On 15 Oct 2015, at 19:57, Brent Meeker wrote:
On 10/15/2015 12:11 AM, Bruno Marchal wrote:
If arithmetic is false, Church-Turing thesis makes no more sense.
?? It will make sense as an axiom in a certain branch of
mathematics.
Then it is no more the classical Church's thesis. It will be
something like intuitionist Church's thesis. That would again be
just a change of subject. It is like saying that 1 cloud + 1 cloud
= 1 cloud can refute 1+1=2. That is not good logic.
But it's a good example of the limitations of axiomatic inferences.
Why?
Cloud are not the intended notion when we talk about natural numbers.
It makes no sense to ask oneself of what is the successor of a cloud,
or to accept that for all clouds x e have s(x) ≠ 0.
The fact that some mathematical structure are not group is not a
limitation of the theory of group, nor of the axiomatic or semi-
axiomatic approach.
Bruno
BRent
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