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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? (repleteness)
Date: Sun, 15 Sep 2024 15:18:01 -0700
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On 9/15/2024 3:07 PM, FromTheRafters wrote:
> on 9/15/2024, Ross Finlayson supposed :
>> On 09/15/2024 11:03 AM, FromTheRafters wrote:
>>> After serious thinking Ross Finlayson wrote :
>>>> On 09/14/2024 09:27 AM, Ross Finlayson wrote:
>>>>> 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.
>>>>>
>>>>>
>>>>
>>>> "What, no witty rejoinder?"
>>>
>>> What you said has no relation to the 'nextness' of elements in discrete
>>> sets. What is 'next' to Pi+2 in the reals?
>>
>> In the, "hyper-reals", it's its neighbors,
>> in the line-reals, put's previous and next,
>> in the field-reals, there's none,
>> and in the signal-reals, there's nothing.
>>
>> Or, you know, "noise".
>>
>> Of course the hyper-reals are said to be only
>> a "conservative" meaning "meaningless" extension
>> to the standards, so, only the line-reals say
>> anything about it at all.
>>
>> Except nothing, ....
>>
>>
>> I wonder what you think of something like Hilbert's
>> "postulate of continuity" for geometry, as with
>> regards to that in the course-of-passage of
>> the growth of a continuous quantity, it encounters,
>> in order, each of the points in the line.
>>
>> Just ignore it?
> 
> What is the successor function on the reals? Give me that, and maybe we 
> can find the 'next' number greater than Pi.

define a granularity and you can explore some numbers. However this is 
finite and misses infinite points. They are not countable and are 
infinitely dense. This is just a finite view based on a given granularity:

// kind of small... ;^)
granularity = (10^(-696), 0, 0)

p0 = (-1, 0, 0)
p1 = (1, 0, 0)
pdif = p1 - p0;

cur = p0;

// start at cur and go off into infinity and beyond! ;^)
for (;;)
{
    next = cur + granularity
    plot_line(cur, next);
    cur = next;
}


;^)