Abstract
A continuous ultraproduct is the model for a language constructed as follows: it is the set of all “continuous” maps in the product of a “continuous family” of discrete models for the language over a zero-dimensional space X, modulo an ultrafilter U in the Boolean algebra of all clopen subsets of X. The Main Theorem is a proper extension of the classic Fundamental Theorem of Ultraproducts.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ailing, N. L. (1965) Rings of continuous integer-valued functions and nonstandard arithmetic. Trans. Amer. Math. Soc. 118, 498–525.
Banaschewski, B. (1955) Über nulldimensionale Räume. Math. Nachr. 13, 129–140.
Barwise, J. Ed. (1977) Handbook of Mathematical Logic. North-Holland, Amsterdam.
Boole, G. (1847) The Mathematical Analysis of Logic. Cambridge.
Bourbaki, N. (1951) Theorie des ensembles. Hermann 846–1141.
Chang, C. C., and Keisler, H.J. (1991) Model Theory. North-Holland, Amsterdam.
Comfort, W. W. and Negrepontis, S. (1974) The Theory of Ultrafilters. Springer-Verlag, New York.
Gillman, L. and Jerison, M. (1960) Rings of Continuous Functions. Van Nostrand, Princeton.
Hardy, G. H. and Write, E. M. (1956) An Introduction to the Theory of Numbers. Oxford.
Halmos, P. R. (1960) Naive Set Theory, van Nostrand, New York.
Kelley, J. L. (1955) General Topology, van Nostrand, New York.
Lang, S. (1965) Algebra. Addison-Wesley, Reading.
Lamport, L. (1994) LaT EX. Addison-Wesley, Reading.
Los, J. (1955) Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. Mathematical interpertations of formal systems, N-H, 98–113.
Mendelson, E. (1987) Introduction to Mathematical Logic. Wadsworth, Pacific Grove.
Pierce, R. S. (1961) Rings of integer-valued continuous functions. Trans. Amer. Math. Soc. 100, 371–394.
Stone, M. H. (1936) The theory of representations of boolean algebras. Trans. Amer. Math. Soc. 40, 37–111.
Stone, M. H. (1937) Applications of the theory of boolean rings to general topology. Trans. Amer. Math. Soc. 41, 375–481.
Zariski, O. and Samuel, P. (1958) Commutative Algebra, I. van Nostrand, Princeton.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Alling, N.L. (1997). Continuous Ultraproducts. In: Holland, W.C., Martinez, J. (eds) Ordered Algebraic Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5640-0_3
Download citation
DOI: https://doi.org/10.1007/978-94-011-5640-0_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6378-4
Online ISBN: 978-94-011-5640-0
eBook Packages: Springer Book Archive