Hello Stephen,
On 01 Dec 2011, at 13:16, Stephen P. King wrote:
Hi Bruno,
Could you eleborate a bit about how "Computability is the only
notion immune to Cantor's diagonalization"?
Cantor proved that infinity is sensible to diagonalization. Given an
infinite set, you can find a bigger set by diagonalization.
Gödel proved that any effective provability system (theory) is
incomplete. Given a theory rich enough to talk on numbers, you can
build a richer provability system by diagonalization.
Tarski proved that any system of definition will lack the expressive
power to define some notion, notably its truth notion. Again, this
follows from diagonalization.
Diagonalization is a sort of transcendental operation in mathematics.
If you have about anything pretending to be a universal notion in the
domain, you can diagonalize against it.
That is why Stephen C. Kleene was skeptical when Church told him that
his lambda-calculus defined a universal notion of computability.
At first, it looks like computability is sensible, NOT immune, to
diagonalization. Imagine that there is a universal language for
computability L. Consider all the computable function defined on N and
with value in N, enumerated from their code in that language:
f_0, f_1, f_2, f_3, f_4, etc.
Let g be defined on n by g(n) = f_n(n) + 1 (g is said to be defined
by diagonalization)
Each f_i are computable function from N to N, so f_n(n) is well
defined, and "+ 1" is obviously computable, so g seems to be computable.
But if L is universal, the g should be in the list. So g = f_k, for
some k.
But then
g(n) = f_k(n), given that g = f_k
In particular, with n = k
g(k) = f_k(k)
But by definition of g, we know that
g(k) = f_k(k) + 1
By Leibniz identity rule
f_k(k) = f_k(k) + 1
And by using the fact that f_k is a function from N to N, we know that
f_k(n) is a number for any n, so f_k(k) is a number, and this number
can be subtracted at the left and right hand side of the equality
above, so
0 = 1.
Church's pretension that L is universal seems to be refuted.
But that proof is wrong. Do you see why?
Take the time to find the mistake by yourself before reading the
solution below.
------
What is wrong is that the language L might define more than the
computable functions from N to N, but can also define functions from
from subset of N to N. In that case, the reasoning just shows that
g(k) = f_k(k) is not defined.
This shows that an universal machine can crash (run in a loop without
ever giving an output), and this necessarily so to be universal.
Worse, there will be no effective means (and thus no complete theory
of universal machine or language) to decide if some f_i is defined on
N or a proper subset of N. If that was the case, we would be able to
filter out the functions from a proper subset of N to N from the
functions from N to N, and then the diagonalization above would lead
to 0 = 1.
Let us call a function partial if that function if either from N to N,
or from a subset of N to N, and a function is total if it is defined
on N. The reasoning above shows that the set of total functions is not
immune to diagonalization. But the superset of the partial functions
is, and that is a deep strong argument for Church thesis.
As far as I know, in mathematics, only computability, and
computability related notion (like their relativization on oracles)
are immune for the diagonalization. Gödel, in his Princeton lecture
called that immunity a miracle.
Everything that I explain depends on this. UDA works thanks to the
universality of the UD, which relies on that diagonalization closure,
and AUDA uses the fact that self-reference is build on the
diagonalization procedure. Unfortunately, or fortunately, all
diagonalizations are hidden in Solovay's theorem on the arithmetical
completeness of G and G*.
You might search on "diagonalization" in the archive of this list to
find more on this.
Bruno
On 12/1/2011 4:21 AM, Bruno Marchal wrote:
I have an original thesis on that. Not only Babbage discovered the
universal machine, but he discovered the equivalent of Church
thesis, which is the key notion to understand that the universal
machine is truly universal.
Universality = universality with respect to computing, or any
digital processes. Computing does not need to be restricted on
numbers, but it happens that the natural numbers together with
addition and multiplication is Turing universal, so that the
numbers' restriction is an apparent restriction.
Computability is the only notion immune to Cantor's
diagonalization, and that gives a conceptual very deep argument for
Church thesis.
http://iridia.ulb.ac.be/~marchal/
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