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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)
Date: Sat, 19 Oct 2024 17:32:15 -0000 (UTC)
Organization: muc.de e.V.
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WM <wolfgang.mueckenheim@tha.de> wrote:
> On 19.10.2024 16:24, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:

>>>> rather than the standard mathematical concept of a mapping from N ->=
 N
>>>> where n is mapped to 2n.  In this standard notion, all numbers are
>>>> doubled, and we encounter no undoubled even natural numbers.

>>> Therefore the standard notion is wrong, if the natural numbers are a =
set.

>> You mean it's wrong because it doesn't gel with your intuition?

> No, it does not comply with mathematics.

How would you know?  You don't have a degree in maths, and aren't willing
to take lessons from those who do.

> When multiplying all natural numbers by 2, then the number of numbers
> remains the same but the density is reduced and therefore the interval
> is doubled.

That's not mathematics.  It's merely your intuition, derived from finite
sets and misapplied to infinite sets.

> 2 > n. Hence either natural numbers are created which have not been
> multiplied, then =E2=84=95 is not a set, or other numbers are created, =
then =E2=84=95
> is a set.

Again, not mathematics, but merely your intuition.  We're talking about
infinite sets here and maps between them.  There is no notion of
"created" involved at all.

>> If you think you can obtain an "undoubled" number in that mapping,
>> please feel free to give an example.

> I can prove it by 2n > n.

You can't.  You don't even understand the meaning of the word prove as it
pertains to mathematics.  As I say, if you maintain there is such a
doubled number which wasn't in N to begin with, you should produce it or
shut up.

>> You can't, of course, you'll just say that all such are "dark
>> numbers",

> Either dark numbers or natnumbers which have not been processed. There=20
> is no other way because 2n > n.

Again, not mathematics.  There is no notion of "processed" any more than
there's one of "created".  There's a map between two infinite sets.

>> Note that I haven't talked about "sets which change" - that's entirely
>> your idea.  I talked about a map from N -> N, where n maps to 2n.

> This Bourbaki-notion can be applied to potentially infinite sets only.=20
> Try to understand the correct mathematics.

In mathematics, there is no meaningful distinction between what you think
of as two different forms of infinity.  That's why such a distinction has
fallen out of use in serious mathematical discourse.

> Regards, WM

--=20
Alan Mackenzie (Nuremberg, Germany).