PHIL30043: The Power and Limits of Logic

March 2019

at the University of Melbourne

PHIL30043: The Power and Limits of Logic is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems (proved not once but twice, first for a tableaux proof system for predicate logic, then a Hilbert proof system), through the Deduction Theorem, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability 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 Vimeo.

Outline

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

Prelude

• Logical Equivalence
• Disjunctive Normal Form
• Why DNF Works
• Prenex Normal Form
• Models for Predicate Logic
• Trees for Predicate Logic

Completeness

• Introducing Soundness and Completeness
• Soundness for Tree Proofs
• Completeness for Tree Proofs
• Hilbert Proofs for Propositional Logic
• Conditional Proof
• Hilbert Proofs for Predicate Logic
• Theories
• Soundness and Completeness for Hilbert Proofs for Predicate Logic

Compactness

• Counting Sets
• Diagonalisation
• Compactness
• Non-Standard Models
• Inexpressibility of Finitude
• Downward Löwenheim–Skolem Theorem

Computability

• Functions
• Register Machines
• Recursive Functions
• Register Machine computable functions are Recursive
• The Uncomputable

Undecidability and Incompleteness

• Deductively Defined Theories
• The Finite Model Property
• Completeness
• Introducing Robinson’s Arithmetic
• Induction and Peano Arithmetic
• Representing Functions and Sets
• Gödel Numbering and Diagonalisation
• Q (and any consistent extension of Q) is undecidable, and incomplete if it’s deductively defined
• First Order Predicate Logic is Undecidable
• True Arithmetic is not Deductively Defined
• If Con(PA) then PA doesn’t prove Con(PA)