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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions?
Date: Sun, 13 Oct 2024 20:04:10 -0400
Organization: Peripheral Visions
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Chris M. Thomasson explained :
> On 9/18/2024 7:31 AM, Jim Burns wrote:
>> On 9/18/2024 8:39 AM, WM wrote:
>>> On 16.09.2024 19:30, Jim Burns wrote:
>>>> On 9/15/2024 3:47 PM, WM wrote:
>> 
>>>>> I don't believe in gaps on the real line.
>>>>
>>>> There aren't gaps and there aren't next.numbers
>>>> in numbers.situating.splits of rationals with
>>>> countable.to.numerators.and.denominators
>>>
>>> So what is next instead?
>> 
>> What is between one and the next?
>> A gap.
>
> With regard to a very _strict) line of thinking (no mixing and matching), say 
> 100% natural numbers... There is NO "gap" between, say:
>
> 3 and 4

No gaps. But there are gaps between these naturals with respect to the 
rationals. That is, there are rationals which fill the distance between 
natural numbers (and integers) as embedded in Q.

With some algebraic numbers not being in Q, we have reals filling the 
'gaps' in Q with respect to the reals but there are no 'gaps' in Q.