Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-zzcdp Total loading time: 0.57 Render date: 2021-11-27T10:08:13.002Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER

Published online by Cambridge University Press:  01 May 2018

FREDERIK S. HERZBERG
Affiliation:
CENTER FOR MATHEMATICAL ECONOMICS (IMW) INSTITUTE FOR INTERDISCIPLINARY STUDIES OF SCIENCE (I2SOS) BIELEFELD UNIVERSITY, UNIVERSITÄTSSTRAßE 25 D-33615 BIELEFELD, GERMANY and MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY (MCMP) LUDWIG MAXIMILIAN UNIVERSITY OF MUNICH GESCHWISTER-SCHOLL-PLATZ 1 D-80539 MUNICH, GERMANY E-mail: fherzberg@uni-bielefeld.de
VLADIMIR KANOVEI
Affiliation:
LABORATORY 6 IITP, MOSCOW, RUSSIA and DEPARTMENT OF MATHEMATICS INSTITUTE OF ECONOMICS AND FINANCE MIIT, MOSCOW, RUSSIA E-mail: kanovei@googlemail.com
MIKHAIL KATZ
Affiliation:
DEPARTMENT OF MATHEMATICS BAR ILAN UNIVERSITY RAMAT GAN 5290002, ISRAEL E-mail: katzmik@macs.biu.ac.il
VASSILY LYUBETSKY
Affiliation:
LABORATORY 6 IITP, MOSCOW, RUSSIA E-mail: lyubetsk@iitp.ru

Abstract

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2018 

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

Baumgartner, J. E. and Laver, R., Iterated perfect-set forcing. Annals of Mathematical Logic, vol. 17 (1979), no. 3, pp. 271288.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory, third ed., Studies in Logic and Foundations of Mathematics, vol. 73, North Holland, Amsterdam, 1992.Google Scholar
Davis, M.Applied Nonstandard Analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons] New York–London–Sydney, 1977.Google Scholar
Feferman, S., Some applications of the notions of forcing and generic sets. Fundamental Mathematics, vol. 56 (1965), pp. 325345.Google Scholar
Herzberg, F. S., A definable nonstandard enlargement. Mathematical Logic Quarterly, vol. 54 (2008), no. 2, pp. 167175.CrossRefGoogle Scholar
Herzberg, F. S., Stochastic Calculus with Infinitesimals, Lecture Notes in Mathematics, vol. 2067, Springer, Heidelberg, 2013.CrossRefGoogle Scholar
Hurd, A. E. and Loeb, P. A., An Introduction to Nonstandard Real Analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Orlando, FL, 1985.Google Scholar
Jech, Th., The Axiom of Choice, North-Holland Publishing Company, Amsterdam, London, 1973.Google Scholar
Kanovei, V. and Katz, M., A positive function with vanishing Lebesgue integral in Zermelo–Fraenkel set theory. Real Analysis Exchange, vol. 42 (2017), no. 2, pp. 16.Google Scholar
Kanovei, V. and Lyubetsky, V., Problems of set-theoretic non-standard analysis. Surveys Russian Mathematical Surveys, vol. 62 (2007), no. 1, pp. 45111.CrossRefGoogle Scholar
Kanovei, V. and Reeken, M., Nonstandard Analysis, Axiomatically, Springer Monographs in Mathematics, Springer, Berlin, 2004.CrossRefGoogle Scholar
Kanovei, V. and Shelah, S., A definable nonstandard model of the reals, this Journal, vol. 69 (2004), no. 1, pp. 159–164.Google Scholar
Kanovei, V. and Uspensky, V. A., On the uniqueness of nonstandard extensions. Moscow University Mathematics Bulletin, vol. 61 (2006), no. 5, pp. 1–8.Google Scholar
Keisler, H. J., The hyperreal line, Real Numbers, Generalizations of Reals, and Theories of Continua (Ehrlich, P., editor), Kluwer, Dordrecht, 1994, pp. 207237.CrossRefGoogle Scholar
Keisler, H. J., Foundations of Infinitesimal Calculus. Instructor’s Manual. Prindle, Weber & Schmidt. The online 2007 edition is available at the site http://www.math.wisc.edu/∼keisler/foundations.html.Google Scholar
Kunen, K., Set Theory, An Introduction to Independence Proofs, North-Holland Publishing Company, Amsterdam, London, 1980.Google Scholar
Luxemburg, W. A. J., A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (International Symposium, Pasadena, California, 1967) (Luxemburg, W. A. J., editor), Holt, Rinehart and Winston, New York, 1969, pp. 1886.Google Scholar
Luxemburg, W. A. J., What is nonstandard analysis? American Mathematical Monthly, vol. 80 (1973), Supplement, pp. 3867.CrossRefGoogle Scholar
Nelson, E., The virtue of simplicity, The Strength of Nonstandard Analysis (van den Berg, I. P. and Neves, V., editors), Springer, Vienna, 2007, pp. 2732.CrossRefGoogle Scholar
Pincus, D. and Solovay, R. M., Definability of measures and ultrafilters, this Journal, vol. 42, (1977), pp. 179–190.Google Scholar
Repicky, M., A proof of the independence of the axiom of choice from the Boolean prime ideal theorem. Commentationes Mathematicae Universitatis Carolinae, vol. 56 (2015), no. 4, pp. 543546.CrossRefGoogle Scholar
Robinson, A. and Zakon, E., A set-theoretical characterization of enlargements, Applications of Model Theory to Algebra, Analysis, and Probability, (International Symposium, Pasadena, California, 1967) (Luxemburg, W. A. J., editor), Holt, Rinehart and Winston, New York, 1969, pp. 109122.Google Scholar
Solovay, R. M., A Model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, vol. 92 (1970), pp. 156.CrossRefGoogle Scholar
Stroyan, K. D. and Luxemburg, W. A. J., Introduction to the Theory of Infinitesimals, Pure and Applied Mathematics, vol. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976.Google Scholar
4
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.

MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER
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.

MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER
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.

MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER
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? *