Patrick Stevens

Ilia Zaichuk Thanks for the edit! I made a couple of linguistic changes, and made the "uniqueness of $\varnothing$" a bit less compact.

Ilia Zaichuk: I've made the appropriate changes to the markup to make text display in MathJax (which is the LaTeX-syntax markup language used for maths on this site and on Stack Exchange). However, I think it's a bug in Arbital (which I've just pointed out to the developers) that it's not rendering correctly. (EDIT: I've altered the markup into a form that works around the bug. It just makes the markup look a bit less nice.)

In general, you can use \text{text here}; if you want to put maths inline with the text here, you can use dollar signs:

 \text{Heinz 

This is not universally agreed-upon, but I use "$A$ decides whether or not $B$ holds" to mean "$A$ outputs $1$ if $B$ holds, and outputs $0$ otherwise".

If I said "$A$ decides if $B$ holds", I would consider that ambiguous: it might mean "$A$ outputs $1$ if $B$ holds" without the requirement on $A$'s behaviour if $B$ doesn't hold.

Looks good to me!

The non-existence of a total order on $C$ is fun and interesting, I think, and also not very difficult. An excellent exercise in proof by contradiction.

I think the answer is no. Indeed, there are uncountably many $S$, but only countably many machines which can access oracles.