Re: An uncountable countable set

In article <1157193952.357765.296410@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
*** T. Winter schrieb:
In article <1156842504.966630.131290@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> muecke=
nh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:

[ talking about {1, 2, 3, ..., w} ]

> > Let us remove omega from that set. What is the resulting
> > cardinality
>
> Not an actually infinite one, because without omega there are only
> finite natural numbers.

But what then?

It is a potentially infinite set, as described by Peano and by all the
mathematicians before Cantor.

Yes. I asked you about the cardinality, not what it was. What is the
cardinality of that set?

> > As there is a bijection between that set and the set including
> > omega, their cardinalities are the same.
>
> No. Without omega there is no actual infinity.

But there is a bijection. And two sets that are in bijection with each
other have the same cardinality. You may think that is a stupid
definition, but nevertheless, it is a definition that works.

Look at my last post concerning the final proof that 0.111... with
actually infinitely many digits does not exist.

That is not an answer to the question I asked. I ask about the cardinality
of a set and you come back with something completely unrelated. (But see
my rebuttal of that article.)

> You should know that the n-th natural number is defined by the sum of
> the number n-1 plus 1 (Peano). This can be retraced to the sum of n
> 1's. What is there to be defined? If there are infinitely many naturals
> (and if infinity is a number), then there is at least one infinite sum
> of 1's (one from each natural).

Assuming that that sum terminates. Which it does not as there is no last
natural number. Not even set theory makes that a terminating sum. And
this is regardless of whether infinity is a number or not.

Cantor defined aleph_0 in this manner: "Da aus jedem einzelnen Elemente
m, wenn man von seiner Beschaffenheit absieht, eine "Eins" wird, so ist
die Kardinalzahl [von M] selbst eine bestimmte aus lauter Einsen
zusammengesetzte Menge, die als intellektuelles Abbild oder Projektion
der gegebenen Menge M in unserm Geiste Existenz hat." But if you think
he was wrong, there is no need to discuss this topic.

Where is he talking about addition? "when man von seiner Beschaffenheit
absieht"; I would translate this as "when you disregard all properties".
And so, in the transformation the ability to add is lost. And so you
get a set (the current, proper, term is multi-set, I believe) consisting
of only ones.


> > OK, I kind of understand. I do not state that infinity is a number
> > aleph_0. There exists a host of infinities. And aleph_0 is the
> > infinity that gives the equivalence class of sets that are in
> > bijection with the set of natural numbers; the canonical set with
> > cardinality infinity.
>
> If aleph_0 > n for n e N, and if there are even numbers x > aleph_0,
> then aleph_0 is a number. Cantor stated that. If you do not do so, then
> we are in agreement in that point.

You may call it a number, or not. I prefer to call it a cardinal number.
I prefer not to use the word "number" singly, unless there is context
that shows what kind of numbers I am talking about. So I may be talking
about the numbers in the ring Q(sqrt(-3)), or even about the integers in
that ring. Or about the 5-adic numbers, or the Cayley numbers, but
always with context. I think that in his early papers he indeed called
them "Zahlen", but in later papers he did call them "Mächtigkeit". If
you read his papers you may have found that he changes terminology
between papers. Understandable, because it was an early development.
On the other hand, you will see letters by Kronecker where he calls
the elements of Q(sqrt(-3)) or somesuch "Zahlen" (while he already
much earlier tried to remove all non-integers from proofs).

My point is only that: aleph_0 is not in trichotomy with natural
numbers and other cardinals.

Prove it.

What is the cardinality of the set of all finite natural numbers?

It has none.

Why not? Cardinality has been defined so it applies to *each* set.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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