Re: Uncountable sets

On Jul 22, 9:22 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
On Jul 22, 3:15 pm, The World Wide Wade <aderamey.a...@xxxxxxxxxxx>
wrote:

In article <1185079000.688698.217...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,

Proginoskes <CCHeck...@xxxxxxxxx> wrote:
On Jul 21, 10:18 pm, S J Kissane <skiss...@xxxxxxxxx> wrote:
Hi

My intuition says that uncountable sets do not exist. I have no
problem with the existence of aleph-null; I just have a problem with
the existence of anything bigger. Is there a name for this intuition?

Yes, around here it's called "being an anti-Cantor crank".

Many "anti-Cantor" mathematicians - and certainly they are not all
cranks - allow for the existence of uncountable sets.

I think you're not referring to the notion that Cantors's naive set
theory (with a universe), the Mengenlehre, is not ZF(C), the Zermelo-
Fraenkel set-theoretic (by default in definition) theory of sets,
where the axioms of ZF variously amended with choice/well-ordering
(the C) describe a collection of all things that might be sets.

A primary difference between naive set theory, which might by itself
include fanciful notions of sets containing themselves (non-well-
founded, irregular sets), and the set theory of Zermelo is the
axiomatization, or fiat, that there are not irregular sets, via the
axiom of regularity, also known as the axiom of well-foundedness.
Where each of the other axioms of ZF set theory was enlisted to enrich
or broaden the set-theoretic universe (domain of discourse), in
expanding comprehension, the axiom of regularity instead is to
restrict comprehension. The regularity of sets was axiomatized, in a
sense made fiat or demanded, because otherwise Russell's specification
of a set, of one of the paradoxes named after Bertrand Russell in set
theory, would enable that given the existence of a universe of sets,
there would be a set that while specified to contain only sets that
don't contain themselves, would thus contain itself, the Russell
paradox, where there is also the Russell-Myhill paradox to consider,
which is another case showing that universal quantification, of a
sort, over a comprehensive (or comprehended) universe of objects,
would lead to that variously sets of a given largeness were irregular,
that infinite sets bijected to infinite powerset, and so on.

Where it might be so that those antinomies are seen to afflict
basically any theory rich enough to describe itself, instead perhaps
the emphasis should be upon the notion that those results illustrate
something deeper about the fundamental structures themselves: they do
not fit the ideal naive intuition as has seen place since (and in many
ways since long before) the specification of Peano's axioms, for
example, but that instead those INFINITE collections have various
properties that are quite dissimilar to those of finite ones, for
example, that the set of natural integers contains itself, in
compactification of the set towards completion of the numbers, and in
particular resolution of Zeno's paradoxes.

ZF, Zermelo-Fraenkel set theory, partially eponymous for Fraenkel who
called transfinite cardinals a disease from which mathematical
foundations would one day recover, has as its domain of discourse some
few elements of a less restricted universe. (So, add Fraenkel to the
list of those disagreeing with the conclusions of transfinite
cardinals, not "anti-Cantor cranks".) Then, where the specification
of those elements is exactly the same as the specification of the
Russell set of the soi-disant "paradox" of Russell, the collection is
seen to contain itself.

So, ZF is inconsistent. Quantify over sets or don't.

Ross

Congratulations to The World Wide Wade for taking my comment out of
context. Right after

Yes, around here it's called "being an anti-Cantor crank".

I wrote:

But seriously ...

indicating that I was joking, but this was clipped.

--- Christopher Heckman

.

Relevant Pages

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  • Re: Uncountable sets
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