Here we go again.
Arithmetic is complex because it can not be captured in a small set of axioms.
Then the universe doesn't use that arithmetic in implementing physics, and it doesn't have the significance you claim it does. Like I said just above, it uses the kind of arithmetic that can be captured in a small set of axioms. And like I said in our many exchanges, it's true that modern computers can't answer every question about the natural numbers, but they don't need to. Neither does the universe.
Your favorite set of axioms fails to specify arithmetic in the same way that the statement "bricks are rectangular" fails to specify bricks; there are lots of other things that are also rectangular.
Yes, but you only need finite space to specify bricks well enough to get the desired functionality of bricks. Your argument would imply that bricks are infinitely complex because we don't have a finite procedure for determining where an arbitrary object "really" is a brick, because of e.g. all the borderline cases. ("Do the stones in a stone wall count as bricks?")
Then the universe doesn't use that arithmetic in implementing physics,
How do you know?
Like I said just above, it uses the kind of arithmetic that can be captured in a small set of axioms.
What kind of arithmetic is that? It would have to be a kind of arithmetic to which Godel's and Tarski's theorems don't apply, so it must be very different indeed from any arithmetic I've ever heard of.
A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).
ETA: It would seem that rationality quotes are no longer desired. After several days this thread stands voted into the negatives. Wolud whoever chose to to downvote this below 0 would care to express their disapproval of the regular quotes tradition more explicitly? Or perhaps they may like to browse around for some alternative posts that they could downvote instead of this one? Or, since we're in the business of quotation, they could "come on if they think they're hard enough!"