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

In article <48f13c5e$0$28906$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Virgil" wrote
Salviati wrote:

Can a quantitatively unlimited set be given by a quality and
simultaneously by the entity of all of its elements?
In the words of the OP: "Is one-to-one mapping valid for comparing"
what one bottle without bottom contains with what another
bottle without bottom contains?

Valid in what sense? It is certainly possible to note whether
injections, surjections and bijections between sets exist, even when
those sets are infinite,

Salviati objects:
Countability and infinity are two different properties.
There are no different degrees of infinity.

There are different "degrees" of countability: being countable and not
being countable.

and to define a relative size based on those
conditions.

This was Cantor's naive mistake.

This is both logically and mathematically unexceptional.

I see it not at all justified.

Many people are able to distinguish between sets to which there exist
surjections from the naturals and sets to which no such surjections can
exist. If you are not, that is your problem, not ours.

Cantor ignored the 4th logical
possibility of being neither > nor = nor < but simply not quantifiable.

To do that one must reject the axiom of choice even in its weaker axiom
of countable choice form.

That it may conflict with your notion of "size" is irrelevant.

Cantor's transfinite cardinality conflicts with the axiom of Archimedes.

To which "axiom of Archimedes do you refer? The uncountable field of
reals certainly has the Archimedean property that for every real there
is a larger "natural".

The only drawback that I can see is that without the axiom of choice,
there may be pairs of sets which cannot be compared in this way.

AC is pure intention. While one can strive to compare pi
with rational numbers but will never reach absolute accuracy.

There is, however, no limit to potential accuracy other than absolute.
Thus, no matter how close two real numbers are, one can ultimately
distinguish between them.

Salviati
.

Relevant Pages

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