Primer año de posgradoFoundationsPlanificado
Model theory
Structures and their theories: elementary equivalence and elementary embeddings, types, ultraproducts and Łoś's theorem, and quantifier elimination, with examples from orders, fields and algebraic geometry.
math.found.model-theory
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 17
Aprenderás a
- Prove Łoś's theorem and use ultraproducts to give a second proof of compactness.
- Prove quantifier elimination for dense linear orders without endpoints, and state it for algebraically closed fields.
- Use the theory of algebraically closed fields to explain the Lefschetz principle and the Ax–Grothendieck theorem.
Definición
- Structures are elementarily equivalent if they satisfy the same first-order sentences.
- An elementary embedding preserves the truth of every formula with parameters.
- A theory has quantifier elimination if every formula is equivalent, modulo the theory, to a quantifier-free formula.
- An ultraproduct ∏ Mᵢ / U is the quotient of a product of structures by an ultrafilter U on the index set.
Teorema
- Łoś's theorem: a formula holds in ∏ Mᵢ / U exactly when the set of indices i at which it holds in Mᵢ belongs to U.
- The theory of dense linear orders without endpoints is ℵ₀-categorical and has quantifier elimination.
- The theory of algebraically closed fields has quantifier elimination, a result closely related to Chevalley's theorem on constructible sets.
- Ax–Grothendieck: every injective polynomial map ℂⁿ → ℂⁿ is surjective.
Errores frecuentes
- Thinking elementarily equivalent structures must be isomorphic: (ℚ, <) and (ℝ, <) are elementarily equivalent but not isomorphic.
- Taking the existence of many models as a sign that a theory is vague; it is typical of first-order theories with infinite models.
Descubrimientos opcionales
- Algebraic geometry Quantifier elimination for algebraically closed fields gives a logical proof that projections of constructible sets are constructible.
- Forcing and independence Forcing builds new models of ZFC from old ones, a model-theoretic method at the heart of independence proofs.
- ← Structure-preserving maps: homomorphisms and isomorphisms
- ← Axiomatic set theory: hierarchy, cardinal arithmetic and L
Lecturas complementarias
- David Marker, Model Theory: An Introduction (2002)
- Wilfrid Hodges, Model Theory (1993)
- C. C. Chang and H. J. Keisler, Model Theory (1973)