Model theory studies structures through what first-order formulas can detect. The distinction between a substructure and an elementary substructure is an early place where algebraic intuition needs checking. Pillay's short notes begin with embeddings, compactness and types; Marker's book provides a longer graduate route connecting these constructions with algebraic applications. The short route suits a focused review, while the longer route needs time for examples and the additional theory.
What to work on
Compare substructures, elementary substructures and elementary equivalence.
Apply the Tarski-Vaught test in a concrete example.
Use compactness to realize a finitely satisfiable set of formulas in an extension.
Explain how quantifier elimination describes definable sets.
A starting text
Lecture Notes - Model Theory (Math 411) — Anand Pillay. Section 1, Notation and review, printed pages 1-14: emphasize elementary embeddings, the Tarski-Vaught test and compactness. Then read the opening of section 2, Types, omega-saturation and quantifier elimination, beginning on page 15, through the back-and-forth criterion.
Choosing what to read
Evaluate the same formula in a structure and a candidate substructure before applying an elementary-substructure test. For a type, separate finite satisfiability, consistency and realization in the particular model. Work through dense linear orders with a finite partial map, and explain how each extension step uses density and the lack of endpoints. Bring the elementary diagram into a compactness construction explicitly when the desired extension must preserve more than the original theory.
Review the Tarski-Vaught test and the beginnings of quantifier elimination.
How it reads
Compact lecture notes moving quickly from elementary embeddings to types.
What it expects
First-order syntax and semantics, compactness, sets and algebraic examples.
What to keep in mind
A short review leaves less space for developing unfamiliar logical foundations.
Practice
Exercises provided; public solutions not verified. Exercises 1.2, 1.5, 1.7, 1.8 (Tarski-Vaught), 1.11 and 1.14 within section 1.
Complete public PDF.
Edition, reading range and source
Section 1, Notation and review, printed pages 1-14: emphasize elementary embeddings, the Tarski-Vaught test and compactness. Then read the opening of section 2, Types, omega-saturation and quantifier elimination, beginning on page 15, through the back-and-forth criterion.
Elementary-substructure and type exercises. Exercises 1.2, 1.5, 1.7, 1.8 (Tarski-Vaught), 1.11 and 1.14 within section 1.. Exercises provided; public solutions not verified.
Try these before moving on
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
In the language with one binary relation <, use back-and-forth to show that any two countable dense linear orders without endpoints are isomorphic.
A hint
Alternate between including the next unused element of the first order in the domain and the next unused element of the second in the range.
Work through the solution
Let A and B be nonempty countable dense linear orders without endpoints. They are infinite; enumerate them as a0,a1,… and b0,b1,…. Start with the empty increasing finite partial bijection. At an even stage, take the first ai outside its domain. The finitely many existing domain points place ai in a cut: between two consecutive points, below all of them, or above all of them. Choose an unused point of B in the corresponding cut and extend the map. Density and absence of endpoints make every such interval or ray infinite, so avoiding the finitely many used points is possible.
At an odd stage apply the same argument to the inverse map, including the first unused bj in the range. Each extension preserves and reflects order. The union of these compatible finite maps is a function defined on all of A and onto all of B, by the alternating enumeration requirement. It is injective and preserves and reflects <, since every pair occurs at some finite stage. It is therefore an isomorphism.
Proofs & Mathematical Writing — Review recursive constructions and why a union of compatible partial functions is a function.
Already comfortable with this? Skip section 1 if you can use elementary diagrams and the Tarski-Vaught test correctly. Skip this entry if you can also prove the back-and-forth result and distinguish realizing a type in the original model from realizing it in an elementary extension.
A concise graduate treatment beginning with structures and languages. The lecture-note format rewards familiarity with logical notation and independent proof work.
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The complete published book is subscription or purchase content. A preview or sample is not the complete book; institutional access depends on your library licence.
Develops model-theoretic methods, including stability and types.
Preparation: First-order logic; abstract algebra.
Use: Progress from elementary logic toward model-theoretic research.
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Purchase or library subscription required; previews are not the complete book.
A broad classical reference on structures, elementary equivalence and model constructions. Compare its notation and organization with a modern introductory course.
Anand Pillay · Textbook · Research · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
Studies stability through geometric structure, dimension and independence. It is a specialist continuation, not a first introduction to logical syntax.
Heinz-Dieter Ebbinghaus and Jörg Flum · Textbook · Research · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
Studies logical properties of finite structures and their computational connections. Compactness-based methods of ordinary model theory must not be transferred blindly to finite models.
Preparation: First-order logic; model theory; discrete mathematics.
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Purchase or library subscription required; previews are not the complete book.
Introduces the non-independence property and its structural consequences. A specialist continuation after a first model-theory course, with links to combinatorics and geometry.
Preparation: A first course in model theory; types and indiscernibles.