Re: Tegmark's TOE & Cantor's Absolute Infinity

From: Brent Meeker <meekerdb.domain.name.hidden>
Date: Sat, 21 Sep 2002 23:50:20 -0700

On 21-Sep-02, Wei Dai wrote:
> 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.

> 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.

I was not aware that 2nd-order logic precluded independent
propositions. Is this true whatever the axioms and rules of
inference?

Brent Meeker
"If a cluttered desk is the sign of a
 cluttered mind, what's an empty desk a sign of?"
         --- Kenneth Arrow
Received on Sun Sep 22 2002 - 00:05:39 PDT