Re: Axiomatic ontology
John,
Before you wanted me to argue about the meaning of "subjective" and
"existence", and now you want me to argue about where the
contradiction is?
Perhaps you will at least accept there is a paradox?
The issue is whether Russell's Paradox presents problems for
formulations of meaning, mathematical or other. If you want to resolve
the paradox by dropping your "axiom #1" or other, that's fine. As you
say there are a number of ways to do it. Too many. The point is that
(directly or indirectly) as a result of Russell's Paradox it was found
necessary to replace Naive Set Theory with Axiomatic Set Theory, and
the essence of Axiomatic Set Theory is that it has irreducibly
multiple formulations.
The multiplicity of alternate axiomatic formulations with which it was
found necessary to limit set theory is a problem for the philosophy of
mathematics. It suggests that at a fundamental level maths is
indeterminate.
Category Theory tries to deal with that indeterminacy by embracing it,
and establishing a philosophical basis for mathematics using not
identities, but symmetries.
If you want to assert this has no consequences for general knowledge
representation so be it, it is possible to reason yourself almost
anywhere, but I think the discovery of fundamental indeterminacy in
mathematics deserves recognition as more than a mere "interesting
historical note".
On Jan 28, 2008 10:12 PM, John F. Sowa <sowa@bestweb.net> wrote:
> Rob,
>
> There is no contradiction there:
>
> RF> I thought Russell's paradox might provide a hint where any
> > complete conception could tie itself in knots. A set that
> > both is and is not a member of itself. Maybe "the universe"
> > is that way.
>
> There is no contradiction whatever in the assumption that a set
> could be a member of itself. The contradiction arises between
> two other assumptions:
>
> 1. The axiom that for every monadic predicate P(x), there
> exists a set of all the x's for which P is true.
>
> 2. The construction of a special set:
>
> {x | ~memberOf(x,x) }
>
> In English, the set of all x's such that x is not a member of x.
>
> There are two straightforward methods for getting rid of the
> contradiction:
>
> 1. The simplest is to drop axiom #1. That means that there may be
> predicates such as P(x) for which there is no corresponding set.
>
> 2. The other is to define a method of constructing sets that makes
> it impossible to form a set that contradicts axiom #1.
>
> Both of these methods are used in various theories.
>
> Finally, I have no idea what you mean by "that way".
Self-contradictory.
> One thing
> that is certain is that anything that exists can be described by
> a list (possibly a very long list) of simple statements that do
> not contain any negations. Since it is impossible to have a
> contradiction without having at least one negation, everything
> that exists must be describable by a list of consistent statements.
>
> Therefore, nothing that exists can be self contradictory.
A conclusion Descartes himself would be proud of.
This harks back to the recent discussion on the Corpora list, in which
you participated, where the conclusion drawn from the same evidence
was exactly the opposite. More specifically that while a particular
observation of "the universe" cannot be contradictory in itself, it
can contradict any given theory:
http://listserv.linguistlist.org/cgi-bin/wa?A2=ind0801&L=corpora&D=1&F=&S=&P=3750
But I perceive once again it is just the two of us arguing. Perhaps we
should wait to see if anyone else is interested before we waste any
more space with these fine distinctions.
-Rob