On 26 Feb 2013, at 21:40, meekerdb wrote:
On 2/26/2013 1:24 AM, Bruno Marchal wrote:
How did number arise? We don't know that, but we can show that if
we don't assume them, or equivalent (basically anything Turing
Universal), then we cannot derive them.
I'm not sure how you mean that?
I meant that you cannot build a theory, simpler than arithmetic in
appearance, from which you can derive the existence of the numbers.
All theories which want talk about the numbers have to be turing
universal.
So I meant this in the concrete sense that if you write your axioms,
and want talk about numbers, you need to postulate them, or
equivalent. You can derive the numbers from the equational theory:
Kxy = x
Sxyz = xz(yz)
+ few equality rules,
But that theory is already Turing universal, and assume as much the
number than elementary arithmetic.
We know that we experience individual objects and so we can count
them by putting them in one-to-one relation with fingers or notches
or marks. So what are you calling an "assumption" in this?
A theory is supposed to abstract from the experiences. The experiences
motivates the theory, but does not justify it logically.
Bruno
Brent
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