Arithmetic is the branch of mathematics which studies the natural numbers — i.e. the numbers 0, 1, 2, 3, and so on — and operations on the numbers — like addition and multiplication. This course explores the features that make arithmetic distinctive, and pose unique philosophical challenges. The path through the course is as follows.

Arithmetic is infinitary, abstract, a priori and apodictic, necessary, completely general, and scientifically indispensable. You will start by surveying these features, and encounter the general idea of a formal theory of arithmetic.
	A common sentiment is that, in mathematics, consistency suffices for existence. You will explore this idea, understanding what it means to describe a theory as "consistent", and how one might establish consistency. This will lead into into a discussion of Hilbert's programme, which aimed to provide proofs that (various) mathematical theories are consistent. Famously, this programme floundered when Gödel discovered his incompleteness theorems.
	You will learn about the technical details behind the incompleteness theorems, including such concepts as: (computable) enumerability, representability, the arithmetization of syntax, Tarski's Diagonal Lemma, Gödel sentences, and consistency sentences.
	Armed with this technical knowledge, you will assess the philosophical significance of these results, both for Hilbert's programme and for other philosophical positions.
	To finish the course, you will consider other approaches to the philosophy of arithmetic, and how they deal with the phenomenon of incompleteness.
The course will be based entirely weekly lectures, backed up with classes. Each lecture/class will have compulsory readings.

Please note that the course combines philosophical and formal elements! Although it is not a formal prerequisite, the course will presuppose introductory logic (at the level of first year Introduction to Logic 1 & 2); at the very least, you will need to be comfortable with how first-order logic works. The course will not presuppose any particular prior knowledge of mathematics; only that you know how to count, and can make sense of expressions like ‘x2 + 3x + 2 = 0’ (even if you cannot quite remember how to solve it). Still, if the very idea of looking at an expression like that fills you with horror, this course is not for you. Half of your final grade will be based on your performance in problem sets, which will help to reinforce your understanding of the technical details behind the incompleteness theorems.

Philosophy Area A