Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-6pznq Total loading time: 0.303 Render date: 2021-03-07T10:10:11.907Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Power set modulo small, the singular of uncountable cofinality

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road Piscataway, NJ 08854-8019, USA. E-mail: shelah@math.huji.ac.ilURL: http://shelah.logic.at/
Corresponding
E-mail address:

Abstract

Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in ℙ = ([μ]μ, ⊇) as a forcing notion we have a natural complete embedding of Levy(ℵ0, μ+) (so ℙ collapses μ+ to ℵ0) and even Levy (). The “natural” means that the forcing ({p ∈ [μ] : p closed}, ⊇) is naturally embedded and is equivalent to the Levy algebra. Also if ℙ fails the χ-c.c. then it collapses χ to ℵ0 (and the parallel results for the case μ > ℵ0 is regular or of countable cofinality). Moreover we prove: for regular uncountable κ, there is a family P of κ partitions Ā = ⟨Aα : α < κ⟩ of κ such that for any A ∈ [κ]κ for some ⟨Aα : α < κ⟩ ∈ P we have α < κ ⇒ ∣AαA∣ = κ.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

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

References

[BaFr87] Balcar, Bohuslav and Franěk, František, Completion of factor algebras of ideals, Proceedings of the American Mathematical Society, vol. 100 (1987), pp. 205–212.CrossRefGoogle Scholar
[BPS] Balcar, Bohuslav, Pelant, Jan, and Simon, Petr, The space of ultrafilters on N covered by nowhere dense sets, Fundamenta Mathematical vol. CX (1980), pp. 11–24.Google Scholar
[BaSi88] Balcar, Bohuslav and Simon, Petr, On collections of almost disjoint families, Commentationes Mathematicae Universitatis Carolinae, vol. 29 (1988), pp. 631–646.Google Scholar
[BaSi89] Balcar, Bohuslav and Simon, Petr, Disjoint refinement, Handbook of boolean algebras, vol. 2, North-Holland, 1989, pp. 333–388.Google Scholar
[BaSi95] Balcar, Bohuslav and Simon, Petr, Baire number of the spaces of uniform ultrafilters, Israel Journal of Mathematics, vol. 92 (1995), pp. 263–272.CrossRefGoogle Scholar
[Ba] Baumgartner, James E., Almost disjoint sets, the dense set problem and partition calculus, Annals of Mathematical Logic, vol. 9 (1976), pp. 401–439.CrossRefGoogle Scholar
[KjSh:720] Kojman, Menachem and Shelah, Saharon, Fallen cardinals, Annals of Pure and Applied Logic, vol. 109 (2001), pp. 117–129, math.LO/0009079.CrossRefGoogle Scholar
[Sh:g] Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.Google Scholar
[Sh:506] Shelah, Saharon, The pcf-theorem revisited, The mathematics of Paul Erdős, II, Algorithms and Combinatorics, vol. 14, Springer, 1997, math.LO/9502233, pp. 420–459.Google Scholar
[Sh:589] Shelah, Saharon, Applications of PCF theory, this Journal, vol. 65 (2000), pp. 1624–1674.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: 11 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 7th 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.

Power set modulo small, the singular of uncountable cofinality
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.

Power set modulo small, the singular of uncountable cofinality
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.

Power set modulo small, the singular of uncountable cofinality
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *