Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-5rzhg Total loading time: 0.477 Render date: 2021-12-03T20:07:47.451Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

THE IMPLICITLY CONSTRUCTIBLE UNIVERSE

Published online by Cambridge University Press:  10 June 2019

MARCIA J. GROSZEK
Affiliation:
DEPARTMENT OF MATHEMATICS DARTMOUTH COLLEGE 6188 KEMENY HALL HANOVER, NY03755-3551, USA E-mail: marcia.groszek@dartmouth.edu
JOEL DAVID HAMKINS
Affiliation:
FACULTY OF PHILOSOPHY UNIVERSITY COLLEGE, OXFORD HIGH STREET, OXFORD OX1 4BH, UK E-mail:joeldavid.hamkins@philosophy.ox.ac.uk

Abstract

We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe Imp, which they introduced in [5]. Specifically, we show that it is relatively consistent with ZFC that $$Imp = \neg {\rm{CH}}$$, that $Imp \ne {\rm{HOD}}$, and that $$Imp \models V \ne Imp$$, or in other words, that $\left( {Imp} \right)^{Imp} \ne Imp$.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

Abraham, U., A mimimal model for $\neg $CH: Iteration of Jensen’s reals.Transactions of the American Mathematical Society, vol. 281 (1984), pp. 657674.Google Scholar
Baumgartner, J. and Laver, R., Iterated perfect set forcing. Annals of Pure and Applied Logic, vol. 17 (1979), no. 3, pp. 271288.Google Scholar
Groszek, M., Applications of iterated perfect set forcing. Annals of Pure and Applied Logic, vol. 39 (1988), no. 1, pp. 1953.CrossRefGoogle Scholar
Groszek, M. and Jech, T., Generalized iteration of forcing. Transactions of the American Mathematical Society, vol. 324 (1991), pp. 126.CrossRefGoogle Scholar
Hamkins, J. D. and Leahy, C., Algebraicity and implicit definability in set theory. Notre Dame Journal of Formal Logic. Advance publication, 20 April 2016. doi: 10.1215/00294527-3542326. http://projecteuclid.org/euclid.ndjfl/1461157794.Google Scholar
Sacks, G. E., Forcing with perfect closed sets, Axiomatic Set Theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1971, pp. 331355.CrossRefGoogle Scholar
1
Cited by

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 IMPLICITLY CONSTRUCTIBLE UNIVERSE
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 IMPLICITLY CONSTRUCTIBLE UNIVERSE
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 IMPLICITLY CONSTRUCTIBLE UNIVERSE
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? *