On 02 Dec 2011, at 05:16, Stephen P. King wrote:
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:
On 01 Dec 2011, at 13:16, Stephen P. King wrote:
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
But if L is universal, the g should be in the list. So g = f_k, for
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
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
Part of the Non-halting result... OK.
It is easier than the non halting.
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
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.
Time is not relevant here. We need only the natural numbers.
It reminds me strongly of the problem of the axiom of choice,
The axiom of choice is not relevant here. This can be made precise: ZF
and ZFC proves the same arithmetical truth. I don't use set theory at
where the existence of unmeasurable sets cannot be excluded inducing
such things as the Banach-Tarski paradox. The same problem that
occurs in the result you discuss above: there is a function/object
that cannot be exactly defined.
Which one? You confuse computation and definition.
How do we get around this impasse?
I don't see an impasse. I explain old and basic uncontroversial notions.
What if there is a way to sequester the pathological parts of the
function without having to define the function?
This is too vague.
(Something like this is discussed here
Nothing there is relevant with the issue I was talking about.
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.
? (forcing is a technic in set theory or in model theory. It does not
concern us here). Modal logic, which have a role in computability (but
not here) generalizes forcing in some way.
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?
It is not scary, and it plays some role in some chapter. but we
generalize it by using any recursive function on the inputs and
programs. It is hard to make sense of your comment. You introduce
difficulties where they don't exist, I think.
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 so that we do not have to resort to the
ansatz of Platonic walls to explain where universal computation is
You seem to know a something about this idea given your previous
reference to amenable groups ,
OK, but that has nothing to do with Church thesis.
but it is as if you are afraid to look over the edge and stare into
the abyss. I think that the Tennenbaum theorem offers us a clue. It
states that "no countable nonstandard model of Peano arithmetic (PA)
can be recursive".
I think that you mix many things. We were not in Model 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?
PA. The non standard models are models of PA. I have no clue what you
are trying to say. You should not mention more technical stuff when
simpler one are presented to you, unless you can make a specific point.
You might search on "diagonalization" in the archive of this list
to find more on this.
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
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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