Daniel, thanks for your very detailed reply. I shouldn't have said 'Self-evident Truths'. The only self evident truth that I know is 'all men are created equal', and that's not true. So I was using it to mean 'assertions'. Which was a stupid thing to do. I am ever tempted to rhetorical slyness over clarity.

I don't find myself forced to believe in uncomputable things in the same way that I find myself forced to accept the existence of countable infinite sets . I've always been really sceptical about the 'real numbers'.

People win the lottery every day!

No it wouldn't. A universe of finite size would work. For example, Conway's game of life on a torus.

Yes, that would be interesting, especially if the torus were very large. I retract that sentence. I think what's bugging me is that a TM with a finite tape seems more complicated than a TM with an infinite tape. If the finiteness is large, then there's more information in the size of the constant than there is in the machine. And in fact I believe that some very simple systems are turing equivalent.

I think my argument is something like 'If you let countability in at all, then you've probably got everything. Over and over again.'