Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness)

On 09/15/2024 03:07 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.

Ah, good sir, then I'd like you to consider a representation
of real numbers as with an integer part and a non-integer
part, the integer part of the integers, and the non-integer
part a value in [0,1], where the values in [0,1], are as
of this model of (a finite segment of a) continuous domain,
these iota-values, line-reals, as so established as according
to the properties of extent, density, completeness, and measure,
fulfilling implementing the Intermediate Value Theorem, thus
for if not being the complete-ordered-field the field-reals,
yet being these iota-values a continuous domain [0,1] these
line-reals.
Then, in these, there is previous and next, yet of course
they're quite indistinguishable from otherwise the topology's
neighborhoods' points except being present in all neighborhoods.
So, in this manner, is made more _replete_, what results from
domains that are _complete_ (with extent, density, completeness,
and measure), models of continuous domains, real numbers, as
regards to the Linear Continuum, and Hardy's usual equivalence
of "real numbers" and "points", here variously _in_, _on_,
or _about_, the line.
These real numbers as integer-part/non-integer-part aren't
simply exchangeable, yet the measure, completeness, density,
extent maintain being so for each.
Of course I could have told you this about thirty years
ago, yet it's about 2014 since when the formalism is
quite all in place.