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More on proper forcing

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
Affiliation:
The Hebrew University, Jerusalem, Israel University of California, Berkeley, California 94720

Extract

§1. A counterexample and preservation of “proper + X”.

Theorem. Suppose V satisfies, , and for some Aω1, every Bω1, belongs to L[A].

Then we can define a countable support iterationsuch that the following conditions hold:

a) EachQiis proper andPiQi, has power1”.

b) Each Qi is -complete for some simple1-completeness system.

c) Forcing with Pα = Lim adds reals.

Proof. We shall define Qi by induction on i so that conditions a) and b) are satisfied, and Ci, is a Qi-name of a closed unbounded subset of ω1. Let : ξ < ω1› ∈ L[A] be a list of all functions f which are from δ to δ for some δ < ω1 and let h: ω1ω1, hL[A], be defined by h(α) = Min{β: β > α and Lβ[A]⊨ “∣α∣ = ℵ0”}.

Suppose we have defined Qj for every j < i; then Pi is defined, is proper (as each Qj, j < i, is proper, and by III 3.2) and has a dense subset of power ℵ (by III 4.1). Let GiPi be generic so clearly there is Bω1, such that in V[Gi] every subset of ω1 belongs to L[A, B], The following now follows:

Fact. In V[Gi], every countableN ⥽(H(ℵ2), ∈, A, B) is isomorphic toLβ[Aδ, Bδ] for some β < h(δ), where δ = δ(N) = ω1, ∩ N.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1984

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References

[1]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[2]Jensen, R. B. and Solovay, R., Some applications of almost disjoint forcing, Mathematical logic and the foundations of set theory (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1968, pp. 84104.Google Scholar
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