On Tue, Jan 17, 2012  Bruno Marchal <marc...@ulb.ac.be> wrote:

"You don't need to assume them. They already exists at the natural numbers'
> epistemological level."

Then in addition to the natural numbers the non computable numbers are
fundamental too.

"Just rememeber that when I use the term "number", I mean a natural number."

I have remembered and that's why I have a problem.

"Together with the laws of addition and multiplication, they are. The rest
> is numbers dreams (themselves recovered by number relations, definable with
> addition and multiplication"

No they are not. Turing proved in 1936 that you can NOT come arbitrarily
close to most real numbers using only the natural numbers and  addition and

"This comes from the fact that elementary arithmetic (on integers) is
> Turing universal."

But integers are very rare.

"You need to postulate the trigonometrical function to recover the natural
> numbers from the real."

And neither trigonometrical functions nor any other deterministic thing
will help you get  arbitrarily close to most real numbers, in fact such is
the nature of infinite sets that if you picked a point at random on the
real number line there is a 100% chance it will be non computable and a 0%
chance it will be a natural number.

"Someone doubting mechanism is not necessarily solipsist."

Why not?

 "very competent people can begin to believe that their are intelligent,
> and that's leads to stupidity."

It seems to me that both modesty and conceit leads to stupidity, if you're
intelligent and you believe you are intelligent then your belief is true.

 John K Clark

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