Models of ZFC

Model theory is really meta, so you will have to pay attention here. Roughly, a ``model of $latex {\mathsf{ZFC}}&fg=000000$'' is a set with a binary relation that satisfies the $latex {\mathsf{ZFC}}&fg=000000$ axioms, just as a group is a set with a binary operation that satisfies the group axioms. Unfortunately, unlike with groups, it is very… Continue reading Models of ZFC


(Standard post on cardinals, as a prerequisite for forthcoming theory model post.) An ordinal measures a total ordering. However, it does not do a fantastic job at measuring size. For example, there is a bijection between the elements of $latex {\omega}&fg=000000$ and $latex {\omega+1}&fg=000000$: $latex \displaystyle \begin{array}{rccccccc} \omega+1 = & \{ & \omega & 0… Continue reading Cardinals

Cauchy’s Functional Equation and Zorn’s Lemma

This is a draft of an appendix chapter for my Napkin project. In the world of olympiad math, there's a famous functional equation that goes as follows: $latex \displaystyle f : {\mathbb R} \rightarrow {\mathbb R} \qquad f(x+y) = f(x) + f(y). &fg=000000$ Everyone knows what its solutions are! There's an obvious family of solutions… Continue reading Cauchy’s Functional Equation and Zorn’s Lemma

Set Theory, Part 2: Constructing the Ordinals

This is a continuation of my earlier set theory post. In this post, I'll describe the next three axioms of ZF and construct the ordinal numbers. 1. The Previous Axioms As review, here are the natural descriptions of the five axioms we covered in the previous post. Axiom 1 (Extensionality) Two sets are equal if… Continue reading Set Theory, Part 2: Constructing the Ordinals

Set Theory, Part 1: An Intro to ZFC

Back in high school, I sometimes wondered what all the big deal about ZFC and the Axiom of Choice was, but I never really understood what I read in the corresponding Wikipedia page. In this post, I'll try to explain what axiomatic set theory is trying to do in a way accessible to those with… Continue reading Set Theory, Part 1: An Intro to ZFC