Meeting times: T,Th from 10:30–11:45am
Meeting place: MB 124
Textbook: Kunen, The Foundations of Mathematics
My office: MB 238-B
Office hours: Monday 10:30–11:30, Thursday 1–2, and by appointment
We often say that mathematics is the study of certain abstract truths, but in what sense is that the case? In this course we will study the foundations of mathematics, which means the systematic way in which we justify mathematical thought. The course will be divided into two sections: set theory and model theory.
Informally, a set is an unordered collection of other sets. It may be somewhat hard to understand what that really means, so the formal definition of set involves axioms. Most familiar mathematical objects are defined by axiom systems, but the axioms for sets are somewhat more complex. We will introduce and discuss each axiom one by one. We will then explore how sets are used to build nearly all other mathematical objects. Along the way we will study many concepts that are useful in other areas: ordinals, cardinals, and the mathematics of the infinite.
A model is a universe that satisfies some axiom system called a theory. We use models to study the nature of formal proof itself. This study culminates with Godel’s completeness theorem, which has numerous applications in logic as well as other areas of mathematics.
Homework exercises will be assigned each week and collected the following week. You may submit revised solutions any number of times. Homework may occasionally include additional items such as reading responses, definitions, calculations, or conjectures.
You may also optionally complete a series of supplemental problems (see the grading system below). I will regularly post new problems for you to choose from, and you may also suggest your own problems in consultation with me.
You are encouraged to collaborate with your classmates, and you are welcome to use outside resources when you are stuck. But please: do not select the same series of supplemental problems as someone else, only turn in solutions that you fully understand, and most importantly write them in your own words.
You will keep a portfolio (paper or electronic) of all your written work for this class. The portfolio will contain your homework exercises, any additional required homework items, and any number of supplemental problems.
To earn a C-, you will complete each homework assignment on time, and keep your portfolio up-to-date with original and corrected exercises, and any other required homework items. You do not need to complete any supplemental problems. I will drop a couple unexcused late assignments and a few incorrect exercise solutions.
To earn above a C-, you may additionally complete a series of supplemental problems. Most supplemental problems will be counted as 1 point for a complete, correct solution. Challenging supplemental problems will be counted as 2 or more points. For an A collect 16–20 points, for a B collect 6–15 points, for a C collect 3–5 points. (I may adjust this guidance early in the semester.)
The portfolio will be evaluated periodically, on the following dates: