Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-5k9ck Total loading time: 0.216 Render date: 2022-06-25T11:14:59.413Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Decisive creatures and large continuum

Published online by Cambridge University Press:  12 March 2014

Jakob Kellner
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090 Wien, Austria, E-mail: kellner@fsmat.at, URL: http://www.logic.univie.ac.at/~kellner
Saharon Shelah
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel Department of Mathematics, Rutgers University, New Brunswick, Nj 08854, USA, E-mail: shelah@math.huji.ac.il, URL: http://shelah.logic.at

Abstract

For f, gωω let be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often.

It is consistent that for ℵ1 many pairwise different cardinals κ and suitable pairs (f, g).

For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

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.)

References

[1]Baumgartner, James E., Iterated forcing, Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 159.Google Scholar
[2]Goldstern, Martin, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991), Israel Mathematical Conference Proceedings, vol. 6, Bar-Ilan University, Ramat Gan, 1993, available at http://info.tuwien.ac.at/goldstern/, pp. 305360.Google Scholar
[3]Goldstern, Martin and Shelah, Saharon, Many simple cardinal invariants, Archive for Mathematical Logic, vol. 32 (1993), no. 3, pp. 203221.CrossRefGoogle Scholar
[4]Kellner, Jakob, Even more simple cardinal invariants, Archive for Mathematical Logic, vol. 47 (2008), no. 5, pp. 503515.CrossRefGoogle Scholar
[5]Rosłanowski, Andrzej and Shelah, Saharon, Norms on possibilities. I. Forcing with trees and creatures, Memoirs of the American Mathematical Society, vol. 141 (1999), no. 671, pp. xii + 167.CrossRefGoogle Scholar
[6]Shelah, Saharon, Proper and improper forcing, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
11
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@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 saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved 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.

Decisive creatures and large continuum
Available formats
×

Save article to Dropbox

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Decisive creatures and large continuum
Available formats
×

Save article to Google Drive

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Decisive creatures and large continuum
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *