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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 18:39:02 -0000 (UTC)
Organization: muc.de e.V.
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WM <wolfgang.mueckenheim@tha.de> wrote:
> On 17.10.2024 00:39, Jim Burns wrote:

>> No natural number is
>> the first to not.have a natural.number.double.

> True.

>>>> When doubling all natural numbers
>>>> we obtain only natural numbers.

>>> That is impossible.

>> There is no first natural number from which we obtain
>> (by doubling) anything not.a.natural.number.

> True.

>> The only set of natural numbers with no first
>> is the empty set..

> No, the set of dark numbers is another set without smallest element.

No.  It is the empty set (which I proved many posts ago).  All empty sets
are the same - there is only one of them.

>> There is no =E2=96=92=E2=96=92=E2=96=92=E2=96=92=E2=96=92 natural numb=
er from which we obtain
>> (by doubling) anything not.a.natural.number.

> Correct is: There is no such _definable_ natnumber.

That is the only sort of natural number there is.

> There is a general rule not open to further discussion:
> When doubling natural numbers we obtain even numbers which have not bee=
n=20
> doubled.
> In potential infinity we obtain more even natural numbers than have bee=
n=20
> doubled.
> In actual infinity we double =E2=84=95 and obtain neither =E2=84=95 or =
a subset of =E2=84=95.

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

> Regards WM

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