Dear Bruno,
Thank you very much for this explicit remark. It is very helpful
for my research. I have some comments and a question.
On 12/1/2011 11:34 AM, Bruno Marchal wrote:
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.
------
It seems that g is not necessarily specified by g(k), since k
assumes too much.
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.
Part of the Non-halting result... OK.
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.
But it still seems that there is something missing in this
conclusion about the Church thesis. It is that for computation all that
one needs to consider in a model is the N -> N and n /subset N -> N
maps/functions. The problem of time that I have complained about is part
of this problem that I see. It reminds me strongly of the problem of the
axiom of choice, where the existence of unmeasurable sets cannot be
excluded inducing such things as the Banach-Tarski paradox
<http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox>. The same
problem that occurs in the result you discuss above: there is a
function/object that cannot be exactly defined. How do we get around
this impasse? What if there is a way to sequester the pathological parts
of the function without having to define the function? (Something like
this is discussed here
http://ndpr.nd.edu/news/24157-on-preserving-essays-on-preservationism-and-paraconsistent-logic/)
We see a similar process in physics where the abstract spaces are
defined to be well behaved ab initio. What if this "good behavior" is
emergent, like what happens when we couple a large number of chaotic
systems to each other. The entrainment process between them forces them
to behave as if they are nice and linear! (L. M. Pecora and T. L.
Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. *64*,
821 (1990). )
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.
Immune from diagonalization but not from forcing
<http://en.wikipedia.org/wiki/Forcing_%28mathematics%29>. It seems to
me, and I admit that this is just an intuition, that we are constraining
the notion of computation to too small of a box to realize its full
potential. Why is the notion of time so scary for computer scientists
and logicians?
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*.
OK, but I want to go further. We need for logics the equivalent of
a principle of variation
<http://en.wikipedia.org/wiki/Variational_principle> so that we do not
have to resort to the ansatz of Platonic walls to explain where
universal computation is implemented. You seem to know a something about
this idea given your previous reference to amenable groups
<http://en.wikipedia.org/wiki/Amenable_group> , but it is as if you are
afraid to look over the edge and stare into the abyss. I think that the
Tennenbaum theorem <http://en.wikipedia.org/wiki/Tennenbaum%27s_theorem>
offers us a clue. It states that "no countable </wiki/Countable_set>
nonstandard model </wiki/Non-standard_model_of_arithmetic> of Peano
arithmetic </wiki/Peano_axioms> (PA) can be recursive
</wiki/Recursion_theory>". We need to retain the property of
countability and recursivity for computation but need the spectrum of
possibilities that non-standard models allows to act as a range of
variation. *Is there a version or model of arithmetic that would allow
this but retain the expressiveness of PA?*
You might search on "diagonalization" in the archive of this list to
find more on this.
Bruno
Onward!
Stephen
PS.
Some related papers:
http://arxiv.org/abs/1110.5456 "Meaning in Classical Mathematics: Is it
at Odds with Intuitionism?" A good discussion with examples of consequences.
http://arxiv.org/abs/1109.5886 "Non-Archimedean Whitney-stratifications"
Another good example of a similar idea.
http://arxiv.org/abs/1108.5062 "A Non-Standard Semantics for Kahn
Networks in Continuous Time" The picture that this paper paints is very
close to my intuition.
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