Titlebook: An Invitation to Mathematical Logic; David Marker Textbook 2024 The Editor(s) (if applicable) and The Author(s), under exclusive license t

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Languages, Structures, and TheoriesWe introduce the fundamental concepts of mathematical logic: languages, structures, satisfaction, theories, logical consequences, and definable sets.

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Gödel’s Completeness TheoremUsing the ideas of Henkin, Gödel’s Completeness Theorem is proved, showing that our proof system completely captures the notion of logical consequence.

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UltraproductsThe ultraproduct construction is introduced and used to give an alternative proof of the Compactness Theorem.

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Model Theory of the Real FieldWe study the model theory of the real field, proving Tarski’s quantifier elimination and decidability results and studying its consequences. We include a brief discussion on more recent work on o-minimal expansions of the real field and exponentiation.

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Models of ComputationRegister machines are introduced as a machine-based model of computation. We show that register machines compute exactly the class of general recursive functions and formulate the Church–Turing thesis that this is exactly the collection of computable functions.

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Universal Machines and UndecidabilityTuring’s universal machine is constructed and used to prove the undecidability of the Halting Problem and the undecidability of validity in first-order logic. We include a brief discussion of the Recursion Theorem.

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Computably Enumerable and Arithmetic SetsWe introduce the computably enumerable sets and the arithmetic sets and show that the form a hierarchy. These results, and the existence of computably inseparable computably enumerably sets, will be used in our approach to the Incompleteness Theorem. We briefly study Kolmogorov randomness as another avatar of incompleteness phenomena.