Model theory

前置知识

17 项必需的前置知识（一直追溯到底）

你将学会

• 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.

定义

• 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.

定理

• 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.

常见误解

• 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.

可选发现

• 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
• ← Non-Euclidean geometry: spherical and hyperbolic
• ← Axiomatic set theory: hierarchy, cardinal arithmetic and L

延伸阅读

• David Marker, Model Theory: An Introduction (2002)
• Wilfrid Hodges, Model Theory (1993)
• C. C. Chang and H. J. Keisler, Model Theory (1973)