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Niveau: Supérieur, Doctorat, Bac+8

POSITIVE MODEL THEORY AND COMPACT ABSTRACT THEORIES ITAY BEN-YAACOV Abstract. We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such. Introduction Trying to extend the classical model-theoretical techniques beyond the strictly first- order context seems to be a popular trend these days. In [Hru97], Hrushovski defines Robinson theories, namely universal theories whose class of models has the amalgama- tion property. He subsequently works in the category of its existentially closed models, which serves as an analogue of the first order model completion when this does not exist. In [Pil00], Pillay generalises this to the category of existentially closed models of any universal theory. In both cases, one works rather in an existentially universal domain for the category, which replaces the monster model of first order theories. The present work started independently of the latter, trying to use ideas in the former in order to define a model-theoretic framework where hyperimaginary elements could be adjoined as parameters to the language, the same way we used to do it with real and imaginary ones since the dawn of time: as the type-space of a hyperimaginary sort is not totally disconnected, we need a concept of a theory who just

POSITIVE MODEL THEORY AND COMPACT ABSTRACT THEORIES ITAY BEN-YAACOV Abstract. We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such. Introduction Trying to extend the classical model-theoretical techniques beyond the strictly first- order context seems to be a popular trend these days. In [Hru97], Hrushovski defines Robinson theories, namely universal theories whose class of models has the amalgama- tion property. He subsequently works in the category of its existentially closed models, which serves as an analogue of the first order model completion when this does not exist. In [Pil00], Pillay generalises this to the category of existentially closed models of any universal theory. In both cases, one works rather in an existentially universal domain for the category, which replaces the monster model of first order theories. The present work started independently of the latter, trying to use ideas in the former in order to define a model-theoretic framework where hyperimaginary elements could be adjoined as parameters to the language, the same way we used to do it with real and imaginary ones since the dawn of time: as the type-space of a hyperimaginary sort is not totally disconnected, we need a concept of a theory who just

- existentially universal domain
- positive model
- free variable
- positive fragment
- then there
- fragment ∆
- generated positive
- partial order
- without any
- than ∆-homomorphisms

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Published by | mijec |

Reads | 11 |

Language | English |

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