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

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
news:Virgil-68A25C.17322702102008@xxxxxxxxxxxxxxxxxxxxxxxxx
In article <48e5519f$0$28910$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
news:Virgil-073440.12360202102008@xxxxxxxxxxxxxxxxxxxxxxxxx

Why should infinite sets be different?

Because nobody manages to set an infinite set.
Is there really a complete set of natural numbers?
You may believe it.
I am only aware of the Archimedean principle of mathematics:
There is no limit to numbers.
A set is something already given for good.

A set is almost anything given by a property, and being a natural is
such a property.

I wonder if there is a vailid definition of an (infinite) set.
Cantor's definition has been declared invalid.

So the ideal set of natural numbers is not identical with
any amount of natural numbers but it is rather something
quite different, something that is fictitious in the sense of
unreachable if seen from the perspective of discrete,
unity based numbers.

Almost everything mathematical is fictitious and physically unreachable,
including small natural numbers.

I did not write physical. I meant by a finite number of additions and
divisions.

Cantor's abandoned definition of an infinite set is
mainly hidden within the axiom of extensionality.

What is *Cantor's* definition of an infinite set?

Read Fraenkel 1923.

I know of two such definitions, which given the axiom of choice are
equivalent, but I was not aware that either was Cantor's.
They are:
(1) Due to Dedekind: a set is infinite if and only if there exists an
injection from it to one of its proper subsets.

As you correctly wrote, this is (a somewhat modified form of) Dedekinds
definition of infinity: The whole is not larger than its proper part.

(2) A set is infinite if for every natural n, the set of naturals less
than n injects into it.( alternately, a set is finite if there is a
natural n such that the set of naturals less than n surjects onto it).

I do not know who added the injection-surjection consideration to (1) and
(2).

Neither (1) nor (2) substitute the first sentence of the book by Hausdorff:
"Eine Menge ist eine Zusammenfassung von Dingen zu einem Ganzen, d.h. zu
einem neuen Ding." Hausdorff accepted in 1914 Cantor's naive notion of an
infinite set composed of single elements.

Salviati


.

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