PHIL30043: The Power and Limits of Logic

March 2021

at the University of Melbourne

PHIL30043: The Power and Limits of Logic (or, as I like to call it, Kurt Gödel’s Greatest Hits) is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability, the Indefinability of Truth and the Undecidability of Predicate Logic, and ending up at Gödel’s Incompleteness Theorems and Löb’s Theorem.

The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. I make use of video lectures I have made freely available on YouTube.

Outline

The course is divided into four major sections, taught over 12 weeks. Here is a list of all of the videos, in case you’d like to follow along with the content.

Soundness and Completeness

• The Language of Predicate Logic
• Proofs for Predicate Logic
• Models for Predicate Logic and Soundness
• Completeness for Predicate Logic

Countability and Compactness

• Identity and Functions
• Countability and Diagonalisation
• Compactness and Countable Models

Computability

• Recursive Functions
• Register Machines
• Register Machines compute the Recursive Functions
• Non-Recursive Functions

Undecidability and Incompleteness

• Theories of Arithmetic
• Diagonalisation and its Consequences
• Provability Predicates, and Beyond