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; 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?.
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.
So to find a math technique powerful enough to decide Godel
sentences , we look for a reasoning technique which is non-axiomatic,
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
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
(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.
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Received on Fri Aug 28 2009 - 22:15:32 PDT