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

In article <48f73ae7$0$6580$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"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.


You cannot get those numbers without pairing off the members of the
given set with the members of a set of naturals.

The counting of a finite set to give it a number of elements is no more
that constructing an appropriate bijection with a subset of N.

So the (finite) number of elements in a set comes from a bijection, not
the other way round.


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.

Then every set could be made equally continuous merely by removing to
one side all its gaps. But since all the spaces in Cantor's "dust" are
open, and the dust itself is closed, such a "remove the spaces to the
side" is impossible.


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.

Since all counting of finite sets requires the establishment of
bijections, counting itself is an application of cardinality.

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.

Then you show yourself to be unarmed.
.

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