Re: Re :The empty set

On Jan 16, 9:32 am, G. Frege <nomail@invalid> wrote:
On Wed, 16 Jan 2008 11:10:35 GMT, Aatu Koskensilta

<aatu.koskensi...@xxxxxxxxx> wrote:

That is to say, there are other set theories, for example Cantor's
original Mengenlehre ("set theory"), with a universal set.

Don't be too modest. This "Mengenlehre" should properly be credited to you.

Especially since it is well known that _Cantor_ didn't consider the
"totality of everything thinkable" to be a set (but an "inconsistent
multiplicity").

F.

--

E-mail: info<at>simple-line<dot>de

I could see how he would have considered it inconsistent, as for
example the universe is a counterexample to the powerset result, yet
there is as well the "domain principle" that each element of the
domain exists as a collection, uniquely defined by its elements (i.e.
purely a set). Cantor had a universe in his set theory, where every
thing is a set and everything is thus a set. Then, Cantor's
"paradox": that a universal set would be its own powerset, and in
terms of a subset (regular sets, the sets of ZF) Russell's: that the
set of all sets that don't contain themselves would contain itself,
led to the axiomatization of well-foundedness/regularity by Zermelo,
which has seen some technical reissuances.

Some people argue that Cantor's later misgivings about his development
of set theory arise from a struggle with the continuum hypothesis,
which is perhaps expressed in correspondence or colloquially, I wonder
if as well "Cantor's paradox", in light of his assurance that the
universe does exist, led him to rethink the fundamentals of his
argument.

I think that Cantor's acceptance of the "domain principle" is
acknowledgment of a universe, in his theory. Ready contradictions are
thus exhibited, where infinite sets are considered to be regular, yet
the inexistence of the universe and thus everything in it is absurdly
trivial, reinforcing the null axiom theory.

That's a good point, it illustrates that the Mengenlehre has some
aspects that are today considered obvious inconsistencies. Consider
as above "union of sets is a set" vis-a-vis "union of sets is not a
set."

I encourage you to consider that perhaps in the infinite, that
infinite collections as sets are plainly not regular/wellfounded,
instead somehow upon an irregular/nonwellfounded substrate.

Ross

--
Finlayson Consulting
.

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