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

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
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In article <4911bb41$0$31864$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
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"Salviati" wrote:

Galilei understood that this was a fallcy because infinite quantities
must
not be quantitatively compared with each other.

That Galileo Gallilei couldn't figure out how to do it does not mean
that it cannot be done. After all, he was not even a mathemetician,
merely an early physicist.

His reasoning is still valid and cannot be invalidated by Cantor's
silly illusion.

Eckard is not sufficciently competent at mathematics to judge what is or
is not valid in mathematics.

Please point me to any compelling evidence that proves Galilei's conclusion
wrong. Cantor not even tried it.

According to Cantor's
interpretation of DA2, there are also "more" integers as compared to
>> >> just positive integers.

According to Cantor, their cardinalities are the same. And it is quite
possible for a set to have the same cardinality as a proper subset of
itself.

I did not object to the countability of the union of positive and
negative
countable numbers. I objected to Cantor's interpretation. He explicitely
concluded that the reals must have more elements as compared to the
rationals because they do neither have the same nor a smaller number of
elements. Actually, they are incomparable

Since Cantor defined precisely what he meant by such comparisons in
terms of injections and bijections,

When I read Cantor's original papers, I did not come across with these
"precise definitions".
When were bijection and injection first mentioned?
Maybe it was by Bernstein, maybe much later?
The principle, not the word, bijection was already used by Galilei.
Cantor wrote in Grundlagen einer allgemeinen Mannigfaltigkeitslehre 1883 §1:
"Von groesster Bedeutung scheint mir zunaechst die Einfuehrung der neuen
ganzen Zahlen für die Entwicklung und Verschaerfung des
Maechtigkeitsbegriffes. Jeder wohldefinierten Menge kommt ... eine
bestimmete Maechtigkeit zu, wobei zwei Mengen dieselbe Maechtigkeit
zugeschrieben wird, wenn sie sich gegenseitig eindeutig, Element für Element
einander zuordnen lassen". Instead of considering bijection the property of
any countable numbers and accept Galilei's insight that bijection does not
allow a quantitative comparison of infinite variables, Cantor continued to
postulate countable higher "Maechtigkeiten". In §4 he even denied the actual
Unendlichkleine.

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




.

Relevant Pages

  • Re: Is one-to-one mapping valid for comparing infinite-sized sets?
    ... Cantor's "countably infinite" requires bijection, ... One can easily PROVE it from the surjective definition of countability ... You are at least as naive as Dedekind and Cantor together. ...
    (sci.math)
  • Re: Attempts to Refute Cantors Uncountability Proof?
    ... Zermelo, but perhaps I am not looking good enough. ... I nearly come to the conclusion that that argument is not from Cantor. ... There are two diagonal proofs by Cantor, ... that paper shows that there is no bijection between a set ...
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  • Re: Is one-to-one mapping valid for comparing infinite-sized sets?
    ... injectability defines a partial ordering on sets. ... Cantor did never disprove Galilei. ... By defining cardinality in terms if injectability. ... Which is a standard test for countability. ...
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  • Re: Cantor Confusion
    ... The countability of the set W of all the words of a finite alphabet is ... impossible to construct a bijection between W and N. Why should it be ... Cantor insisted on well-ordering of all sets. ... condition for the identity function on S to exist. ...
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  • Re: Is one-to-one mapping valid for comparing infinite-sized sets?
    ... in a paper by Cantor himself. ... Cantor's definition of cardinality agrees with what I said above. ... Which is a standard test for countability. ... countables and uncountables. ...
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