The notion of a randomization of a first-order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new structure whose elements are random elements of the original first-order structure. In this paper we treat randomizations as continuous structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the earlier results show that the randomization of a complete first-order theory is a complete theory in continuous logic that admits elimination of quantifiers and has a natural set of axioms. We show that the randomization operation preserves the properties of being omega-categorical, omega-stable, and stable.
Itaï Ben Yaacov 1 ; H. Jerome Keisler 1
@article{CML_2009__1_2_197_0,
author = {Ita{\"\i} Ben Yaacov and H. Jerome Keisler},
title = {Randomizations of models as metric structures},
journal = {Confluentes Mathematici},
pages = {197--223},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {1},
number = {2},
year = {2009},
doi = {10.1142/S1793744209000080},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744209000080/}
}
TY - JOUR AU - Itaï Ben Yaacov AU - H. Jerome Keisler TI - Randomizations of models as metric structures JO - Confluentes Mathematici PY - 2009 SP - 197 EP - 223 VL - 1 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744209000080/ DO - 10.1142/S1793744209000080 LA - en ID - CML_2009__1_2_197_0 ER -
%0 Journal Article %A Itaï Ben Yaacov %A H. Jerome Keisler %T Randomizations of models as metric structures %J Confluentes Mathematici %D 2009 %P 197-223 %V 1 %N 2 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744209000080/ %R 10.1142/S1793744209000080 %G en %F CML_2009__1_2_197_0
Itaï Ben Yaacov; H. Jerome Keisler. Randomizations of models as metric structures. Confluentes Mathematici, Tome 1 (2009) no. 2, pp. 197-223. doi : 10.1142/S1793744209000080. https://cml.centre-mersenne.org/articles/10.1142/S1793744209000080/
[1] I. Ben Yaacov, Model Theory with Applications to Algebra and Analysis, Vol. 2, London Math Society Lecture Note Series 350, eds. Z. Chatzidakis (Cambridge Univ. Press, 2008) pp. 315-427.
[2] I. Ben Yaacov, Continuous and random Vapnik-Chervonenkis classes, Israel J. Math., to appear , arXiv:0802.0068 .
[3] I. Ben Yaacov, On theories of random variables , arXiv:0901.1584 .
[4] I. Ben Yaacov, Israel J. Math. 153, 157 (2006), DOI: 10.1007/BF02771782 .
[5] I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc., to appear , arXiv:0801.4303 .
[6] P. R. Halmos , Measure Theory ( D. Van Nostrand , 1950 ) .
[7] H. J. Keisler, Adv. Math. 143, 124 (1999), DOI: 10.1006/aima.1998.1793 .
[8] W. Rudin , Real and Complex Analysis ( McGraw-Hill , 1966 ) .
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