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

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
news:Virgil-D17D24.11532514102008@xxxxxxxxxxxxxxxxxxxxxxxxx
In article <48f4b637$0$17128$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Virgil" <Virgil@xxxxxxxxx> wrote
It means the sets are of the same cardinality, whereas lack of any
bijection means they are of different cardinalities.

Finite cardinalities are perhaps pointless.

How else can you say that two finite sets are of the same size or that
one is smaller that the other without seeing how their members pair off,
either with each other or with the standard, the finite initial segments
of naturals?

Finite sets are countable. So I just have two numbers to be compared.


At least I am not aware of any
genuine application.
Aleph_0 means infinite but countable.
Aleph_1 means uncountable, in particular continuous.

There is no necessity for a set of cardinality aleph_1 to be in any
sense continuous.

I wrote "in particular". Even Cantor's dust is in some sense
continuous if one does not arbitrarily exclude the spaces
but just removes them to the left axis.


Any other "cardinality" obviously lacks justification.

That you re unable to see such justification could as easily be your own
deficiency rather than any lack of justification.

No. I repeatedly asked for examples. Nobody was able to point me to at least
a single one except for maybe application as something to learn for
students.
I meant genuine applications.

The terminology finite/infinite, countable/uncountable is traditionally
anchored, sufficient and easily understandable even to
non-mathematicians.
Accordingly I recommend to abandon all alephs.

And I recommend you stop pontificating

I have no authority at all.
My only weapons are my arguments.
You are invited to seriously deal with them.

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


.

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