On 29 Jun 2017, at 20:17, John Clark wrote:
On Thu, Jun 29, 2017 at 10:58 AM, Bruno Marchal <[email protected]>
wrote:
> you just said to Adrian Chira: "Assuming there is a largest
prime number leads to a logical contradiction, but assuming free
will exists is like assuming Klogknee exists."
Yes, gibberish in gibberish out.
> But Robinson Arithmetic, a weak but already sigma_1 complete
theory (and thus Turing universal) is consistent with "there is a
biggest prime number".
If Robinson says there is a largest prime number then Robinson
is an ass.
It just means that there is no proof that such larger prime number
don't exist in the theory RA: classical logic +
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
But there is such a proof in the theory
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
+
the infinitely many induction axioms:
(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x),
with F(x) being a formula in the arithmetical language (with "0, s, +,
* and the logical symbols),
RA is already Turing universal.
PA is Turing universal and Löbian, which means that she can prove p ->
Beweisbar('p') for all p sigma. (sigma means provably equivalent with
a formula having the shape "it exists x such that P(x)" and P
decidable/recursive). (church-turing universality is equivalent with
that proving ability).
> You were probably assuming Peano Arithmetic (which is
Robinson arithmetic + the induction axiom),
No I was not making that assumption, and neither was Euclid when
he showed 2500 years before Peano or Robinson were born that there
is no largest prime number.
Then you were talking of the intuitive arithmetical reality, that the
logicians "model" with the notion of standard model of (Peano)
Arithmetic. No problem, that is what all mathematicians does, but that
entails the two theories above. (and requires soronger theories to be
developed, like second order logic or first order set theory).
Euclid notariously use the induction scheme above, and was reasoning
intuitively in second-order logic (like everybody, consciously or
unconsciously).
> the here-and-now 1p experience
To hell with peepee!!
Eliminativism.
To hell the inconvenient data.
Bruno
John K Clark
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,
send an email to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
