On 18 Oct 2015, at 22:37, Brent Meeker wrote:
On 10/18/2015 1:53 AM, Bruno Marchal wrote:
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.
Exactly. Axiomatic inference only works to the degree you can be
sure the axioms apply.
No problem with this.
And the degree we can be "sure" the axioms apply (and that the
inference rules preserve the application) determines the choice on
axioms on which we can agree on what we are talking about.
So, about the natural numbers, all what is asked is something like do
you agree with the following axioms:
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
(+ do you agree that the modus ponens rule preserve the intuitive
meaning we want to refer about).
In that case clouds are excluded for example. But the existence of a
biggest number is not excluded, and you can add the axiom ExAy(x is >=
y). That formula is consistent with the axiom above, as you can see by
building (exercise) a mathematical structure satisfying all axioms
above + ExAy(x is >= y).
(>= means greater or equal).
Bruno
Brent
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