Re: Relative Cardinality



Virgil wrote:

> > 1) A number exists, if a fundamental set or an n-adic representation
> > are available.
>
> A number exists if the axiom system in which one is working allows it to
> exist.

It is defined then. But it does not yet exist. I.e., there is not yet
shown that this number can be put in order < with other numbers.

> > 3) A number cannot exist, if this is impossible.
>
> In
> another post, he has insisted that there exist distinct real numbers
> with no rationals between them.

You are wrong. Ty to better understand simple written text. I said that
there can *not* exist two irrationals without a rational between them
(in normal order). This proves that there are not more irrationals than
rationals (+1).

> > No, it cannot be compared with sqrt(2) but only with a rational which
> > consists of the first 10^20 (or even some more) digits of sqrt(2).
>
> Does WM claim that if x < y and y < z one cannot deduce that x < z?

No. But sometimes x < y cannot be obtained, neither by decimal
representation nor by fractions or continued fractions.

> > The same arguing is deluding the antidiagonal as
> > different from every line number.
>
> Which "line number" is it the same as?

Always the next one in this never ending sequence.
>
> > 1) We find the antidiagonal different from every line number *which
> > can be tested *.
>
> But since one test serves *all* *simulteneously*, that is no restriction.

There is no consistent test of all simultaneously. (And a bijection is
valid only for finite sets. Its validity for infinite sets is not
supported by any axioms.)

Regads, WM

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