MA 225: Foundations of Advanced Mathematics


The title of this site and the course it serves are both Foundations of Advanced Mathematics. That's for two reasons. First, because the course provides a first look at what proofs-based math looks like, that is, what it means to be in a mathematics course where the focus is on proving rather than computing. Second, the subject matter here (logic and set theory) are the first pass at what's called foundations of mathematics (which is a robust subfield in its own right).

Most of what we'll learn here, you'll use in mathematics courses throughout the remainder of your undergraduate and graduate career. Some of it you'll eventually outgrow, though which parts depend on what path through mathematics you take and where you end up.

What is this site and how do I use it?

Most of the text on this site started as a set of notes I wrote, assisted by the NCSU Libraries' Alt-Textbook Project. But I decided that a book format was a little less pretty and a little less navigable than a site like this.

Most everything is hyperlinked, so follow your nose. You can always find your way back to this page, which serves as a table of contents. "Chapters" are also tagged, so you can navigate that way if you prefer.


  1. What are we doing here?
  2. Logic
  3. The Natural Numbers and Induction
  4. Sets
  5. Relations
  6. Functions
  7. Cardinality

Typographical Conventions

  • Where possible, I've used red-printed text to indicate a sentence we want to consider as an object in its own right. For example, The man bit the dog is a simple declarative sentence.
  • Blue-printed text is for running commentary within a proof. (Who needs running commentary? Just shut up and get on with it already!)
  • The words Definition, Theorem, Claim, and Proof in boldface denote the start of a block of text and are usually indented along with that entire block. This is more or less the standard convention for mathematical texts.
  • All math symbols and diagrams are set using LaTeX via the QuickLaTeX plugin for WordPress.