Re: Uncountable sets in CZF?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 08/17/04 

Date: 17 Aug 2004 16:00:17 -0700

fishfry <BLOCKSPAMfishfry@your-mailbox.com> wrote in message news:<BLOCKSPAMfishfry-324166.22494716082004@netnews.comcast.net>...
> In article <88ee1d13.0408161814.cec506a@posting.google.com>,
> agamemnon_atheos@yahoo.com (Agamemnon) wrote:
>
> > Hello everyone. I am a philosophy student with an interest in
> > mathematics and I was hoping someone here could answer a set theory
> > question that I have. When I first heard about uncountable sets, I
> > thought it was very strange that some infinite sets are "larger" than
> > others. It troubled me at first, but eventually I thought to myself
> > "All this means is that if you define size (cardinality) a certain
> > way, and use these particular axioms to capture the logic of sets,
> > then it follows that uncountable sets exist. Don't read too much into
> > it."
> >
>
> Yes, but these axioms and definitions have a certain intuitive appeal.
> You agree that some infinite sets can be put into one-one correspondence
> with the natural numbers; and other infinite sets, can't. You must still
> regard that as strange and wondrous, no? Because the definitions and
> axioms are in no way artificial or forced.

Hi fishfry. Strange and wondrous? This is mathematics. Cut it open
and look inside. There are rare cases where that makes a difference.
There are no paradoxes. Is thus everything a paradox? I disagree
about the axioms.

I've read recently, as have probably you, that in a model of an
intuitionist zet theory, IZF for intuitionist Zermelo-Fraenkel, that
it is not inconsistent for there to be a mapping between set and
powerset. That agrees with some of the things I say.

We had a good discussion within a month about that, see the thread
"'Uncountable' doesn't exist." That contains some further discussion
of this issue by various authors that send copies of their words to
sci.math, among them highly skilled amateur and professional
mathematicians.

Imagine if a set theory had a set of all sets. Then, that set would
be its own powerset.

We've been discussing models of ubiquitous ordinals. In one of those
models EF is bijection between N and the each element of the unit
interval.

I'd like to know more about what Loewenheim-Skolem is saying. Maybe
it can explain to Banach-Tarski that a point on a line has two sides,
but only one on either side. Plain language?

Menzel's description of the halting problem (The Halting Problem, THP)
on sci.logic was worth reading.

Unitize the analog!

Ross

Relevant Pages

  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... you claim that set theory is ... theory in which to express virtually all of mathematics. ... S (call this function 'omega pre S'). ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... > This is not a job the axioms were ever meant to do. ... other person's interpretation require a winning strategy, no more, no ... I'm pretty sure than any model of set theory is intuitively ... figuring out how I tell what is real in mathematics. ...
    (sci.math)
  • Re: Set Theory: Should You Believe
    ... Why, in your opinion, is the orthodoxy in set theory ... mathematics, ignoring the misgivings of geniuses like Poincare, Weyl, ... When NW said that "You don't need axioms", ... NAFL theories (but infinite proper classes, ...
    (sci.logic)
  • Re: Set Theory: Should You Believe
    ... questioning of the status quo. ... the foundations of logic and mathematics as exists today. ... Why, in your opinion, is the orthodoxy in set theory ... and proofs do require axioms as starting points. ...
    (sci.logic)
  • Re: Two results of set geometry
    ... dependencies among theorems and choice principles. ... I think the need to assert this principle for ALL sets was an admission that it didn't follow from more elementary principles, and that it didn't NECESSARILY apply to uncountable sets. ... It is largely from AC that the paradoxes of set theory arise. ... you cannot infer from the axioms ...
    (sci.math)