Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-28jzs Total loading time: 0.314 Render date: 2021-03-01T05:43:35.142Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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

Access options

Get access to the full version of this content by using one of the access options below.

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

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 7 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 1st March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

More on proper forcing
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

More on proper forcing
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

More on proper forcing
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *