The n-Category Café

Axiomatic Set Theory 8: Well Ordered Sets

Posted by Tom Leinster

Previously: Part 7.

By this point in the course, we’ve finished the delicate work of assembling all the customary set-theoretic apparatus from the axioms, and we’ve started proving major theorems. This week, we met well ordered sets and developed all the theory we’ll need. The main results were:

• every family of well ordered sets has a least member — informally, “the well ordered sets are well ordered”;

• the Hartogs theorem: for every set $X$, there’s some well ordered set that doesn’t admit an injection into $X$;

• a very close relative of Zorn’s lemma that, nevertheless, doesn’t require the axiom of choice: for every ordered set $X$ and function $\varphi$ assigning an upper bound to each chain in $X$, there’s some chain $C$ such that $\varphi(C) \in C$.

I also included an optional chatty section on the use of transfinite recursion to strip the isolated points from any subset of $\mathbb{R}$. Am I right in understanding that this is what got Cantor started on set theory in the first place?