Set Theory
How big can infinity get?
Infinity was a single word until someone asked whether two infinite collections could be compared. They can, the answer is not what anyone expected, and the machinery built to handle it turned out to be the foundation everything else sits on.
Sets
Can a collection contain anything at all?
The collection of everything, and why it cannot exist
Naive set theory is one sentence long and takes about four lines to destroy. Russell's paradox is where axioms stop being bureaucracy and start being necessary.
Operations, and what a function really is
A function is usually described as a rule, which is a description rather than a definition. Rebuilding it as a set of pairs costs nothing and settles several arguments in advance.
Cardinals
Are some infinities larger than others?
Counting without numbers
Pairing off is more primitive than counting, and it is the only tool that survives contact with infinite sets. The consequences are immediate and unwelcome.
The diagonal argument
One page of Cantor's shows the reals cannot be listed. The film makes the list and breaks it; the document gives the argument in three forms and shows they are the same.
Ordinals
What comes after all the counting numbers?
Past infinity, and then keep going
Ordinals are what you get by insisting that every collection of steps has a next step. The film builds the first few transfinite ones by hand.
The axiom of choice
Is it obvious, or is it the strangest thing in mathematics?
Set Theory, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume II ↓