Re: Cantor Confusion

In article <45756F17.1010409@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

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.

EB conflates "list" with "set". Nothing in any axiomatic set theory I am
aware of requires sets to be lists, or even listable.

Isn't this "coexistence" on the same low level of abstraction a basic
though hard to unveil intentional mistake by Dedekind?

What "coexistence"?

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.

As Dedekind (and others) demonstrated precisely how to construct each of
these extensions, his arguments conclude with Q.E.F.

Being mislead by the idea of a dotted line of numbers, he overlooked two
aspects. First of all, the irrationals cannot be located numerically.

The "cuts" can be defined precisely.

Locating the numbers exactly on a number line may be a a problem for
engineers, but is not one for mathematicians.

Secondly, the irrationals are not an addendum to the reals but the other
way round, the reals vanish completely within the sauce of irreals.

That "sauce of irreals" is unknown to mathematics.
What engineers cook up is not our problem.

The irrationals are at best fictitious numbers because they do not have
an exact numerical representation available.

All numbers are equally fictitious, mere creations of the mind.
.

Relevant Pages

  • Re: Rational numbers, irrational numbers: each dense in real numbers
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  • Re: Cantor Confusion
    ... but not if you apply the generally accepted definition. ... extended to rationals in order to allow division and include fractions, ... the irrationals cannot be located numerically. ... the irrationals are not an addendum to the reals but the other ...
    (sci.math)
  • Re: Cantor Confusion
    ... though hard to unveil intentional mistake by Dedekind? ... extended to rationals in order to allow division and include fractions, ... the irrationals cannot be located numerically. ... the reals vanish completely within the sauce of irreals. ...
    (sci.math)
  • Re: Rational/Irrational Numbers
    ... The consideration of rationals and irrationals, ... other sets that are NCD2 in the reals from "On Well-Orderingof Sets ... number line, or algebraics and transcendentals, or rationals, ...
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  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
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