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Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (repleteness)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 14 Sep 2024 09:27:48 -0700
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On 09/13/2024 04:05 PM, FromTheRafters wrote:
> WM explained :
>> On 13.09.2024 17:52, Richard Damon wrote:
>>> On 9/13/24 11:41 AM, WM wrote:
>>
>>>> Between [0, 1] and (0, 1] there is nothing, there is not a spot or
>>>> point of the interval.
>>>>
>>> But that doesn't mean there is a lowest most point in (0, 1] as any
>>> point you might want to call it will have another point between  it
>>> and 0.
>>
>> I will not call any point but consider all points. There is no point
>> smaller than all points in the open interval but a smallest one. Only
>> 0 is smaller than all.
>>>
>>> Note, I said between the point your THINK is the first, there is no
>>> such point, and thus you are agreeing to that fact.
>>>
>>> You can only have a first point in the open interval if the interval
>>> has only a finite number of points,
>>
>> No, that is your big mistake. In the interval [0, 1] there is a point
>> next to 0 and a point next to 1, and infinitely many are beteen them.
>
> Define 'next' in this context.

The context is "continuous domains",
there are multiple models of continuous domains,
one of them is "iota-values" or "line-reals",
which is a model of a contiguity so fine as
a model of continuity, where it's, "EF(1)".

Of course, the models of continuous domains are
distinct as with regards to their definitions of
continuity and completeness of operations, so
it entails a bit of book-keeping to keep things.

Oh, you don't have one of those, ..., well, you
can always look to Aristotle, who has at least
two, and Zeno's always looking for how to arrive
at not being a fool, then fast-forward to Bishop
and Cheng who constructively go about making it
so, and for topology there's Vickers who helps
reflect that in topology there are various topologies
not necessarily the standard open topology, in case
you're thorough about these matters and want to
help square away various models of continuity,
continuous domains, continuous topologies their
own first and final, Cantor space, and law(s) of
large numbers.