves to code pγ .

An important restriction in coding arguments is that for inaccessible for only a nonstationary set of γ′ < γ. The reason is that otherwise there are conflicts between the restraint imposed by the different and the need to code extensions of pγ below γ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 4 , December 1989 , pp. 1145 - 1159
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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