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

In article <48f4d35f$0$17136$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
news:Virgil-EE48ED.13391313102008@xxxxxxxxxxxxxxxxxxxxxxxxx
In article <48f2e1a9$0$28920$9b4e6d93@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Salviati" <eckard.blumschein@xxxxxxxx> wrote:

"Virgil" <Virgil@xxxxxxxxx> schrieb im Newsbeitrag
news:Virgil-089CF2.21052511102008@xxxxxxxxxxxxxxxxxxxxxxxxx
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.

I see this an alternative. You may write 0 or 1 but not 2.
Something can be either finite or infinite, countable or uncountable.
Uncountability is not a higher degree of infinity.

It is certainly a different one.

For finite sets A and B, if A injects to D but does not surject to B, no
one objects when one says "A is smaller than B" or "B is larger than A".

And no one objects when A is finite but B is not.

But all sorts of kooks object when both are infinite.

Necessarily, because infinite objects cannot be quantified. Cantor's naive
idea does not have a sound basis. The size is where something ends. Infinite
objects do not have an end and therefore no size.

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.

I do not have problems with this terms.

Then why do you object to them?

I do not object to injection and surjection but to non-admissible
applications of them.

Whatever is wrong about saying that the naturals can be injected to the
reals but not surjected?


First we have to clarify: What does "uncountable field of reals" mean?
Archimedian implies countability.

How so? The Archimedean property says, in one of its forms, that for
every positive real, there is a larger natural.

I wrote "implies". If there is no upper limit to naturals, then there is
consequently also no lower limit to their reciprocal.

Sure there are such lower limits. Lots of them
Every negative rational is such a lower limit.

I know of no formulation
of that property that says anything directly about how many reals or how
many naturals exist.

I know that mathematician do not appreciate 0=1/oo but I like this duality.

The your arithmetic must be all messed up, since the consequences of
imposing this on arithmetic are fatal.

One may deduce that countably many naturals are
enough, but not that countably many reals aer enough.

Reals are not countable.


That is precisely what we have been trying to get you to admit.

The combination countably many suggests adependency
that is not correct. Uncountably many would be wrong.
Uncountably belongs to much, not to many.

Maybe in the mathematics of your native language, but not in English
mathematics. Countablity, and its lack, are properties of manyness, not
of muchness, at least in English.



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.

No.
0.999... and 1.000... are rational as long as one can distinguish between
them.
One can always distinguish between different numerals (names for
numbers), but if as in this case, they represent the same real, you
cannot distinguish between their values.

"0.999..." and "1.000..." are different names for the same number.

Only if we agree on reading ... as the fiction of infinitely much.
Notice:
Reals are something irreal.

It is Salviati's mathematical philosophy that is "irreal"
.

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