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Re: [ontolog-forum] Universal Basic Semantic Structures

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Chris Menzel <chris.menzel@xxxxxxxxx>
Date: Sun, 30 Sep 2012 00:50:08 -0500
Message-id: <CAO_JD6PFX-XGG1PjoqPQWfWQFe8yzGh4tSz0BXMPCg4tWw36zg@xxxxxxxxxxxxxx>
On Sat, Sep 29, 2012 at 8:51 PM, John F Sowa <sowa@xxxxxxxxxxx> wrote:
Chris and Avril,

> virtually every text on set theory presents the axioms with just a
> single binary predicate "∈" for membership. The subset relation is
> always defined; using "isin" instead of "∈":

I agree.  But the theory of Boolean collections is much simpler
and has much less structure than Cantor's set theory.  If you
consider the lattice of all possible first-order theories, then
both set theory and many versions of mereology can be considered
specializations of the theory of Boolean collections.

I'm not sure how this addresses the claim you made to which I objected — that set theory has two operations (better, relations) and mereology one, which I took to imply that set theory has two primitive relations. That's just not faithful to the way set theories are almost always developed.

With Cantor's elementOf operator, you have an enormous amount of
power for building up arithmetic, the hierarchies of infinity, etc.

The membership relation has no power at all in itself. The power in set theory comes from the axioms you associate with the membership relation. For example, without the axiom of infinite you don't get an infinite set; without the Powerset axiom you don't get higher infinities.
Lesniewski considered that to be too much power. He claimed that
it is clearer and simpler to start with mereology, which cannot,
by itself, be used to construct arithmetic.

And lacking appropriate axioms, you can't construct it in set theory either.
> If we consider only sets with rank 1, that is, no inner sets, then
> mereology and set theory become identical.

> That's a pretty misleading way to put it as the membership relation
> in set theory does not correspond to anything in mereology, so there
> will be many truths of set theory that don't correspond to anything
> in mereology, e.g., "a ∈ {a}", "a ≠ {a}", etc.

I agree with Chris.  But I would add that some programming
languages that support things they call sets define "a = {a}".

And in some set theories, too. I was a bit careless in using "set theory" above to mean Zermelo-Fraenkel set theory, which includes the axiom of foundation that rules out "a = {a}". Without foundation, you can consistently assert the existence of such "non-well-founded" sets. Indeed, there are well-developed set theories with anti-foundation axioms that entail the existence of a wide variety of non-well-founded sets.


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