On Sat, Sep 21, 2002 at 10:26:45PM -0700, Brent Meeker wrote:
> 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.

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Are you aware of the distinction between first-order logic and
second-order logic? Unlike first-order theories, second-order theories can
be categorical, which means all models of the theory are isomorphic. In a
categorical theory, there can be undecidable propositions, but there are
no semantically independent propositions. That is, all propositions are
either true or false, even if for some of them you can't know which is the
case if you can compute only recursive functions. If you add a false
proposition as an axiom to such a theory, then the theory no longer has a
model (it's no longer *about* anything), but you might not be able to tell
when that's the case.
Back to what Hal wrote:
> 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.
This needs to be qualified a bit. Mathematical objects are more than the
formal (i.e., deductive) consequences of their axioms. However, an axiom
system can capture a mathematical structure, if it's second-order, and you
consider the semantic consequences of the axioms instead of just the
deductive consequences.