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MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER

Published online by Cambridge University Press:  01 May 2018

FREDERIK S. HERZBERG ,
VLADIMIR KANOVEI ,
MIKHAIL KATZ  and
VASSILY LYUBETSKY
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
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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.

Keywords

03E3503C20definabilityhyperrealsuperstructure
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 385 - 391
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Baumgartner, J. E. and Laver, R., Iterated perfect-set forcing. Annals of Mathematical Logic, vol. 17 (1979), no. 3, pp. 271–288.Google 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. 325–345.Google Scholar
Herzberg, F. S., A definable nonstandard enlargement. Mathematical Logic Quarterly, vol. 54 (2008), no. 2, pp. 167–175.Google 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. 1–6.CrossRefGoogle Scholar
Kanovei, V. and Lyubetsky, V., Problems of set-theoretic non-standard analysis. Surveys Russian Mathematical Surveys, vol. 62 (2007), no. 1, pp. 45–111.Google 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. 207–237.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. 18–86.Google Scholar
Luxemburg, W. A. J., What is nonstandard analysis? American Mathematical Monthly, vol. 80 (1973), Supplement, pp. 38–67.Google 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. 27–32.Google 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. 543–546.Google 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. 109–122.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. 1–56.Google 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