MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER

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: definability; hyperreal; superstructure; elementary embedding.

[1]  David Pincus,et al.  Definability of measures and ultrafilters , 1977, Journal of Symbolic Logic.

[2]  H. Jerome Keisler,et al.  The Hyperreal Line , 1994 .

[3]  S. Feferman.  Some applications of the notions of forcing and generic sets , 1964 .

[4]  A. E. Hurd,et al.  An Introduction to Nonstandard Real Analysis , 2011 .

[5]  Saharon Shelah,et al.  A definable nonstandard model of the reals , 2004, J. Symb. Log..

[6]  W. A. J. Luxemburg.  What Is Nonstandard Analysis , 1973 .

[7]  K. D. Stroyan,et al.  Introduction to the theory of infinitesimals , 1976 .

[8]  Miroslav Repický.  A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem , 2015 .

[9]  Kenneth H Britten,et al.  The virtue of simplicity , 2006, Nature Neuroscience.

[10]  R. Solovay.  A model of set-theory in which every set of reals is Lebesgue measurable* , 1970 .

[11]  B. Russell.  THE AXIOM OF CHOICE , 2003 .

[12]  Vladimir Kanovei,et al.  A positive function with vanishing Lebesgue integral in Zermelo-Fraenkel set theory , 2017, 1705.00493.

[13]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

[14]  Leif Arkeryd.  Nonstandard Analysis , 2005, Am. Math. Mon..

[15]  Vladimir Kanovei,et al.  Problems of set-theoretic non-standard analysis , 2007 .

[16]  Kenneth Kunen,et al.  Set Theory: An Introduction to Independence Proofs , 2010 .

[17]  Frederik Herzberg.  Stochastic Calculus with Infinitesimals , 2013, Lecture notes in mathematics.

[18]  Frederik Herzberg.  A definable nonstandard enlargement , 2008, Math. Log. Q..

[19]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[20]  Martin Davis,et al.  Applied Nonstandard Analysis , 1977 .

[21]  James E. Baumgartner,et al.  Iterated perfect-set forcing , 1979 .