|
A simple algebraic characterization of nonstandard extensions
Author:
Marco Forti
Journal:
Proc. Amer. Math. Soc. 140 (2012), 2903-2912
MSC (2010):
Primary 03H05, 03C07, 03C20; Secondary 26E35
Posted:
November 28, 2011
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We introduce the notion of functional extension of a set  , by means of two natural algebraic properties of the operator `` '' on unary functions. We study the connections with ultrapowers of structures with universe  , and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.
References
- 1.
Vieri
Benci and Mauro
Di Nasso, Alpha-theory: an elementary axiomatics for nonstandard
analysis, Expo. Math. 21 (2003), no. 4,
355–386. MR 2022004
(2004i:03108), http://dx.doi.org/10.1016/S0723-0869(03)80038-5
- 2.
Vieri
Benci and Mauro
Di Nasso, A purely algebraic characterization of
the hyperreal numbers, Proc. Amer. Math.
Soc. 133 (2005), no. 9, 2501–2505 (electronic). MR 2146232
(2006b:26036), http://dx.doi.org/10.1090/S0002-9939-05-07429-0
- 3.
Marco
Forti, Mauro
Di Nasso, and Vieri
Benci, Hausdorff nonstandard extensions, Bol. Soc. Parana.
Mat. (3) 20 (2002), no. 1-2, 9–20 (2003). MR 2010860
(2004h:54019)
- 4.
Vieri
Benci, Marco
Forti, and Mauro
Di Nasso, The eightfold path to nonstandard analysis,
Nonstandard methods and applications in mathematics, Lect. Notes Log.,
vol. 25, Assoc. Symbol. Logic, La Jolla, CA, 2006,
pp. 3–44. MR 2209073
(2006m:26066)
- 5.
C.
C. Chang and H.
J. Keisler, Model theory, 3rd ed., Studies in Logic and the
Foundations of Mathematics, vol. 73, North-Holland Publishing Co.,
Amsterdam, 1990. MR 1059055
(91c:03026)
- 6.
Mauro
Di Nasso and Marco
Forti, Topological and nonstandard extensions, Monatsh. Math.
144 (2005), no. 2, 89–112. MR 2123958
(2005j:03050), http://dx.doi.org/10.1007/s00605-004-0255-2
- 7.
William
S. Hatcher, Elementary extension and the hyperreal numbers,
Mathematical logic and formal systems, Lecture Notes in Pure and Appl.
Math., vol. 94, Dekker, New York, 1985, pp. 205–219. MR 801912
(87b:03147)
- 8.
Karel
Hrbáček, Axiomatic foundations for nonstandard
analysis, Fund. Math. 98 (1978), no. 1,
1–19. MR
528351 (84b:03084)
- 9.
H.J. Keisler, Foundations of Infinitesimal Calculus, Prindle, Weber and Schmidt, Boston, 1976.
- 10.
Edward
Nelson, Internal set theory: a new approach to nonstandard
analysis, Bull. Amer. Math. Soc. 83 (1977),
no. 6, 1165–1198. MR 0469763
(57 #9544)
- 11.
Siu-Ah
Ng and Hermann
Render, The Puritz order and its relationship to the Rudin-Keisler
order, continuum (Venice, 1999) Synthese Lib., vol. 306, Kluwer
Acad. Publ., Dordrecht, 2001, pp. 157–166. MR 1895391
(2003d:54045)
- 12.
Christian
Puritz, Ultrafilters and standard functions in non-standard
arithmetic, Proc. London Math. Soc. (3) 22 (1971),
705–733. MR 0289283
(44 #6474)
- 13.
Abraham
Robinson and Elias
Zakon, A set-theoretical characterization of enlargements,
(Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston,
New York, 1969, pp. 109–122. MR 0239965
(39 #1319)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
03H05,
03C07,
03C20,
26E35
Retrieve articles in all journals
with MSC (2010):
03H05,
03C07,
03C20,
26E35
Additional Information
Marco Forti
Affiliation:
Dipartimento di Matematica Applicata “U. Dini”, Universitá di Pisa, Via Buonarroti 1C, 56100 Pisa, Italy
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11104-3
PII:
S 0002-9939(2011)11104-3
Received by editor(s):
December 24, 2010
Received by editor(s) in revised form:
January 31, 2011 and March 3, 2011
Posted:
November 28, 2011
Additional Notes:
This work was partially supported by MIUR Grants PRIN 2007, 2009, Italy.
Communicated by:
Julia Knight
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|
