Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
- From: Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx>
- Date: Wed, 11 May 2005 13:41:58 -0400
Virgil said:
> In article <MPG.1cead5a960834f0e989c05@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1cea7e1a9c12de61989bf2@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> > >
> > >
> > > > >
> > > > > > I have provided two proofs. Counting WILL NOT get you to infinity.
> > > > > > It's
> > > > > > a weak tool for trying to study infinty. It's fine for finite
> > > > > > things.
> > >
> > > But the axiom of infinity and the principle of induction do get you to
> > > infinity in finitely many steps. That is what they are for.
> > > > >
> > > > > Indeed. Except that your proofs have been refuted thoroughly.
> > >
> > > > Not in any way that doesn't refute the proofs you hold so dear, and
> > > > not in any substantive way. Only by retreating to your local set of
> > > > confusions in the form of unquestionable contradictory axioms and
> > > > assumptions do you parry your away out of truth's way.
> > >
> > > Without some set of axioms as a foundation, there is no truth, only
> > > chaos. The axioms WE use have been tested and not found wanting. The
> > > informal unaxiomatized logic you are trying to use has been also tested
> > > and has been found wanting. So any contradictions are in your arena, not
> > > ours.
> > >
> > They haven't been tested very well. I have detected several inconsistencies
>
> What you call inconsistencies are clashes between what you want to hold
> in your axiom system and what actually holds in our axiom system, not
> between different things holding within our system.
They are inconsistencies between cardinality and other well-established areas
of mathematics which are not my invention.
> > so
> > I don't know what people have been looking at. I don't know exactly what is
> > wrong with mathematics in the last century, but there is pervasive insanity
> > in
> > this area that needs correction. You are wrong. Comment on my set size
> > method,
> > or shut up.
>
> Your set size method would have the sets of naturals, evens, primes, and
> rationals all of different sizes but the ratioals and the reals of the
> same size. State clearly what rule you have used to produce these
> results or shut up yourself.
I have. I gave you a method for calculating the first three, and explained my
thinking on the rationals, that one can get arbitrarily close to any real, and
that it therefore constitutes an enumeration of the reals. It is certainly
dense in the reals.
>
> Clearly being a proper subset is not enough to make one infinite set
> smaller thatn another in Orlow's definition, since that would force the
> set of rationals to be smaller that the set of reals. It would also not
> allow comparisons between such sets as the set of even naturals and the
> set of odd prime naturals, since these sets are disjoint.
Proper subsets are always smaller than the superset. Axiom.
Show me a real that cannot be expressed to any desired precision as a rational.
If you calculate pi to a million digits, shift them all to the left a million
times to get a whole number, and divide by 10^million, you have a rational
number that approximates pi to the millionth digit of precision. WM has shown
countability of the reals, as I have. That distinction fades into opinion.
>
> So just what is this vaunted definition you want us to evaluate as being
> so much better than injections and bijections?
>
> Give us a clue, or shut up!
>
If you haven't got a clue yet, I am not surprised.
--
Smiles,
Tony
.
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