Hostname: page-component-7d684dbfc8-4nnqn Total loading time: 0 Render date: 2023-09-28T16:09:16.967Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

More on proper forcing

Published online by Cambridge University Press:  12 March 2014

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


§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.

Research Article
Copyright © Association for Symbolic Logic 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[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