Logic and Set Theory

Logic and Set Theory(D)

LOGIC AND SET THEORY (D)
No specific prerequisites.

Ordinals and cardinals

  • Well-orderings and order-types. Examples of countable ordinals. Uncountable ordinals and Hartogs’ lemma. Induction and recursion for ordinals. Ordinal arithmetic. Cardinals; the hierarchy of alephs. Cardinal arithmetic. [5]

Posets and Zorn’s lemma

  • Partially ordered sets; Hasse diagrams, chains, maximal elements. Lattices and Boolean algebras. Complete and chain-complete posets; fixed-point theorems. The axiom of choice and Zorn’s lemma. Applications of Zorn’s lemma in mathematics. The well-ordering principle. [5]

Propositional logic

  • The propositional calculus. Semantic and syntactic entailment. The deduction and completeness theorems. Applications: compactness and decidability. [3]
    Predicate logic The predicate calculus with equality. Examples of first-order languages and theories. Statement of the completeness theorem; *sketch of proof*. The compactness theorem and the L¨owenheim-Skolem theorems. Limitations of first-order logic. Model theory. [5]

Set theory

  • Set theory as a first-order theory; the axioms of ZF set theory. Transitive closures, epsilon-induction and epsilon-recursion. Well-founded relations. Mostowski’s collapsing theorem. The rank function and the von Neumann hierarchy. [5]

Consistency

  • ∗Problems of consistency and independence∗. [1]

Appropriate books

  • B.A. Davey and H.A. Priestley Lattices and Order. Cambridge University Press 2002
  • T. Forster Logic, Induction and Sets. Cambridge University Press 2003
  • A. Hajnal and P. Hamburger Set Theory. LMS Student Texts number 48, CUP 1999
  • A.G. Hamilton Logic for Mathematicians. Cambridge University Press 1988
  • † P.T. Johnstone Notes on Logic and Set Theory. Cambridge University Press 1987
  • D. van Dalen Logic and Structure. Springer-Verlag 1994

Associated GitHub Page
https://jaircambridge.github.io/Logic-and-Set-Theory-D/