Re: Cantor Confusion

On 12/5/2006 1:20 AM, Virgil wrote:
In article <45746B98.5040606@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 12/1/2006 8:20 PM, Virgil wrote:
In article <1164967792.130794.251330@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

May be if you apply your personal definition of potentially infinity,
but not if you apply the generally accepted definition.

What "generally accepted" meaning is that? Most mathematicians do not
accept that a set can be "potentially" infinite without being actually
so.

I see it quite differently: Potentially and actually infinite points of
view mutually exclude each other as do countable and uncountable,
rational and irrational.

Except that countable and uncountable coexist within the same set theory
and rational and irrational coexist within the same real umber field.


Cantor's DA2 illustrates that there is no such field/list of real numbers.
Isn't this "coexistence" on the same low level of abstraction a basic
though hard to unveil intentional mistake by Dedekind?
Dedekind argued: As naturals can be extended to the integers in order to
allow subtraction and include negative numbers, and integers can be
extended to rationals in order to allow division and include fractions,
so rationals can perhaps be extended to reals in order to allow
non-linear operations and include irrationals.
Being mislead by the idea of a dotted line of numbers, he overlooked two
aspects. First of all, the irrationals cannot be located numerically.
Secondly, the irrationals are not an addendum to the reals but the other
way round, the reals vanish completely within the sauce of irreals.
The irrationals are at best fictitious numbers because they do not have
an exact numerical representation available. Kronecker said, they are no
numbers at all. Since the properties of the reals have to be the same as
these of the irrationals, all reals must necessarily also be uncountable
fictions.


.

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