Model Theory I

		Study programme 2019-2020	Français
		Model Theory I
		Learning Activity

Code	Lecturer(s)	Associate Lecturer(s)	Subsitute Lecturer(s) et other(s)	Establishment
S-MATH-023	POINT Françoise			Université de Mons

Language of instruction	Language of assessment	HT(*)	HTPE(*)	HTPS(*)	HR(*)	HD(*)	Term
Français	Français	15	15	0	0	0	Q2

Organisational online arrangements for the end of Q3 2019-2020 assessments (Covid-19)
Production of individual or group work, essay, report, dissertation...
Description of the modifications to the Q3 2019-2020 online assessment procedures (Covid-19)
Review of first-order theories and the corresponding languages (fields, rings, groups, vector-spaces, Boolean algebras, graphs, equivalence relations). Review on the notions of morphisms (isomorphisms, automorphisms, embeddings). Revew on elementary classes. -Chains of structures and theorem on elementary chains. -types and result on realisation of 1-types in an elementary extension. -notion of partial elementary map and link with the property of having the same type and extension of these maps. -Definition and criterium for quantifier elimination (application of the compacity theorem-how to measure the complexity of a formula). -Criterium for elementary substructure and theorems of Lowenheim-Skolem (down and up). -Notion of kappa-categoricity and Vaught theorem. -Cantor theorem on dense linear orders and the go-and-forth construction. -Spaces of types and descrition of the topology (link with the compacity theorem). -Theorem of Ryll-Nardewski (admitting the proof of the omitting type theorem). Notion of an omitted type, atomic models and theorem on the unicity of countable atomic models up to isomorphisms. -------------- written exam of approximated length 3 hours and an half, of the type written synchroneous production.

Content of Learning Activity

Lowenheim-Skolem theorems, elementary substructures, existentially closed ones. Model-complete theories, quantifier elimination (criteria for these properties). Algebraic examples for these notions. Back-and-forth and dense/discrete orders. Equivalence relations. Categoricity and Ryll-Nardweski theorem.

Required Learning Resources/Tools

Marker, D., Model theory. An introduction. Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002.

Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990, 1977, 1973.