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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 20:19:34 -0000 (UTC)
Organization: muc.de e.V.
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WM <wolfgang.mueckenheim@tha.de> wrote:
> On 19.10.2024 20:22, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:

>> An infinite set is one which has a proper subset which can be
>> put into 1-1 correspondence with the original set.  That is the
>> definition.

> According to Dedekind every set {1, 2, 3, ..., n} is in correspondence=20
> with the set {2, 4, 6, ..., 2n} which covers twice the interval,=20
> containing numbers not in the original set.

This is true, though has nothing to do with my point about the
definition of an infinite set.

> This does not change when the whole set =E2=84=95 is multiplied by 2.

It does.  It changes dramatically.

> The result covers twice the interval, ....

It does not, except in the sense that twice infinite =3D infinite.

> .... containing numbers not in the original set =E2=84=95.

No.  If you think that, then give an example of a 2n which "isn't in the
original set N".  You won't and you can't.  But you'll likely come back
to your standard get-out clause about (non existent) "dark numbers".

> Regards, WM

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