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

On 09/28/2024 05:42 PM, FromTheRafters wrote:

Chris M. Thomasson formulated the question :

On 9/28/2024 3:04 AM, FromTheRafters wrote:

Chris M. Thomasson was thinking very hard :

On 9/27/2024 3:13 PM, FromTheRafters wrote:

WM expressed precisely :

On 25.09.2024 19:22, Richard Damon wrote:

On 9/25/24 11:14 AM, WM wrote:

NUF increases. At no point it can increase by more than 1.

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Why not?

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Because there is a finite gap between two unit fractions.

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There are no gaps in the set of unit fractions.

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He might be thinking

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But I rather doubt it.
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of the gap between say 1/1 and 1/2, its there. Think of the
"granularity" of _strict_ unit fractions for a moment.

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There are gaps with respect to a domain other than the unit
fractions. There are gaps in the set of primes with respect to the
set of naturals, that is, these gaps are filled with composite
numbers in the naturals.
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There are gaps in Q with respect to R which are filled with
irrationals thus making R complete (meaning no gaps).
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Then think of trying to fill the gap with unit fractions breaking
the strict rule for sure:
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1/2 + ((1/16*4 + 1/8*3 + 1/4 + 1/8) - 1/2) = 1
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;^)

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Here you've gone from unit fractions to all of Q+ I think. Yes, there
are gaps in Unit Fractions with respect to Q+ yet filling them with
other elements of Q+ does not 'complete' the set with respect to Q or R.
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Here, I'm thinking of gaps as meaning places where a cauchy sequence
does not converge to a member of the set but somehow would fit
between them in an extended domain which included that element.

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Wrt to R, there are no gaps, right? It's infinitely dense.
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However, _strictly_ confining ourselves to the strict _order_ of unit
fractions, well, there are gaps:
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1/1 - 1/2 = 1/2 gap
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No we know that there are infinitely many unit fractions smaller than
the gap for sure. However, using a strict order and strictly unit
fractions,
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1/1, 1/2, 1/3, 1/4, ...
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Well, we can't put anything between 1/1 and 1/2 for this would _break_
the rules wrt the word, strict...
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Humm... Sound okay to you? Thanks.

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If I understand you correctly, that seems fair. A gap would mean that an
element is missing from the sequence, since none are missing, there is
no gap.

Is Mueckenheim really worth mucking out his stall?
I don't know if that's the same word, yet the idea
is that if you have a horse and it's housed in a stall
it naturally fills it with its regular excrement from
getting feed and all, then as with regards to that of
course according to mechanical translation, Muecken
is mosquitoes, sort of like old J.G. was Chinese ideograms
for "infinite foul toot", same old bots.
Then, for iota-values, the idea of there being standard
infinitesimals, and of course if you've studied the
development of the theories of real analysis you know
that the raw differential muchly involves the notion
of the mathematical infinitesimal and it's certainly
part of the intuition of the continuous nature and
certainly no different in the calculus than what
arises as from vanishing differences, there are the
thousands of years of "The Philosophers" as with regards
to the "infinitely-divisible" and "infinitely-divided",
or the infinitely-Divisible or INfinitely-divisible,
about the continuous and atomism,
that it's so that "clock arithmetic" is a most usual
sort of notion of the course-of-passage of time,
and that modularity, is so built in equal elapsed times,
vis-a-vis, breaking down the integral, the integers,
and that modularity is both ways.
Then, it seems so easy to argue the one way, and it is,
in fact for example you can look to Aquinas with regards
to Aristotle's "un-moved mover" as with regards to things
moving at all, or Scotist's and their "yes, there sort of
is a first and last in the infinite if there's either",
that it needn't result contradiction nor impasse, when
a greater holistic theory isn't missing out either,
continuity or atomism.
Then that for mathematics there is "real non-standard analysis"
which is as of "standard infinitesimals", isn't merely the
geometric series of Zeno any-more, with that being yet classical,
and isn't merely Dirac's delta, a function/functional/distribution /
limit-of-functions/modeled-as-a-limit-of-functions / a function,
everyone's first and favorite non-standard function with real
analytical character, and isn't merely about the hypergeometric
and its three regular singular points 0, 1, and infinity,
and not just Jordan measure for the line segment before Vitali,
and not merely Vitali's what some have as non-measurable and
some have as doubling-measure, that it's quite all simple then
that when a function like simply for naturals n/d so standardly
modeled as a limit-of-real-functions then having real analytical
character as via a doubling-measure exactly between the discrete
and infinite and the continuous and unital, then that's where
it's perceived those are objects of mathematics.
The infinite, living, working Hilbert's Museum of All Mathematics,
is an idea after the "Hilbert's Hotel", which is more or less
into exercises of Dirichlet principle / pigeonhole principle,
has that, as you keep mucking, it's sort of resulting a
blocking and hazard to brighter and more central halls.
Or, it's like, the thought experiment about the staves and
the boat. So, the boat is made of staves, and over time,
the staves are replaced until each is un-original.
Is it the same boat?
Kind of like these posts. If you add them all up, is it non-zero?
Then, the usual idea is that canon and "The Philosophers"
is the existing non-zero, then whether anything's ever really new.
Here what's new is sort of lining up
clock arithmetic
Zeno's geometric series
Dirac's unit impulse function
Jordan measure of line integral, not Vitali's
Vitali's doubling measure
natural/unit equivalence function, Int f = 1
if merely helping out that it's clock arithmetic
again, and helping explain that doubling-measures
are integral in atomism, about the continuous
and discrete, according to The Philosophers.
(Mathematics)