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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Mon, 14 Oct 2024 15:27:14 -0700
Organization: A noiseless patient Spider
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On 10/14/2024 9:37 AM, WM wrote:
> On 14.10.2024 18:04, joes wrote:
>> Am Mon, 14 Oct 2024 15:40:01 +0200 schrieb WM:
>>> On 14.10.2024 14:15, joes wrote:
>>>
>>>> No, we are taking the complete, actually infinite set which reaches to
>>>> "before" w.
>>> and fills the space between 0 and ω evenly. Same happens with the
>>> doubled set between 0 and ω2.
>> No, there is no consequent infinity. The even numbers do not go
>> 0, 2, 4, ..., w, w+2, w+4, ..., w*2
> 
> Either the doubled numbers are natural, then half of them have not been 
> among the original set, or all natural numbers have been doubled, then 
> the result contains infinite numbers.
> 
> That doubling _all_ natural numbers only yields _all_ natural numbers is 
> impossible.

Take the index, using natural numbers

1: { 1 }
2: { 1, 2 }
3: { 1, 2, 3 }
4: { 1, 2, 3, 4 }
....

vs:

1: { 2 }
2: { 2, 4 }
3: { 2, 4, 6 }
4: { 2, 4, 6, 8 }
....