Randomizations of Scattered Theories H. Jerome Keisler January 9, 2015 H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 1 / 1
1. Overview Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T . In many cases, T R has properties similar to those of T . H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 2 / 1
1. Overview Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T . In many cases, T R has properties similar to those of T . In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 2 / 1
1. Overview Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T . In many cases, T R has properties similar to those of T . In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. This means that up to isomorphism, every separable model is determined in a simple way by assigning a probability to each isomorphism type of countable models of T . H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 2 / 1
1. Overview Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T . In many cases, T R has properties similar to those of T . In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. This means that up to isomorphism, every separable model is determined in a simple way by assigning a probability to each isomorphism type of countable models of T . In the other direction, if T R has few separable models, then T is scattered in the sense of Morley. Assuming the absolute form of Vaught’s conjecture, T has countably many non-isomorphic countable models if and only if T R has few separable models. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 2 / 1
1. Overview Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T . In many cases, T R has properties similar to those of T . In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. This means that up to isomorphism, every separable model is determined in a simple way by assigning a probability to each isomorphism type of countable models of T . In the other direction, if T R has few separable models, then T is scattered in the sense of Morley. Assuming the absolute form of Vaught’s conjecture, T has countably many non-isomorphic countable models if and only if T R has few separable models. I will also discuss what happens without Vaught’s conjecture. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 2 / 1
Some References 1 I. Ben Yaacov, A. Berenstein, C. W. Henson, A. Usvyatsov. Model Theory for Metric Structures. Cambridge U. Press (2008), 315–427. 2 I. Ben Yaacov and H. J. Keisler. Randomizations as Metric Structures, Confluentes Mathematici 1 (2009), 197-223. 3 U. Andrews and H. J. Keisler. Separable Models of Randomizations. To appear, JSL. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 3 / 1
Some References 1 I. Ben Yaacov, A. Berenstein, C. W. Henson, A. Usvyatsov. Model Theory for Metric Structures. Cambridge U. Press (2008), 315–427. 2 I. Ben Yaacov and H. J. Keisler. Randomizations as Metric Structures, Confluentes Mathematici 1 (2009), 197-223. 3 U. Andrews and H. J. Keisler. Separable Models of Randomizations. To appear, JSL. 4 R. L. Vaught. Denumerable Models of Complete Theories. In Infinitistic Methods, Pergamon Press (1961), 303-321. 5 D. Scott. Logic with Denumerably Long Formulas and Finite Strings of Quantifiers. In Theory of Models, North-Holland (1965), 329-341. 6 M. Morley. The Number of Countable Models. JSL 25 (1970), 14-18. 7 J. Steel. On Vaught’s Conjecture. Cabal Seminar 76–77, 193–208. 8 J. Baldwin, S. Friedman, M. Koerwien, and M. Laskowski. Three Red Herrings around Vaught’s Conjecture. To appear, TAMS. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 3 / 1
2. Continuous Structures Structures live on complete metric spaces. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. · Ψ. Connectives 0, 1, Φ / 2, Φ − Quantifiers sup , inf. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. · Ψ. Connectives 0, 1, Φ / 2, Φ − Quantifiers sup , inf. N | = Φ means the sentence Φ has value 0 in N . H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. · Ψ. Connectives 0, 1, Φ / 2, Φ − Quantifiers sup , inf. N | = Φ means the sentence Φ has value 0 in N . = Φ } , N ≡ N ′ means Th ( N ) = Th ( N ′ ). Th ( N ) = { Φ : N | H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. · Ψ. Connectives 0, 1, Φ / 2, Φ − Quantifiers sup , inf. N | = Φ means the sentence Φ has value 0 in N . = Φ } , N ≡ N ′ means Th ( N ) = Th ( N ′ ). Th ( N ) = { Φ : N | Every prestructure is ≡ its completion. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. · Ψ. Connectives 0, 1, Φ / 2, Φ − Quantifiers sup , inf. N | = Φ means the sentence Φ has value 0 in N . = Φ } , N ≡ N ′ means Th ( N ) = Th ( N ′ ). Th ( N ) = { Φ : N | Every prestructure is ≡ its completion. N is separable if its completion has a countable dense subset. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
2. Continuous Structures Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0 , 1] with 0 meaning true. · Ψ. Connectives 0, 1, Φ / 2, Φ − Quantifiers sup , inf. N | = Φ means the sentence Φ has value 0 in N . = Φ } , N ≡ N ′ means Th ( N ) = Th ( N ′ ). Th ( N ) = { Φ : N | Every prestructure is ≡ its completion. N is separable if its completion has a countable dense subset. By a model of U we mean a prestructure N | = U . H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 4 / 1
3. Randomization of a Theory Let T be a complete first order theory with T | = ∃ x ∃ y ( x � = y ). Let L be the signature of T . L R is the continuous signature with: H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 5 / 1
3. Randomization of a Theory Let T be a complete first order theory with T | = ∃ x ∃ y ( x � = y ). Let L be the signature of T . L R is the continuous signature with: Two sorts, K for random elements and E for events. H. Jerome Keisler Randomizations of Scattered Theories January 9, 2015 5 / 1
Recommend
More recommend
