First-year graduateFoundationsPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 17
You will learn to
- 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.
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Optional discoveries
- 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
Further reading
- David Marker, Model Theory: An Introduction (2002)
- Wilfrid Hodges, Model Theory (1993)
- C. C. Chang and H. J. Keisler, Model Theory (1973)