>>13
Well, we don't really know what the finite or infinite really 'are'. It doesn't make Church-turing thesis false though. Pick any arbitrary system and if it's rich enough, you end up with the basic truths of arithmetic appearing in the system.
I'm not even asking you about any system at all: if you are to completly randomize the formal systems you pick, those that end up not being trivially inconsistent, and are not too trivial, will have computation appearing within them in one form or another and have the same limitations and results apply to them. It's like a contagious truth that you can't get rid of no matter what you'll try.