On Tue, Dec 20, 2005 at 02:34:16PM -0500, John Sowa wrote:
> > 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 use something else. (01)
Right. In fact, you can even avoid ontological commitment to the empty
set consistently with ZF set theory. For all the axioms of ZF say is
that there is some (unique) *thing* that has no members. Since ZF is a
theory of *sets*, it is natural (for mathematicians, at any rate) to
consider the thing with no members to be a set along side everything
else that exists according to the theory. But there is nothing in the
axioms that forces this assumption upon you. So if the idea of a set
with no members leads to unpleasant episodes of cognitive dissonance --
and it has to be admitted that the notion does not really sit
comfortably with the appealing intuitions that ground ZF -- you can,
with perfect mathematical propriety, return to psychological equilibrium
by therapeutically identifying the empty set with any (intuitive)
non-set you please: John Sowa, World War II, God (or maybe better, the
Devil! ;-), the number 0, or whatever.** (02)
> Actually, sets form a Boolean algebra with the empty set corresponding
> to falsehood. (03)
Eh? John, I know you know that there's no complement operation in set
theory, at least, not in ZF. That said, in the class theories
underlying, e.g., OWL and SUMO, and I'm thinking perhaps your own
account, it does appear that classes form a Boolean algebra; so I'm
guessing that's what lies behind your remark here. (04)
Chris Menzel (05)
**Completely non-ONTAC-related postscript below... (06)
**COMPLETELY NON-ONTACRELATED POSTSCRIPT -- DO NOT RESPOND TO THE LIST! :-)
The great philosopher of science Bas van Fraassen once noted significant
similarities between God and the empty set -- notably, both are
uncreated, necessary beings having no parts, etc -- so perhaps that is a
good reason for letting Providence play the august role of the empty set
in the ontology of set theory. Hmm. Come to think of it, have we got a
new version of the ontological argument here? The axioms of ZF are
known a priori; it is an a priori consequence of the axioms of ZF that
there is a (unique) empty set; so it is known priori that the empty set
exists, "and this everyone understands to be God" (cf. Thomas Aquinas,
ST I, Qu2, Art3 ;-) (07)
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