Re: Is one-to-one mapping valid for comparing infinite-sized sets?

In article <490501b1$0$13388$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Klaus Cammin" <netzklaus@xxxxxxxx> schrieb im Newsbeitrag
news:6mgqq3Fgj09lU1@xxxxxxxxxxxxxxxxxxxx
Salviati schrieb:
Why should those mathematicians be wrong who denied the LUB?

Who are they?

To be found via Wikipedia.

If they deny the LUB, they're silly. They could have a continuum,
but they don't want it. They won't get another.

Well, Cantor's paradise looks nice on the first glimpse,
and you are correct in that there is no quadrature of the circle
as is no perpetuum mobile.
Nonetheless Dedekind as well as Cantor, Hilbert, Zermelo, etc. did not
create a sound and fertile basis but they furnished an unnecessary illusion
with perhaps detrimental consequences in physics.

That eckard.blumschein does not consider that a sound and fertile basis,
impresses only eckard.blumschein.


No. As in case of Cantor's DA2, a correct result is no compelling
evidence for correct reasoning.

Great.
Please explain, how one can yield correct results with incorrect
reasoning.

Ask teachers. This happens not rarely.

Consider doing 1/2 times 2/4, by "cancelling" the 2s to get 1/4.



Is proof a matter of taste for you? Funny. I tell you, the strongest
proofs
are those, which everybody must accept even against prior objections.

Be aware that I refuse to discuss whith someone who insults me.

Someone who claims, that Cantor's math was flawed because of his alleged
mental problems, should not expect a too friendly addressing.

I rather see his insanity a consequence of feeling possibly wrong.

Eckard.blumschein, as usual, is seeing what is not there.

Telling apart periodic reals from those without that property is the next
step.

This would not work for different systems of numbers.

Any system based on positional notation, like binary, decimal and
hexadecimal, works that way.



Funny: you and WM keep mixing up elements and their sets.

The genuine continuum cannot be resolved into single elements.

In which case, the set of reals, and geometric lines, are not continua.
But are continuous, which is quite different.

I do not see much justification for calling it a set.

Your sort of continua are not sets, nor anything else mathematical.
The only occur in extra-mathematical considerations.


What you called outer-mathematical must not be excluded
from mathematics if one respects genuine rigorosity.

That's a good one.

No joke.

The point is, even the best understanding was sacrified
to seeming rigorosity after Cantor managed to cheat.

Yes, sometimes the course of science declares previous statements as
invalid. Otherwise we would still believe today, that earth is the center
of the universe. But how can you expect, that your prejudice, that Cantor
cheated, will invalidate his mathematical proofs?

I tenmd to guess that Cantor was just overly naive and ambitious, and he
mainly cheated himself. History of mathematics shows many wrong mathematical
proofs, not just evidence for the existence of god.
When Cantor assumed trichotomy, he could not show something else than
trichotomy. So his proofs were in some sense circular and wrong from the
very beginning.

Oe can prove, entirely without circularity, trichotomy for the Dedekind
cuts and also for the equivalence classes of Cauchy sequences. So for
what sort of reals does eckard.blumschein claim needs it to be assumed?


I am arguing, real numbers are something quite different
Agreed, hence my advice: throw away the old rationals.
If so, why to distinguish between rationals and reals?

You mean, irrationals and rationals, don't you?

No.

Then you are being incoherent, Eckard.

Reals are both, and that's
because YOU agreed to our definition of reals as infinite decimals.

If you (FIRST) don't distinguish between rationals and irrational numbers,
you are able to detect, what they have in common. It's also nice having a
unifying theory for them. The differences are still there, getting visible
after some technical work.

Why do you not even try to understand me?

Insofar as you are talk about anything relevant to mathematics we try,
but when you talk nonsense, as you do so often, we don't.
.

Relevant Pages

  • Re: A wise decision
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    (sci.math)
  • Re: Is one-to-one mapping valid for comparing infinite-sized sets?
    ... Nonetheless Dedekind as well as Cantor, Hilbert, Zermelo, etc. did not ... do not understand what makes the reals different from the rationals. ... characterization of reals as infinite decimals, ... from mathematics if one respects genuine rigorosity. ...
    (sci.math)
  • Re: Galileos Paradox
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    (sci.math)
  • Bye
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  • Re: Skolems Paradox and why is math the way it is?
    ... This is not a job the axioms were ever meant to do. ... It's actually a problem with language generally, not just mathematics. ... just procrastinated the problem of interpretation for one step. ... Consider for example the set of reals you considered in another posting, ...
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