The student will be aquainted with the Zermelo-Fraenkel axiom system ZFC for set theory with the axiom of choice and with how ZFC may serve as a formalization of mathematics.
In the first part, emphasis will be put on the well ordering concept, on ordinal numbers and transfinite recursion and induction and on the equivalence of the well ordering principle, the axiom of choice and Zorn’s lemma.

Advanced Mathematical Logic Fall 2013

• What is the Mathematical Logic?

Propositional calculus and first-order logic. Theorem proving, Model theory, soundness, completeness, and compactness, Herbrand’s theorem, Skolem-Lowenheim theorems, Craig interpolation. Theory of computation and recursive function theory, Church’s thesis, computability and undecidability. Feasible computability and complexity. Peano arithmetic and the incompleteness theorems, nonstandard models.

Distinguishing two levels in studying logic is customary. At the sentential or truth-functional level, simple declarative sentences are the basic units of analysis. Complex statements are built up from these basic units by means of truth-functional connectives. Symbolically, simple statements are represented by sentence letters and the connectives through special symbols.