The nature of number

[Categories+Sheaves]

Yes, you don't have to ask. The pupil will ask questions, but even if these seem silly, I am paying attention.

PS
G.E.B. Gödel, Escher, Bach; no I hadn't just found it my sort of thing thank you.

And G.E.B. seemed right up your alley...
1. Napoleon put up a really cute video on the diagonalization proof in the 'existence and infinity thread', so feel free to use that video (if the jumble I dumped here is too... jumbly) if you want to keep kicking Cantor around.
2. That Coxeter book has a wonderfully short proof that there are only 5 platonic solids:

A convex polyhedron is said to be regular if its faces are regular and equal, while its vertices are all surrounded alike... ...If its faces are {p}'s [that is, p-gons], [with q such faces] surrounding each vertex, the polyhedron is denoted by {p,q}
The possible values for p and q may be enumerated as follows. The solid angle at a vertex has q face-angles, each (1-2/p)*pi... A familiar theorem states that these q angles must total less than 2*pi. Hence 1-2/p < 2/q

Which we can rewrite as (p-2)(q-2)<4. Because p>=3, (there are no 1-gons or 2-gons) and q>=3 (q=1 just gives you a p-gon in 3-space, and q=2 isn't much better) the only admissible values are {3,3}. {3,4}, {4,3}, {3,5} and {5,3} (tetrahedron, octahedron, cube, icosahedron, and dodecahedron respectively)

Man, geometry... I feel so nostalgic...