**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.

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.

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

- 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

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

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

- 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)

I’m *Greg Restall*, and this is my *personal website*. ¶ I am the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews, and the Director of the Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology ¶ I like thinking about – and helping other people think about – logic and philosophy and the many different ways they can inform each other.

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- St Andrews profile.
- My office is in Edgecliffe, The Scores.