Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T10:00:50.084Z Has data issue: false hasContentIssue false
  • English
  • Français

Standardization principle of nonstandard universes

Published online by Cambridge University Press:  12 March 2014

Masahiko Murakami
Masahiko Murakami*
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan, E-mail: muramasa@ms.u-tokyo.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

A bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1645 - 1655
Copyright
Copyright © Association for Symbolic Logic 1999

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

REFERENCES

[1]Bell, J., A characterization of universal complete Boolean algebras, Journal of the London Mathematical Society, vol. 2 (1975), no. 12, pp. 86–88.CrossRefGoogle Scholar
[2]Bell, J., Boolean-valued models and independence proofs in set theory, 2nd ed., Oxford University Press, Oxford, 1985.Google Scholar
[3]Blass, A., The Rudin-Keisler ordering of P-points, Transactions of the American Mathematical Society, vol. 179 (1973), pp. 145–166.Google Scholar
[4]Blass, A., End extensions, conservative extensions, and the Rudin-Frolik ordering, Transactions of the American Mathematical Society, vol. 225 (1977), pp. 325–340.Google Scholar
[5]Chang, C. and Keisler, J., Model theory, 3rd ed., North-Holland, Amsterdam, 1990.Google Scholar
[6]Comfort, W. W. and Negrepontis, S., The theory of ultrafilters, Springer-Verlag, Berlin, 1974.CrossRefGoogle Scholar
[7]Frolík, Z., Sum of ultrafilters, Bulletin of the American Mathematical Society, vol. 73 (1967), pp. 87–91.CrossRefGoogle Scholar
[8]Grigorieff, S., Intermediate submodels and generic extension in set theory, Annals of Mathematics, vol. 101 (1975), no. 2, pp. 447–490.CrossRefGoogle Scholar
[9]Halmos, P., Lectures on Boolean algebras, Van Nostrand, New York, 1963.Google Scholar
[10]Kawai, T., Nonstandard analysis by axiomatic method, Southeast Asian conference on logic, North-Holland, Amsterdam, 1983, pp. 55–76.Google Scholar
[11]Kunen, K., Some application of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179–227.CrossRefGoogle Scholar
[12]Mansfield, R., The theory of Boolean ultrapowers, Annals of Mathematical Logic, vol. 2 (1971), pp. 297–323.CrossRefGoogle Scholar
[13]Nelson, E., Internal set theory, Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 1165–1198.CrossRefGoogle Scholar
[14]Ozawa, M., Forcing in nonstandard analysis, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 263–297.CrossRefGoogle Scholar
[15]Ozawa, M., Scott incomplete boolean ultrapowers on the real line, this Journal, vol. 60 (1995), no. 1.Google Scholar
[16]Robinson, A., Non-standard analysis, North-Holland, Amsterdam, 1966.Google Scholar
[17]Rudin, M., Partial orders on the type in βN, Transactions of the American Mathematical Society, vol. 155 (1971), no. 2, pp. 353–362.Google Scholar
[18]Scott, D., Boolean models and nonstandard analysis, Applications of model theory to algebra, analysis and probability, Pasadena, 1967, pp. 89–92.Google Scholar
[19]Scott, D., Boolean-valued models for set theory, mimeographed notes for the 1967 American Mathematical Society Symposium on axiomatic set theory, 1967.Google Scholar
[20]Scott, D., Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, no. XIII, part I, American mathematical Society, Providence, 1971.CrossRefGoogle Scholar
[21]Scott, D. and Solovay, R., Boolean valued models of set theory, Proceedings of Symposia in pure Mathematics, no. 13, part II.Google Scholar
[22]Sikorski, R., Boolean algebras, Springer-Verlag, Berlin-Heidelberg-New York, 1964.Google Scholar
[23]Takeuti, G., Two applications of logic to mathematics, Princeton University Press, Princeton, 1978.Google Scholar
[24]Takeuti, G. and Zaring, W. M., Axiomatic set theory, Springer-Verlag, New York, 1973.CrossRefGoogle Scholar
[25]Vopěnka, P., The limits of sheaves and applications on constructions of models, Bulletin of the Academy of Polon. Sci. Ser. Sci. Math. Astron. Phys., vol. 13 (1965), pp. 189–192.Google Scholar
[26]Vopěnka, P. and Hájek, P., The theory of semiset, North-Holland, Amsterdam, 1972.Google Scholar
[27]Yasugi, M., Tsujii, Y., and Mori, T., A metatheory of nonstandard analysis, Tsukuba Journal of Mathematics, vol. 17 (1993), no. 1, pp. 251–265.Google Scholar