×

Standard sets in nonstandard set theory. (English) Zbl 1067.03071

Summary: We prove that standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the \(\in\)-language.

MSC:

03H05 Nonstandard models in mathematics
03E70 Nonclassical and second-order set theories
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1016/0003-4843(71)90005-2 · Zbl 0225.02044
[2] Standardization principle of nonstandard universes 64 pp 1645– (1999) · Zbl 0948.03062
[3] Axiomatic set theory XIII pp 33– (1974)
[4] Sibirski Matematicheski Zhurnal 30 pp 64– (1989)
[5] An axiomatic presentation of the nonstandard methods in mathematics 67 pp 315– (2002)
[6] Model theory (1990)
[7] Fundamenta Mathematicae 123 pp 199– (1987)
[8] Contemporary Mathematics 176 (1994)
[9] An axiomatics for nonstandard set theory, based on von Neumann-Bernays-Gödel theory 66 pp 1321– (2001)
[10] Vestnik Moskovskogo Universiteta, Ser. 1, Mat., Mech. pp 68– (1997)
[11] DOI: 10.4213/mzm1225
[12] DOI: 10.1007/s001530050155 · Zbl 1002.03044
[13] Mathematica Japonica 45 pp 555– (1997)
[14] Stadia Logica 55 pp 227– (1995)
[15] Russian Mathematical Surveys 46 pp 1– (1991)
[16] Set theory (1978)
[17] Proceedings of the Special Session on Nonstandard Methods, AMS Annual Meeting in Baltimore 2003 (2004)
[18] DOI: 10.1016/S0168-0072(01)00039-2 · Zbl 0985.03065
[19] Fundamenta Mathematicae 98 pp 1– (1978)
[20] Nonstandard methods in commutative harmonic analysis (1997) · Zbl 0873.43001
[21] No elementary embedding from V into V is definable from parameters 64 pp 1591– (1999) · Zbl 0946.03061
[22] Applications of model theory to algebra, analysis and probability pp 109– (1969)
[23] Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 308 pp 301– (1989)
[24] Osaka Journal of Mathematcs 29 pp 267– (1992)
[25] DOI: 10.1090/S0002-9904-1977-14398-X · Zbl 0373.02040
[26] Sibirski Matematicheski Zhurnal 30 pp 89– (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.