Re: abundance of irrationals!)

In article <1115296950.103586.101090@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Randy Poe wrote:
> > mueck...@xxxxxxxxxxxxxxxxx wrote:
> > > Virgil wrote:
> > > > The definition of 'cardinality' is an equivalence class under the
> > > > equivalence relation of being in one to one correspondence
> (having
> > a
> > > > bijection from one to the other).
> > > >
> > > > Thus any set has a well defined cardinality.
> > > >
> > > Obviously this is a meaningless and self-contradictive definition,
> > > which easily cshould be abandoned because it (and the actual
> > existence
> > > of any infinite set) does not follow from any axiom.
> >
> > A classic example of what I just wrote about a minute
> > ago. Mueck doesn't understand it, therefore it's
> > meaningless.
>
> It is not a matter of understanding, but a matter of contradiction.

WE have a number of axiom systems, such as the Zermelo-Frankel system,
in which infinite sets are REQUIRED by the axioms. Unless Wm can show
that one of these contains an internal contradiction, he has no case,
since there is never any requirement that any axiom system coforms to
any physical reality.

If there were such physical reality requirements, how is it we have
several mutually exclusive axiom systems for geometry?


> 0. a b c ...
> 1 2 3
>
> The digit-positions of my listed numbers are enumerated by natural
> numbers. Can you find a number n e N which does not enumerate a
> digit-position? No. So *every* natural number enumerates a digit
> position. Two cases are conceivable:
> Either there are not infinitely many natural numbers, then there are
> not infinitely many digits.

I can conceive of "infinitely many" natural numbers, each marking the
last digit of one of your listed numbers, and none of your listed
numbers having more that finitely many non-zero digits.


> Or there are infinitely many natural numbers, then there are infinitely
> many digit positions.

There are infinitely many digit positions possible, but in WM's list,
each listed number uses only finitely many of them.

If this were not so, WM would be able to point to one which needs
infinitely many decimal places.

Claims of existence, such as WM keeps making, require proofs. The most
direct proof of existence is to produce one of the things alleged to
exist.

WM has not done this, and cannot do this.

We, on the other hand, have produced mathematically valid proofs that
such numbers cannot exist. For example, we have pointed out that the nth
number in WM's list never extends beyond the nth decimal place, so there
cannot be any n for which infinitely many decimal places are required.

Does WM dispute this and claim the there is an nth number in his list
requiring more than n decimal places? Again, let him produce such an n
and its corresponnding number.



> One without the other is impossible and nothing
> but an arbitrary and wilful attempt to save a rotten theory. But you
> attempt that.

Handwaving is not evidence.
>
> Further I understand that Cantor exchanged the quantifiers, to obtain,
> from what today is called axiom of infinity or induction or
> Archimedean, grossly
> An Em : n < m,
> his
> Em An : n < m
> with m called aleph_0.
> A typical quantifier mismatch.

That is entirely WM's mishmash, and none of Cantor's.
Cantor NEVER claims that aleph_0 is a member of N, that falsehood is
entirely WM's.

> And all of you who accused me to do so, all of you don't realize that.

We realize that what ever you think you are talking about is riddled
with internal contradictions, but what we are talking about has been
examined by experts for decades without finding any internal
contradictins.
.

Relevant Pages

  • Re: abundance of irrationals!)
    ... >> It is not a matter of understanding, but a matter of contradiction. ... > in which infinite sets are REQUIRED by the axioms. ... The axiom requires n+1 for given n. ... > last digit of one of your listed numbers, ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> WE have a number of axiom systems, ... such set meets Cantor's definition of an infinite set. ... digit position in the nth listed number. ... >> with internal contradictions, but what we are talking about has been ...
    (sci.math)
  • Re: A puzzle for Cantorists
    ... Any universe of sets used as a model of ZF ... For example, I "believe" in the Axiom of Choice, ... comprehensive line of argument about why and how infinite sets are ... the hierarchy" is not mathematics. ...
    (sci.math)
  • Re: An uncountable countable set
    ... index digit number n of K and so cover K up to digit number n. ... of index positions is infinite. ... The number of digit positions is irrelevant. ... My basic assumption is the existence of this axiom. ...
    (sci.math)
  • Re: Inconsistency of the usual axioms of set theory
    ... exists a Dedekind-infinite set" implies the existence of N? ... one can prove that "there exists an infinite ... particular axiom of infinity ...
    (sci.math)
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