Set theory is the standard foundational discipline for mathematics, with sets being used to define number, functions, structures like groups and topological spaces, etc.
In this discussion, we will review the axioms and basic properties of:
- Zermelo-Frenkel - the standard axiomization of the theory of sets
- NBG - ZF extended to include "very large" classes
- Morse-Kelley - An NBG-like system with a slightly stronger comprehension axiom
- NF - an unorthodox theory that disproves choice, proves infinity, is non-wellfounded, and generally nuts
- ML - A strengthening of NF with proper classes, with which NF is equiconsistent
Discussion will begin with a brief discussion of set theory's origins, aims, and basic concepts, followed by a quick recap of Russel's and Cantor's paradoxes. After that we will move on to discuss specific axiom systems (primarily ZF and NF, and their variants as time permits) and how they avoid the paradoxes and provide the sets important to mathematics (natural numbers, ordered pairs, and relations). Where possible, fundamental differences in how the theories handle their subject matter will be discussed.
If we manage to run through the above, I hope to include some brief comments on large cardinal axioms, and certain oddities in the relationship between NF and ML.
Note: I'm not going to lie, I totally heart NF and will probably be a little heavier in coverage on that system.