On 1/21/2014 8:13 AM, Jason Resch wrote:
Why would you want to do that? It seems like an unnecessary extra axiom that doesn't
have any purpose or utility.
It prevents the paradoxes of undeciability, Cantor diagonalization, and it corresponds
more directly with how we actually use arithmetic.
I'm not sure it helps. What you may gain from avoiding paradoxes makes many of our
accepted proofs false. E.g. Euclids proof of infinite primes. Or Euler's identity. Most
of math would be ruined. A circle's circumference would not even be pi*diameter.
Would this biggest number be different for different beings in different universes? What
is it contingent on?
You're taking an Platonic view that there really is an arithmetic and whether there's a
biggest number is an empirical question. I'm saying it's an invention. We invented an
system in which you can always add 1 because that was convenient; you don't have to think
about whether you can or not. But if it leads to paradoxes or absurdities we should just
modify our invention keeping the good part and avoiding the paradoxes if we can. Peano's
arithmetic will still be there in Platonia and sqrt(2) will be irrational there. But the
diagonal of a unit square may depend on how we measure it or what it's made of.
Brent
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