Date: Dec 12, 2012 12:51 PM
Author: Alan Smaill
Subject: Re: fom - 01 - preface

WM <> writes:

> On 12 Dez., 12:07, Alan Smaill <> wrote:
>> WM <> writes:
>> > On 11 Dez., 12:54, Shmuel (Seymour J.) Metz
>> > <> wrote:

>> >> In <virgil-2FC1D7.13530210122...@BIGNEWS.USENETMONSTER.COM>, on
>> >> 12/10/2012
>> >>    at 01:53 PM, Virgil <> said:

>> >> >In order to define the product of a real number times a
>> >> >transfinite, the definition must hold for all reals and all
>> >> >transfinites.

>> >> There's a more fundamental problem; she/he/it is conflating cardinals,
>> > Cantor was a male. So "he" would be appropriate.
>> And the problem is passed over in silence by WM.

> No, I mentioned the problem that Shmuel cannot even calculate limits
> of sipmle sequences.

As I said, you passed over the problem at issue in silence.

> And again I mention the problem that you cannot
> read simple texts:
> See: Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leipzig
> 1883)] There he writes: "da doch auf diese Weise eine bestimmte
> Erweiterung des reellen Zahlengebietes in das Unendlichgroße erreicht
> ist" My translation: Since in this manner a definite extension of the
> real domain of numbers into the infinitely large has been
> accomplished.

A bad translation; it's not the domain that is real, but the numbers:
better is "a definite extension of the region of real numbers into the
infinitely large".

And what does this give? an ordered set; but no multiplication
defined here, of course!

Indeed, when (ordinal) multiplication is introduced (section 3), it is
in the context of Cantor's number classes:

"The first number-class (I) is the set of finite integers
1,2,3, ...,nu,..., which is followed by a second number-class
consisting of certain infinite integers following each other
in a determined succession; after defining the second number-class,
the third is reached, then the fourth etc."

(translation George Bingley)

3.14159... does not make an appearance anywhere in these number classes.

> Regards, WM

Alan Smaill