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

On 09/07/2024 09:27 AM, Richard Damon wrote:

On 9/7/24 12:14 PM, Ross Finlayson wrote:

On 09/07/2024 08:34 AM, Ross Finlayson wrote:

On 09/07/2024 05:57 AM, Richard Damon wrote:

On 9/6/24 11:00 PM, Ross Finlayson wrote:

On 09/06/2024 07:01 PM, Richard Damon wrote:

On 9/6/24 8:07 AM, WM wrote:

On 06.09.2024 05:08, Richard Damon wrote:

On 9/5/24 10:08 AM, WM wrote:

>

NUF(x) must grow. It cannot grow by more than 1 at any x.

>

This is NOT the "ancient" idea of infinity,

>
NUF(x) must grow. It cannot grow by more than 1 at any x.
Right or wrong in your opinion?
>
Regards, WM
>

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Only if it exists.
>
If it does, it must be counting some sub-finite values as "unit
fractions" that are not the reciprocal of the Natural Numbers (since
there is no smallest of those unit fractions to count from).
>
So, either it is counting some sub-finite values (actually a lot of
them
a countable infinity of them) or it just doesn't exist.
>
Maybe that is your dark numbers, these sub-finite numbers that are
reciprocals of some post-finite values above the infinite set of
Natural
Numbers (which have no upper bound) and are below Omega.

>
That's a remarkable supposition, I wonder how you'd imagine
both to satisfy to yourself and others that thusly is a
"consistent" form, of course which only requires "internal
consistency" for its own sake, then besides, to suffer the
running of the gauntlet, of those who'd insist it contradicted
theirs. For, their are simple inductive arguments that nothing
ever happens or is, at all.
>
I sort of appreciate the sentiment, though, that "infinite"
is big enough to have quite a range.
>
>

>
The problem with "consistancy" is that WM's mathematicss isn't
consistent with the full Natural Numbers, and unlikely to be helped
with
the addition of something even more esoteric.
>
His work doesn't define the set well enough to actually define how it
must work, and the best answer is likly to just adopt one of the
existing set of sub-finite number, it just needs to have a countably
infinite subset of values that can be reasonable defined as "unit
fractions".

>
>
Maybe it'd do better with less.
>
How about this, imagine it was your duty to convince a panel of
mathematicians that something that made for the most properties
possible of the notion of an iota-value or least-positive-rational,
had a way to define this thing. Is it any different than f(1) for
n/d, d-> oo, n -> d, modeling a not-a-real-function as a limit of
real functions?
>
Or does that just mean crazy-town to you? The crazy-town here
is actually sort of crazy-town, like, you walk out on the streets
and at various intervals encounter derelict indigents who are
entirely insane, on most given Tuesdays.
>
How do you keep your sanity in crazy-town, or help rehabilitate
crazy-town? Among the ideas that nothing can be crazy if it's
all consistent, in the infinite, get into things like Hausdorff's
constructible universe, and Skolem's countable universe, or,
"model of ZF", if "universe" makes no sound to you.
>
>

>
You might aver "nobody does that" and that
would be wrong - every day there's Dirichlet
and Poincare and Dirac and there's time-ordering
and making the derivation of Fourier series and
most usually an equi-partitioning of a unit interval
as according to a large number N in a sort of
reverse delta-epsilonics, in fact it suffices to
say that without such a notion of equi-partitioning
infinitely a unit interval there'd be no modularity
of distance at all.
>
In fact it was in the physics courses where it was
introduced "not-a-real-function yet with real
analytical character as modeled as a (continuum)
limit of real functions", Dirac delta, which is only
so _in the limit_ and just like calculus only perfect
and so _in the limit_, that it's first little bit is just
like its last little bit.
>
And non-zero, ....
>
>
And if you've never heard of that then congratulations,
here's a new word for your vocabulary.
>
>

>
Oh, I've heard, and used, the Dirac Delta Function, which is a "strange"
beast that falls at the edges of normality.
>
The issue that WM seems to miss is that as you move from the very normal
finite world of very simple arithmetic to the more esoteric world some
of the common just assumable properties go away, and intuition gets
broken, and you need to decide, do I keep my intuition, and need to stay
out of that realm, or am I willing to accept that some things break the
rules I am used to in order to gain the higher order properties of the
system.

Yes, WM has long demonstrated inconstancy and not meeting
the maxims of coming to terms for mathematical formalists.
("WM: the sockpuppet appelled Wolfgang Muckenheim of
the linguist's menagerie of mechanical cranks,
that is a soft-ball straw-man waste of the
conceptual space of the super-classical in the non-standard,
soft-ball in doing things stupid and straw-man in
doing things stupid in the higher conceptual space
of the super-classical in the non-standard.")
The envelope, of differential equations, is at once, for
example, undefined for differential equations and the
bounds for integral equations. While integral equations
and differential equations both build toward each other
and result in infinite limits some completions of each
other, like a regulus can draw a hypoerboloid sheet,
has that just like points and spaces, neither is the
other and according to plain inductive neither ever
arrives the other.
Then, f(n) = n/d, d -> oo, n -> d pretty much
"is" "standard" "infinitesimals".
Or, one of Aristotle's continuums if like many you hold
that Aristotle's always been right. I.e., Aristotle
already has at least _two_ models of continuous domains.
Then to Archimedes and Eudoxus and so on is given the
one there.
That there are multiple law(s) of large numbers, infinities,
in terms of there being more than one space of all sequences
of 0's and 1's, _logical_ models, _mathematical_ models,
_geometrical_ models, is a good thing.
At least you don't compound your ignorance with invincibility,
not that there's anything wrong with invincible.