Re: Fwd: SUO Quo Vadis
Avril,
This is not a debate. It's a simple technical matter.
> I hope we can reach a consensus here.
You only need to reach a consensus when you are trying
to decide what to do. In my theories.htm paper, I was
giving a tutorial. The question of what to do is very
important, but that is not the topic of that paper.
> You could classify collection systems in many ways...
Of course.
> Empty set, as it is in ZF, causes misunderstandings and
> is conceptually incoherent.
I agree that it causes confusion among students. But it is
a perfectly reasonable mathematical assumption. You are free
to say that you don't like it, and you don't intend to use it.
But please note that I explicitly said that the infinite
lattice contains all the elegant theories and all the truly
ugly theories. You may have a different opinion about what
theories are elegant or ugly, but that opinion is irrelevant
to the question of where those theories are situated in the
lattice.
> I'd put mereology under the category of Boolean algebras,
> and all collection formalisms that have the memberOf
> operator under the category Set theories. But this is not
> a very important issue.
Actually, sets form a Boolean algebra with the empty set
corresponding to the false element. Those versions of mereology
that don't have a null or empty element don't have a natural
analog of F.
> I don't quite understand what you mean by a closure of
> axioms.
That is standard terminology in logic, and I defined it in
the theories.htm paper:
closure(S) = {p | p is provable from S]
The closure of any set S of axioms is nothing more nor less
than the set of all propositions that are provable from S.
> That sort of a belief is similar to the belief that a man
> washes himself....
So what? Some relations are reflexive and some aren't.
If you like the axiom, accept it. If you don't ignore it.
Your preferences have no affect on the lattice.
> D.M.Armstrong's Theory of Universals 1978 19.VI:
>
> "One hand washes another; both wash the rest of the body.
> Perhaps the trickiest sort of case is that where a person
> loves or hates himself. But even here genuine self-relation
> seems avoidable. If a man loves himself, then it is not
> that self-loving state which he loves, but other aspect
> of himself. It is possible that he should love the self-
> loving state, but this seems to demand a new, second-
> order, loving state which is distinct from the original one."
That's an interesting solution to the apparent paradox of
loving and hating being antonyms. You could also solve that
problem by saying that loving and hating are not true antonyms.
In any case, that's irrelevant to the structure of the lattice.
> BOT describes a contradiction, a round square.
Yes. It includes all possible contradictions. So what?
> ... but in that case [a plan for something that doesn't yet
> exist] the category would describe something that exists inside
> the brains of a human being. And if that person dies or forgets
> what the category describes, the category would be like a circle
> in sand in some deserted island.
So what? What's wrong anything on a deserted island? Unlike
Bishop Berkeley, I believe that a tree that falls when nobody
is around to hear it still makes exactly the same vibration
in the air.
> BOT is very problematic: you cannot divide by 0.
So what? The problem only exists if you try to do that.
The solution is not to do that.
John