On 21-Sep-02, Hal Finney wrote:
> However we know that, by Godel's theorem, any axiomatization
> of a mathematical structure of at least moderate complexity
> is in some sense incomplete. There are true theorems of that
> mathematical structure which cannot be proven by those
> axioms. This is true of the integers, although not of plane
> geometry as that is too simple.
> This suggests that the axiom system is not a true definition
> of the mathematical structure. There is more to the
> mathematical object than is captured by the axiom system. So
> if we stick to an interpretation of Tegmark's TOE as being
> based on mathematical objects, we have to say that formal
> axiom systems are not the same. Mathematical objects are more
> than their axioms.
I don't see how this follows. If you have a set of axioms, and
rules of inference, then (per Godel) there are undecidable
propositions. One of these may be added as an axiom and the
system will still be consistent. This will allow you to prove
more things about the mathematical structures. But you could
also add the negation of the proposition as an axiom and then
you prove different things. So until the axiom set is
augmented, the mathematical structures they imply don't exist.
"One way (of designing software) is to make it so simple that
there are obviously no deficiencies and the other way is to
it so complicated that there are no obvious deficiencies."
--- Tony Hoare