@1roomstudio – I think where his incompleteness theorem does come in, is that we could never prove that statement about the nature of the infinitude of it, because in order to do that, we’d need to get to the end.
But from within the confines of the system, where it doesn’t even make sense to talk about outside of the system; we can’t.
So, in a sense, there is no meaningful circumstance in which a violation of the theorem can apply.
“It’s a cheat” is my point…by attempting to avoid representing “unimaginably large or small numbers” the fallacy of infinity continues to confuse a (potentially) better representation of reality.
I believe Godel would agree… but this is where I admit to getting bogged down… until I remind myself “it’s not real … is just a useful concept.”
Then I point to Sagittarius 
in the night sky and thereby orient myself to the center of the galaxy and declare “that is the best this mere mortal can do… for now…”
Indeed, though reluctantly. It’s interesting, before, he believed in the perfection of mathematics, but couldn’t help but arrive at the conclusion of its ultimate inconsistency.
This is where the space of constructible mathematics has formed – I wonder what he’d think of the computationalist models of physics.
Isn’t it so.
It’ll be interesting further to see what the super advanced generally intelligent machines of the future think about the world. I wonder what philosophical conclusions they’d arrive at acting as perfectly rational agents with intelligence far exceeding our own.
Soon enough even the like’s of Von Neumann will be humbled by the mere smartphone of the future. Or, should I say, supergeniusphone.
No… wait… the iQPhone+ 
I’ve enjoyed the discussion! Thanks for chiming in and challenging my conceptions of the world. 