Foundations of Math

My (long term) goal for these notes is to build a first course in higher math, starting from the foundations of logic and set theory and building towards comfort with a lot of the general tools and techniques which are helpful to have when beginning any course in the “upper division” (analysis, algebra, topology, etc.). I’ll include problems and (perhaps) solutions along with my notes.

The goal of these notes is not to be a discussion of the philosophy of math, although we will undoubtedly encounter moments when such a discussion would be possible or even natural. I will digress at times to have that discussion, but not always; my experience is not with philosophy of math but with using the foundations to prove things rigorously.

• Preliminaries: Philosophy, Rigour, and Logic

  (These first few paragraphs are mostly a discussion of philosophy; the impatient reader can safely skip to Logic (A Practical Primer)

  • Problems

    Here are some problems designed to get you started thinking in terms of first order logic.
• Introduction to Sets

  Now we’ll discuss the backbone of all modern math, set theory. Math can be thought of as, in it’s most basic form, the study of collections; a series of advanced ways of counting. A set is effectively a well-define collection of objects; however, it is actually quite difficult to specify what “well-defined” means. There was a time when a set was taken to be any collection of objects which could be described; however, this leads to paradoxes. For example, let $S$ denote the collection of all collections which do not contain themselves. Is $S$ in $S$? If it is, then $S$ contains itself, and so $S$ cannot be an element of $S$, a contradiction. But if it is not, then $S$ does not contain $S$, so $S$ must be in $S$, another contradiction. The only possibility is that “membership”, the relation that specifies if $S$ is in the collection $S$, is not well defined, and so the collection is not well-defined.
• Relations
• Functions