Number
What has to exist between the marks?
Nobody builds the number systems, and so nobody notices that they were built. This subject constructs each of them from the one before, and at every stage the construction is forced by a specific equation that had no solution.
Naturals & integers
Where do the counting numbers come from?
Counting, axiomatised
Five axioms produce the natural numbers, and the fifth one is induction. The film shows what goes wrong if you leave any of them out.
Subtraction demands new numbers
Three minus five has no answer in the naturals, so either the question is banned or the system grows. Watching the extension get built makes negative numbers stop being a convention.
Rationals
Is division enough to fill the line?
Every gap you can name, and one you cannot
The rationals are dense — between any two there is another — which feels like it should be enough. The film shows that density and completeness are entirely different things.
The diagonal that broke a school
√2 in four lines, and then the harder question the four lines leave behind: if it is not a ratio, what is it?
The real numbers
Can the holes be filled in more than one way?
Two roads to the continuum
The reals built twice — as Dedekind cuts and as Cauchy sequences — and then a proof that the two constructions are the same object in every respect that matters.
Why the line has no gaps
Completeness is usually an axiom you accept. Here it arrives as the answer to a bounded set with no visible least upper bound.
Number, 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 III ↓