Topics without videos

Here are some video ideas. Please take them and make videos out of them.

Polite numbers

Make a video on the polite numbers. Show why no power of two can be written as the sum of two or more consecutive natural numbers. Do this by pairing off, as show here.

Proving the basic properties of arithmetic

A video series that proves all the basic properties of arithmetic. It would start with Peano arithmetic, then proceed by constructing the integers via a congruence relation on natural numbers, then construct the rationals via a congruence on the integers, and finish with one of the constructions of the reals, or at least describe the axiomatization of the reals.

Different types of proof by induction

Make several videos on different types of proof by induction. I haven't seen this done yet, but I haven't looked very hard either. Here's a link with a couple types of proof by induction.

Total functions

What is a total function?
A total function always terminates.
Proving a function non-total
A function can be proven non-total by finding a cycle when evaluating the function.
Explain how subtraction can be modified to become total. This is often called "proper subtraction."

Unions and intersections of countably infinite sets

This may be helpful.

Is the union of two countably infinite sets countably infinite?
Proving the union of countably many infinite sets is countably infinite
Proving the intersection of two countably infinite sets countably infinite
Proving the intersection of two countably infinite sets is countably infinite

Other