SoSe 2010 » Introduction to Mathematical Logic
Ziegler

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Synopsis

Mathematical logic is the mathematical discipline that investigates mathematical statements and their meaning in a setting of a formalised language with a well-defined semantics. In this setting one can investigate fundamental notions such as definability, provability and consistency with mathematical means. In this sense mathematical logic is a technical branch of mathematics that at the same time provides a framework for meta-mathematical investigations. Apart from other diverse applications within mathematics and outside (notably in Computer Science), mathematical logic in particular sheds light on the foundations of mathematics. In this introduction to mathematical logic we shall concentrate on set theory and classical first-order logic. Axiomatic set theory provides a foundation for ordinal and cardinal numbers. The syntax, semantics, expressive power of first-order logic are investigated, along with a discussion of its role in the foundations of mathematics. Formal proofs in the framework of a proof calculus are connected with an account of logical consequence (Gödel's Completeness Theorem). Among further key results to be covered are the compactness theorem for first-order logic and the lower Löwenheim-Skolem Theorem. We close with Gödel's Incompleteness Theorems.

  1. Introduction and Motivation
    1. The Axiomatic Approach
    2. Peano's Axioms for the Integers
  2. Ordinal Numbers
    1. Well-Ordered Sets
    2. The Ordinals
    3. Operations on Ordinals
    4. Integers and Transfinites
  3. Cardinality
    1. Subpotence and Equipotence
    2. Cantor-Schröder-Bernstein
    3. Finite Sets
    4. Countable Sets
    5. Uncountable Sets
  4. Applications
    1. Algebraic Reals and Transcendence Degree
    2. Recursive Sets
    3. Borel Hierarchy
    4. Wall Street Trading
  5. A Foundation of Set Theory
    1. Zermelo-Fraenkel
    2. (Re)constructing Set Theory
    3. The Axiom of Choice
  6. First-Order Logic
    1. Example: Group Theory
    2. Syntax
    3. Semantics
    4. Downward Löwenheim-Skolem
  7. Propositional Logic
    1. Compactness of Propositional Logic
    2. Proof in Propositional Logic
    3. Metatheorems in Propositional Logic
    4. Completeness of Propositional Logic
  8. Metatheory of First-Order Logic
    1. Proof in First-Order Logic
    2. Completeness Theorem
    3. Compactness
    4. Complete Theories
  9. Computability Theory
    1. Incompleteness of Arithmetic

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