Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-55wx7 Total loading time: 0.452 Render date: 2021-03-01T04:57:34.322Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

The bounded proper forcing axiom

Published online by Cambridge University Press:  12 March 2014

Martin Goldstern
Affiliation:
Institut für Algebra und Diskrete Mathematik, Technische Universität Wien, A-1040 Wien, Austria, E-mail: goldstrn@rsmb.tuwien.ac.at
Saharon Shelah
Affiliation:
Department of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, E-mail: shelah@math.huji.ac.il
Corresponding

Abstract

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, there is a directed set meeting all these antichains.

A regular cardinal κ is called ∑1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) ⊨ ‘φ’” implies “∃δ < κ, H(δ) ⊨ ‘φ’”.

We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a ∑1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).

We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1995

Access options

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

References

[Ba 1]Baumgartner, James E., Applications of the proper forcing axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 913959.CrossRefGoogle Scholar
[Ba 2]Baumgartner, James E., Iterated forcing, Surveys in set theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 159.Google Scholar
[Fu]Fuchino, Sakaé, On potential embedding and versions of Martin's axiom, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 481492.CrossRefGoogle Scholar
[Mi]Mitchell, William, Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (19721973), pp. 2146.CrossRefGoogle Scholar
[Sh 56]Shelah, Saharon, Refuting Ehrenfeucht conjecture on rigid models, Israel Journal of Mathematics, vol. 25 (1976); [= Abraham Robinson Memorial Symposium, Yale, 1975], pp. 273–286.CrossRefGoogle Scholar
[Sh 73]Shelah, Saharon, Models with second-order properties, II: Trees with no undefined branches, Annals of Mathematical Logic, vol. 14 (1978), pp. 7387.CrossRefGoogle Scholar
[Sh b]Shelah, Saharon, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[Sh f]Shelah, Saharon, Proper and improper forcing, Perspectives in Mathematical Logic, Springer Verlag.Google Scholar
[To]Todorcevic, Stevo, A note on the proper forcing axiom, Axiomatic set theory (Baumgartner, J.et al., editors), American Mathematical Society, Providence, Rhode Island, 1984, pp. 209218.CrossRefGoogle 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: 11 *
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.

The bounded proper forcing axiom
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.

The bounded proper forcing axiom
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.

The bounded proper forcing axiom
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *