Re: Uncountable sets

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".

But seriously ... do you have a problem with the definition of
cardinality (when two sets have the same "size"), or are you just
amazed by Cantor's diagonalization proof? The uncountability of the
reals is usually the first non-intuitive result in mathematics that
people run across. (The fact that there is no analog for the quadratic
formula when solving a fifth-degree polynomial equation is another.)

Intuition has to go out the window at this point, because people don't
have day-to-day experience with infinite, let alone uncountable, sets.

Has their been any research done on formulations of set theory which
adopt the axiom "uncountable sets do not exist" (I'd be particular
interested in theories which retain the general flavour of ZFC, e.g.
untyped theories).

If you read through my thread "The Ultimate Refutation of Cantor" here
on sci.math, you'll find out that there is an axiom system where there
are no infinite sets, which boils down to the statement "infinite sets
exist" being an assumption, not a provable statement. You will
probably want to skip over the parts of that thread related to
diagonalization.

Another intuition I have is: only recursively enumerable sets exist.
Is there a name for this intuition?

I would suspect that these sets could be called "potentially
infinite", since you can construct them one element at a time, but I
haven't read up on the literature here.

(Wikipedia has the following paragraph on their "actual infinity"
page: "In metaphysics, Aristotle distinguished between actual and
potential infinities. An actual infinity is something which is
completed and definite and consists of infinitely many elements. A
potential infinite is a sequence which is endless. Whereas all the
elements of an actually infinite set are assumed to exist together
simultaneously, the elements of a potentially infinite sequence exist
only consecutively over time.")

Similarly, are there ZFC-ish
formulations of set theory which only allow only recursively
enumerable sets? (I might allow co-recursively enumerable sets as
well, not sure about that one....)

You're probably getting close to the issues surrounding the Axiom of
Choice. I'm not 100% sure and will let someone else who knows more
continue from this idea.

I would distinguish these views from constructivism or intuitionism.
I've got no problem with the law of the excluded middle, etc., just
with certain types of infinite sets. My justification is that I am a
Platonist of some sort, I believe that mathematical objects have some
sort of real existence; but while I am happy to believe that the
natural numbers (as individuals), and the set of natural numbers as a
whole, and various other "sane" sets, have a real independent
existence, "insane" sets (like uncountable sets) don't in my view
qualify.

I suspect that a good place to look, if you have access to the local
university's library, is K. Kunen's book _Set Theory: An Introduction
to Independence Proofs_. I haven't read it yet, but I suspect they
give examples of alternate set theories which are "consistant" (that
is, given a certain set of axioms, you can't derive a contradiction).

--- Christopher Heckman

.

Relevant Pages

  • Uncountable sets
    ... My intuition says that uncountable sets do not exist. ... the existence of anything bigger. ... only recursively enumerable sets exist. ...
    (sci.math)
  • Re: Uncountable sets
    ... My intuition says that uncountable sets do not exist. ... the existence of anything bigger. ... only recursively enumerable sets exist. ...
    (sci.math)
  • Re: Uncountable sets
    ... My intuition says that uncountable sets do not exist. ... the existence of anything bigger. ... only recursively enumerable sets exist. ...
    (sci.math)
  • Re: Uncountable sets
    ... My intuition says that uncountable sets do not exist. ... the existence of anything bigger. ... only recursively enumerable sets exist. ...
    (sci.math)
  • Re: Galileos Paradox
    ... The existence of the set of natural numbers is used all over the place ... You are just FLAT OUT INCORRECT that non-standard analysis is ... DOES prove the existence of ordinals. ... Do the infinite forms of them jut off in other directions? ...
    (sci.math)
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