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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?
Date: Sun, 22 Sep 2024 12:00:53 -0700
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On 9/22/2024 4:02 AM, FromTheRafters wrote:
> Chris M. Thomasson submitted this idea :
>> On 9/21/2024 4:25 PM, Richard Damon wrote:
>>> On 9/21/24 9:57 AM, WM wrote:
>>>> On 21.09.2024 01:06, Richard Damon wrote:
>>>>> On 9/20/24 2:33 PM, WM wrote:
>>>>
>>>>>> How can 10 points exist in linear order without a first one?
>>>>
>>>>> Who says they can't.
>>>>
>>>> In order to count to 10, you have to start at 1.
>>>>>
>>>>> Its just when they become infinite that there might be ends that 
>>>>> don't exist.
>>>>
>>>> In order to count a countable set, you have to start at 1.
>>>>
>>>> Regards, WM
>>>>
>>>
>>> Right, so the countable numbers have ONE end that can be used to 
>>> count from.
>>
>> What about the signed integers? Well, we can always start from origin, 
>> at 0. ;^)
> 
> You could think of them as a sequence of ordered pairs of naturals.
> 
> ((0,0),(1,0),(1,1),(2,0),(2,1),...)
> 
> First of the pair is the magnitude and the second of the pair is the 
> sign where one equals negation. The sequence is one-ended and perhaps 
> now more obviously countable.
> 
> (0,1,-1,2,-2,3,-3,...)

Agreed. Think of something like:

A = ..., -3, -2, -1, -0
B = +0, +1, +2, +3, ...

(...)A--->(-0+)--->B(...)

Signed zero aside for a moment... lol. ;^)

Make any sense? Or, is it kook ville WM style?

Actually, we can pick any origin we want to. It does not have to be zero...