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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: Thu, 17 Oct 2024 21:22:23 -0000 (UTC)
Organization: muc.de e.V.
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References: <vb4rde$22fb4$2@solani.org>   <vejmkm$e069$1@solani.org> <eb21591a-a60a-4baf-bdbd-afef2a69c230@att.net> <vejte9$e3ds$1@solani.org> <53460f91-4542-4a92-bc4b-833c2ad61e52@att.net> <ventec$255vi$2@dont-email.me> <venunr$2533b$4@dont-email.me> <29ce40e9-f18a-44d4-84d9-23e587cf9dea@att.net> <veor6u$2asus$1@dont-email.me> <2b6f9104-a927-49ee-9cf0-6ee3f82edc23@att.net> <verkkk$2r6kk$1@dont-email.me> <verlk6$4dv$1@news.muc.de> <vermdv$2s24h$1@dont-email.me>
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WM <wolfgang.mueckenheim@tha.de> wrote:
> On 17.10.2024 20:39, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:

>>> There is a general rule not open to further discussion:
>>> When doubling natural numbers we obtain even numbers which have not b=
een
>>> doubled.
>>> In potential infinity we obtain more even natural numbers than have b=
een
>>> doubled.
>>> In actual infinity we double =E2=84=95 and obtain neither =E2=84=95 o=
r a subset of =E2=84=95.

>> All of these "rules" are so loosely and ambiguously formulated, that t=
hey
>> don't actually say anything at all - they are meaningless.

> These rules are basic.

They are not.  They're loosely worded and ambiguous.  They do not form
the basis of any further mathematics.

> You don't understand them.  Perhaps too much at once.

Now we have an ad hominem.  I understand full well how meaningless they
are.

> Start with 2n > n for every natural number.  (0 is not a natnumber.)

Depends on the exact formulation.  0 is frequently regarded as a natural
number.  It makes it easier to build, for example, rings on top of it.

But other than zero, 2n > n for every natural number, yes.  In
particular, for every natural number n, 2n is also a natural number.

> If you can't understand or don't believe, then there is no common basis
> for discussion.

It's not a matter of belief.  It's a matter of correct and rigorous
mathematics.

> Regards WM
>>=20

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