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

"Klaus Cammin" <netzklaus@xxxxxxxx> schrieb im Newsbeitrag
news:6maq2iFfv3gkU1@xxxxxxxxxxxxxxxxxxxx
Salviati schrieb:

Assuming that real numbers are an archimedian ordered field,
I am not sure if this assumption is correct.

See Michael, this can be proved, too.
Actually the chain starts with first order logic.

, <, = means trichotomy but ignores | |.
Already Brouwer understood: the TND is only valid for rationals.

With nested intervals I am catched within the rationals because I will
never arrive at a difference equal to zero for as many steps as you
like.

Since the prove is true (it's based on 2.), your thinking is wrong.
When you quit this thinking, you won't have this problem any longer.
(Hell, put your arm down!)

IIRC 2. refers to the LUB. Irrationals belong to the genuine continuum.
They are uncountable and cannot be distinguished from the rationals there.
Why should those mathematicians be wrong who denied the LUB?


4. and then you can prove, that every decimal expansion is a real
number, and vice versa, that every real number has a decimal expansion.

No objection.

That's a pretty charming property of your reasoning: when you assert 4.,
you must assert 1.-3., too, because it's a direct consequence. When you
deny only 3., then 4. isn't true, and you're no longer entitled to say "No
objection"!

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


Albrecht deserves credit for denying the existence of pi. Hence, when I
had
to decide, who of the two of you is silly and who is insane, I proudly
award you the medal of unsilliness.

Albrecht Storz might be more familiar with modern terminology.
My position is quite clear.
Pi is an irrational and transcendent number and accordingly it "exists"
within the real numbers as the solution to a not numerically solvable task.
My criticism leads to consequences.
Be aware that I refuse to discuss whith someone who insults me.

I do not doubt that with the tiny addition that the limit does not
belong to the rational numbers.

Right, since we're talking about infinite decimal expansions (reals),
no limit belongs to the rational numbers.

That's why I met mathematicians who admitted that they
do not understand what makes the reals different from the rationals.
My answer is as simple as "unmathematical": They are a different
quality. Each single of them is uncountable.


Didn't Meray correctly write "fictitious limit"?

You may do that, but since all reals are fictionous in our new sense, the
term is superfluous. It's advantageous to do so, because emotional
qualifications might be attached to this term, i.e. outer-mathematical
reasoning.

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


Until today nobody has ever offered something better for IR than the
above.
Really? Were not Euclid, Archimedes, Spinoza, Galilei, Gauss, and many
others correct while Cantor's rejection of all rationality was at least
highly questionable?

Do you think math is progressing while going back in time?

It already did so if one compares medival with
highly developed ancient mathematics.

Until now I thought it's the other way round ...
Yes, the modern theory of numbers is better than theirs.

(Yes, they were great guys, but that's irrelevent here.
And yes, they also produced some crap.)

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

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?
Mostly I would not need it.
However I prefer |sign(0)|=1 for reals,
and this is obviously wrong for rationals.


[...] therefore of course not at all necessary to use it
differently as compared to rationals.

You'll see later, that you can save some wanted properties of the former
rationals, but first you MUST throw away the old rationals.
Come on. Give yourself a revolutionary push!

I do not intend spoiling mathematics for nothing.


A pretext is a reason which is pretended to have caused you to do
something. If you insist that your sentence contibuted something worth
to understand than tell me please what you mean by latter and former.

Sorry, I got it wrong with "pretext". I meant to say "bias" or
"prejudice".

You are a German, too. Hopefully Englishmen will ignore our mistakes.

And I meant, what you expressed responding to 3. and what I keep seeing in
my tutorials: the whole crappy babbling about "a sequence approaching its
limit without ever reaching it."

Because of your rude style I will not try to figure out what you meant.

This thinking puts a completely false focus on the matter. In modern math,
the sequences (1,1,1,...) and (1,0.5,0.333...,0.25,...) satisfy the same
definition of a limit. One "reaches its limit" and one doesn't.

Modern definition might be able to hide unresolved questions.

So, it was a very sensible thing to drag away our ancestors' beady eyes
from a completely sterile notion, and instead say for instance, that only
finite many members of a sequence are outside an arbitrary epsilon-
environment.

Much more clearer, simplier, more capable and omitting outer-mathematical
babbling.

When Hilbert's argumentation in 1925 was weak (I would boldly say wrong)
he resorted to inappropriate wording. Is "babbling" your last argument?

With the sympathy of a senior I realize that you didn't realize why this
metaphor is extremely valuable as a hint to still lacking understanding
of a peculiarity of real numbers.

Do you refer to your reasoning about unfullfillable tasks like sqrt(2)?

Heisenberg's cat or Wigner's friend are relevant.

If so, you might easily conclude, that I think it's bull.

When I learned English, I learned that a bull is male animal of the cow
family.
I will not read your time-demanding messages further.

Salviati:
.... in ultima conclusione, gli attributi di eguale
maggiore e minore non aver luogo ne gl'infiniti,
ma solo nelle quantità terminate.
IR >|> IR+ | | R


.

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