On 29 Aug 2009, at 07:15, marc.geddes wrote:
>
>
>
> On Aug 29, 2:36 am, Bruno Marchal <marc....domain.name.hidden> wrote:
>
>>
>> Obviously (?, by Gödel) Arithmetic (arithmetical truth) is infinitely
>> larger that what you can prove in ZF theory.
>
> Godel’s theorem doesn’t mean that anything is *absolutely*
> undecidable;
OK.
Computability is absolute,
Provability is relative.
> it just means that not all truths can captured by
> *axiomatic* methods; but we can always use mathematical intuition (non
> axiomatic methods) to decide the truth of anything can't we?.
In principle. "No ignorabimus" as Hilbert said. Yet no machine or
formal systems can prove propositions too much complex relatively to
themselves, and there is a sense to say that some proposition are
undecidable in some absolute way, relative to themselves.
>
> http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
>
> "The TRUE but unprovable statement referred to by the theorem is often
> referred to as “the Gödel sentence” for the theory. "
>
> The sentence is unprovable within the system but TRUE. How do we know
> it is true? Mathematical intuition.
Not really. The process of finding out its own Gödel sentence is
mechanical. Machines can guess or infer their own consistency, for
example. In AUDA intuition appears with the modality having "& p" in
the definition (Bp & p, Bp & Dp & p).
Those can be related with Bergsonian time, intuitionistic logic,
Plotinus universal soul, and sensible matter.
>
> So to find a math technique powerful enough to decide Godel
> sentences ,
This already exists. The diagonilization is constructive. Gödel's
proof is constructive. That is what Penrose and Lucas are missing
(notably).
> we look for a reasoning technique which is non-axiomatic,
This is the case for the "& p" modalities. They are provably
necessarily non axiomatisable. They lead to the frst person, which,
solipstically, does separate truth and provability.
> by asking which math structures are related to which possible
> reasoning techniques. So we find;
>
> Bayesian reasoning (related to) functions/relations
> Analogical reasoning (related to) categories/sets
Those are easily axiomatized.
I see the relation "analogy-category", but sets and functions are
together, and not analogical imo.
I don't see at all the link between Bayes and functions/relations.
Actually, function/relations are the arrows in a category.
>
> Then we note that math structures can be arranged in a hierarchy, for
> instance natural numbers are lower down the hierarchy than real
> numbers, because real numbers are a higher-order infinity. So we can
> use this hierarchy to compare the relative power of epistemological
> techniques. Since:
>
> Functions/relations <<<< categories/sets
You may use some toposes (cartesian close category with a sub-object
classifier). Those are "mathematical" mathematicians. But assuming
comp, does not let you much choice on which topos you can choose. It
has to be related to the S4Grz epistemic logic (in the "ideal" case).
>
> (Functions are not as general/abstract as sets/categories; they are
> lower down in the math structure hierarchy)
>
> Bayes <<<<<< Analogical reasoning
>
> So, analogical reasoning must be the stronger technique. And indeed,
> since analogical reasoning is related to sets/categories (the highest
> order of math) it must the strongest technique. So we can determine
> the truth of Godel sentences by relying on mathematical intuition
> (which from the above must be equivalent to analogical reasoning).
> And nothing is really undecidable.
The truth of Gödel sentences are formally trivial. That is why
consistency is a nice cousin of consciousness. It can be shown to be
true easily by the system, and directly (in few steps), yet remains
unprovable by the system, not unlike the fact that we can be quasi
directly conscious, yet cannot prove it. Turing already exploited this
in his "system of logic based on ordinal" (his thesis with Church).
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Aug 29 2009 - 14:10:50 PDT