On 02/07/2024 12:38 PM, Mild Shock wrote:
> Lame and fake nonsense.
>
> Ross Finlayson schrieb am Donnerstag, 18. Januar 2024 um 18:12:20 UTC+1:
>> Now I'm very much enjoying your introduced notation, and as well particularly
>> your attention to quantifier disambiguation, I generally frame it in this sort
>> of way: that the universal quantifier has various forms for-any/for-each/for-every/for-all,
>> that these are "four different for's", that in some conditions are same and others different.
>
> If you want four diferent quantifiers, how about
> possibilistic and actual quantifiers:
>
>
> Types, Tableaus and Gödel’s God in Isabelle/HOL
> September 13, 2023 - David Fuenmayor and Christoph Benzmüller
> - Barcan Formulas are satisfied for standard (possibilist) quantifiers:
> - Barcan Formulas not satisfied for actualist quantifiers:
> https://www.isa-afp.org/browser_info/current/AFP/Types_Tableaus_and_Goedels_God/document.pdf
>

An, "ontological argument for the existence of G-d", has,
that the deity is super-logical.

One of the ideas of "humility" or hubris, is that hubris
is one of these words kind of like "entropy". On the one
hand, it usually means one thing, on the other, its opposite.
So, whether "hubris" is "vanity" or "humility", is generally
where vanity's considered bad and humility's considered virtuous,
with respect to concepts of deity.

What there may be is "an ontological argument for the
existence of a teleological argument, and vice-versa",
with the idea that Kant left "sublimity" for us while
Hegel left "a teleological argument", then that the
"neo-Kantians" or "neo-Hegelians" sort of have neither.

Kind of like "neo-Platonists", they're missing the main gist.

If relations are always relevant to a domain, then predicates
are always contingent their domains, as with respect to a
universe of objects, the existence of their relations "truth values".

There's a certain notion of "voluntary submission", which is among
reasons why some thinkers have that the fundamental question of
metaphysics, has its own answer in "axiomless natural deduction".

So I'm reading through this paper,

"This meta-logical concept [existence relativization]
does not appear in our object language."

If you've heard of Kunen inconsistency, it sort of has a
simple notion of a "universal complement", that for a
universe of objects, "the object" is the same thing as
"the universe \ the object".

This works better in an extra-ordinary theory, with
free comprehension of logical objects, after a domain
of discourse, that is a Comenius language where only
and all truisms are well-formed, all and purely logical.

So anyways Kant left us _sublimity_ and Hegel left
us _teleology_ so go away unless you acknowledge
_extra-ordinary_ theory.

Those authors' "computer formalized G-dlike" is
just a contradiction that they hope to assume.

Monism is a usual course after Platonism.
It sort of presumes a, "true theory", what's only,
"a theory of truth". Deductively, such a thing
is a constructivist's perfection.

Now, when you get to Cohen's Independence of
the Continuum Hypothesis, you'll find he makes
us of a neat sort of idea that in the ubiquitous
ordinals, that it's extra-ordinary, helping lever
open the Continuum Hypothesis, and indeed a sort
of approach for things like Kunen Inconsistency
as addressed above, and other universal concerns,
in the ontological and teleological together, logically.

Then, about quantifier disambiguation, it helps leaves
the statements of many forms of relations so disambiguated,
neatly and in-line.

So anyways, Kant left us _sublimity_ and Hegel left
us _teleology_, which makes quite an _extra-ordinary theory_.

Mink Oil / Rub it in your glove

Le 07/02/2024 à 13:29, Richard Damon a écrit :
> On 2/7/24 2:24 AM, WM wrote:

>> They are even points at the real line in linear order. Hence a first
>> exists.
>
> Nope. Since there is no first positive rel point either, there still
> isn't a FIRST unit fraction.

That is wrong in case of actual infinity. Without a first point there are
no points.

Regards, WM

Le 07/02/2024 à 19:59, Jim Burns a écrit :
> On 2/7/2024 2:35 AM, WM wrote:

>> Therefore we have the alternative choice:
>> either
>> every ordered set of points 1/n on
>> positive the real line,
>> having gaps of uncountably many points
>> between each other,
>> has a first point
>> or
>> for every point 1/n
>> there exists a point 1/(n+1).
>> Both together is impossible.
>> I prefer the first.
>
> It is the second alternative,
> | for each ⅟n exists ⅟n⁺¹ < ⅟n
> which supports the claim that
> no unit.fraction is smallest of them,
> which in turn is supported by
> ∀ n ∈ ℕ ∃ n+1 ∈ ℕ and
> n⋅⅟n⋅⅟n⁺¹ < n⁺¹⋅⅟n⋅⅟n⁺¹
>
> Your argument appears to begin and end
> at "I prefer the first".

NUF(x) cannot increase by more than 1 per point because all unit fractions
are points which have giant distances, namely uncountably many points.

This is disproving your choice. However, also your choice implies
infinitely many unit fractions which cannot be identified by any eps.

Regards, WM

On 2/8/24 5:59 AM, WM wrote:
> Le 07/02/2024 à 13:29, Richard Damon a écrit :
>> On 2/7/24 2:24 AM, WM wrote:
>
>>> They are even points at the real line in linear order. Hence a first
>>> exists.
>>
>> Nope. Since there is no first positive rel point either, there still
>> isn't a FIRST unit fraction.
>
> That is wrong in case of actual infinity. Without a first point there
> are no points.
>
> Regards, WM
>

Nope, we KNOW there are points, as we can name them.

Your logic just doesn't works for this case, an infinite unbounded set.

What is the first point on a circle? or do those not exist?

Le 08/02/2024 à 13:35, Richard Damon a écrit :
> On 2/8/24 5:59 AM, WM wrote:
>> Le 07/02/2024 à 13:29, Richard Damon a écrit :
>>> On 2/7/24 2:24 AM, WM wrote:
>>
>>>> They are even points at the real line in linear order. Hence a first
>>>> exists.
>>>
>>> Nope. Since there is no first positive rel point either, there still
>>> isn't a FIRST unit fraction.
>>
>> That is wrong in case of actual infinity. Without a first point there
>> are no points.

> Nope, we KNOW there are points, as we can name them.

You cannot, remember the missing eps for almost all of them.
>
> Your logic just doesn't works for this case, an infinite unbounded set.
>
> What is the first point on a circle? or do those not exist?

Here we have no circle but a linear problem containing points with large
distances.

Regards, WM

Fritz Feldhase schrieb am Mittwoch, 7. Februar 2024 um 16:21:07 UTC+1:
> On Wednesday, February 7, 2024 at 8:24:41 AM UTC+1, WM wrote:
>
> > [The unit fractions] are points at the real line in linear order. Hence a
> first exists.
>
> The integers are points at the real line in linear order. Hence a first exists?

Yes, that is proved by the unit fractions.

Regards, WM

Fritz Feldhase schrieb am Mittwoch, 7. Februar 2024 um 16:43:22 UTC+1:
> On Wednesday, February 7, 2024 at 8:35:37 AM UTC+1, WM wrote:

> | The latter is excluded by
> | ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
> und nun (nach einer Rückfrage):
> | It would be excluded by
> | ∀ n ∈ ℕ ∃ n+1 ∈ ℕ.
> It "would be" excluded?

The first "latter" concerns ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0 excludes an
increase of NUF by more than 1.
The second "latter" concerns ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0 does not
prove that for every n there is n+1.
>
> Hinweis: Es gilt im Kontext der klassischen Mathematik in der Tat:
>
> | ∀n ∈ ℕ: ∃m ∈ ℕ: m = n+1.
>
But in the same context every unit fraction is followed by a gap excluding
the increase of NUF by more than 1 in a point.

> > we have the alternative choice:
> > either
> > every ordered set of points 1/n on the positive real line, having gaps of
> > uncountably many points between each other, has a first point
> > or
> > for every point 1/n there exists a point 1/(n+1).
> > Both together is impossible. I prefer the first.

> Great, Mückenheim. MATHEMATICS prefers the second (see Peano-Axioms).

That is only because no mathematician hitherto has considered the second.

Regards, WM
>

On 02/08/2024 04:35 AM, Richard Damon wrote:
> On 2/8/24 5:59 AM, WM wrote:
>> Le 07/02/2024 à 13:29, Richard Damon a écrit :
>>> On 2/7/24 2:24 AM, WM wrote:
>>
>>>> They are even points at the real line in linear order. Hence a first
>>>> exists.
>>>
>>> Nope. Since there is no first positive rel point either, there still
>>> isn't a FIRST unit fraction.
>>
>> That is wrong in case of actual infinity. Without a first point there
>> are no points.
>>
>> Regards, WM
>>
>
> Nope, we KNOW there are points, as we can name them.
>
> Your logic just doesn't works for this case, an infinite unbounded set.
>
> What is the first point on a circle? or do those not exist?

Any one of them is, if you go all the way around.

If you consider a spiral space-filling curve from the origin,
its outer shell in 2-D is a circle, its first point sort of
connecting the origin to the first and last point (1, 0), say.

Now you might aver "well that's not a real function". Functions,
over time in the history of mathematics, have seen some great
changes in definition from what were classical functions to today
what are Cartesian functions and to what are continuous functions
especially among continuous domains, what are differentiable
functions and integrable functions according to their parametric
forms and their spaces, then these days the most famous "not-a-
real-function" is probably the Dirac delta from the subfield of
the distribution theory, these days, of course as it's usually
"modeled as an infinite or continuum limit of a family of real
functions".

I think that if you want to look like you know about the continuous
and discrete, and function theory, and real analysis, and then
especially that this context is about non-standard real analysis,
and it is a thing and wirklich, then you need some essential
formalisms that fit in explanations/apologetics with ALL the
canon and ALL the wings of "Hilbert's Infinite Living Museum
of Mathematics" and ALL the mathematical objects of ALL the
universe of mathematical objects.

It's sort of necessary to at least have the sorts of terms,
for points and spaces, and the point along out through all
the infinite-dimensional Euclidean spaces, R^N, so that
the simple act of line-drawing from zero to one, is the
first and simplest sort of action that results a tally,
a mark, a counting, the numbers'.

I think you have something better in both your knowledge
of mathematical objects and foundations of both the extra-
ordinary and simply ordinary, than brick-batting and block-butting
in an obstrepious manner.

So, non-standard elements _do_ have a foundational basis,
and non-standard elements _do_ have a milieu and setting,
and non-standard elements _are_ sensible, and quite directly
for example that quantum mechanics and particle/wave duality,
is about the continuous and discrete, and mathematics _owes_
physics the mathematics of this real/wirklich "extra-standard".

And INFINITY.

On 2/8/2024 6:39 AM, WM wrote:
> Le 07/02/2024 à 19:59, Jim Burns a écrit :
>> On 2/7/2024 2:35 AM, WM wrote:

>>> Therefore we have the alternative choice:
>>> either
>>> every ordered set of points 1/n on
>>> positive the real line,
>>> having gaps of uncountably many points
>>> between each other,
>>> has a first point
>>> or
>>> for every point 1/n
>>> there exists a point 1/(n+1).
>>> Both together is impossible.
>>> I prefer the first.
>>
>> It is the second alternative,
>> | for each ⅟n exists ⅟n⁺¹ < ⅟n
>> which supports the claim that
>> no unit.fraction is smallest of them,
>> which in turn is supported by
>> ∀ n ∈ ℕ ∃ n+1 ∈ ℕ and
>> n⋅⅟n⋅⅟n⁺¹ < n⁺¹⋅⅟n⋅⅟n⁺¹
>>
>> Your argument appears to begin and end
>> at "I prefer the first".
>
> NUF(x) cannot increase by
> more than 1 per point
> because
> all unit fractions are
> points which have giant distances, namely
> uncountably many points.
>
> This is disproving your choice.

> However, also your choice implies
> infinitely many unit fractions which
> cannot be identified by any eps.

Each unit.fraction ⅟i′ is preceded by more than
any final.ordinal.k.many unit.fractions ⅟k′
such that each ⅟k′ is NOT followed by more than
_any_ final.ordinal.j.many unit.fractions ⅟j′

∀⅟i′ ∈ ⅟ℕ:
∀k: Final(k) ⇒ ∃ᵏ⁺¹ ⅟k′ ∈ ⅟ℕ:
(0 < ⅟k′ < ⅟i′ ∧
¬∀j: Final(j) ⇒ ∃ʲ⁺¹ ⅟j′ ∈ ⅟ℕ:
(⅟k′ < ⅟j′ < ⅟1))

Each unit.fraction ⅟i′ is
infinitely.preceded by unit.fractions ⅟k′
each ⅟k′ finitely.followed by unit.fractions ⅟j′

A _final ordinal_ k cannot fit Bob in too.
Final(k) ⟺
{α|α<k} ⃒⇇ {Bob}∪{α|α<k} ⟺
¬∃f: {α|α<k}∪{Bob} ⇉ {α|α<k}:
β≠γ ⇒ f(β)≠f(γ)

After all ordinals which Bob cannot fit in,
Bob can fit in.
Final ᣔ<ᣔ ¬Final

For each unit.fraction ⅟j′
Bob can fit in ahead of ⅟j′ and
Bob cannot fit in behind ⅟j′

∀⅟j′ ∈ ⅟ℕ:
{⅟k′|⅟k′<⅟j′}∪{Bob} ⇉ {⅟k′|⅟k′<⅟j′} ∧
{⅟i′|⅟j′<⅟i′} ⃒⇇ {Bob}∪{⅟i′|⅟j′<⅟i′}

On 2/8/2024 6:39 AM, WM wrote:
> Le 07/02/2024 à 19:59, Jim Burns a écrit :
>> On 2/7/2024 2:35 AM, WM wrote:

>>> Therefore we have the alternative choice:
>>> either
>>> every ordered set of points 1/n on
>>> positive the real line,
>>> having gaps of uncountably many points
>>> between each other,
>>> has a first point
>>> or
>>> for every point 1/n
>>> there exists a point 1/(n+1).
>>> Both together is impossible.
>>> I prefer the first.
>>
>> It is the second alternative,
>> | for each ⅟n   exists ⅟n⁺¹ < ⅟n
>> which supports the claim that
>> no unit.fraction is smallest of them,
>> which in turn is supported by
>> ∀ n ∈ ℕ ∃ n+1 ∈ ℕ  and
>> n⋅⅟n⋅⅟n⁺¹ < n⁺¹⋅⅟n⋅⅟n⁺¹
>>
>> Your argument appears to begin and end
>> at "I prefer the first".
>
> NUF(x) cannot increase by
> more than 1 per point
> because
> all unit fractions are
> points which have giant distances, namely
> uncountably many points.

<WM>
> Either NUF increases in one point to 1
> or to more.
> The latter is excluded by
> ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
> No quantifier interchange is used.
<WM/>

Elaborate on why
the unit.fraction.less gap G[⅟k] which
no unit.fraction ⅟k lacks
| ¬∃⅟k ∈ ⅟ℕ: ¬∃G[⅟k]
| ∀⅟k ∈ ⅟ℕ: ∃G[⅟k]

excludes 0 ∉ ⅟ℕ
which lacks a unit.fraction.less gap
| ¬∃G[0]
from being a point at which
NUF(x) increases by more than 1.

Keep in mind that, in general,
nope[ ∀j:∃k:P(j,k) ⊢ ∃k:∀j:P(j,k) ]
and
nope[ ¬∃j:¬∃k:P(j,k) ⊢ ¬∃k:¬∃j:P(j,k) ]

Moreover,
for _anti-symmetric_ P(j,k) ⇒ ¬P(k,j),
(see also j<k ⇒ ¬(k<j))
the opposite of that:
| anti-sym P, ∀j:∃k:P(j,k) ⊢ ¬∃k:∀j:P(j,k)
and
| anti-sym P, ¬∃j:¬∃k:P(j,k) ⊢ ¬∃k:¬∃j:¬P(j,k)

> This is disproving your choice.
> However, also your choice implies
> infinitely many unit fractions which
> cannot be identified by any eps.

On 2/8/2024 4:35 AM, Richard Damon wrote:
> On 2/8/24 5:59 AM, WM wrote:
>> Le 07/02/2024 à 13:29, Richard Damon a écrit :
>>> On 2/7/24 2:24 AM, WM wrote:
>>
>>>> They are even points at the real line in linear order. Hence a first
>>>> exists.
>>>
>>> Nope. Since there is no first positive rel point either, there still
>>> isn't a FIRST unit fraction.
>>
>> That is wrong in case of actual infinity. Without a first point there
>> are no points.
>>
>> Regards, WM
>>
>
> Nope, we KNOW there are points, as we can name them.
>
> Your logic just doesn't works for this case, an infinite unbounded set.
>
> What is the first point on a circle? or do those not exist?

Good one. Usually I get "my" first point wrt a unit circle via:

x = cos(0)
y = sin(0)

;^D

On 2/8/24 7:52 AM, WM wrote:
> Le 08/02/2024 à 13:35, Richard Damon a écrit :
>> On 2/8/24 5:59 AM, WM wrote:
>>> Le 07/02/2024 à 13:29, Richard Damon a écrit :
>>>> On 2/7/24 2:24 AM, WM wrote:
>>>
>>>>> They are even points at the real line in linear order. Hence a
>>>>> first exists.
>>>>
>>>> Nope. Since there is no first positive rel point either, there still
>>>> isn't a FIRST unit fraction.
>>>
>>> That is wrong in case of actual infinity. Without a first point there
>>> are no points.
>
>> Nope, we KNOW there are points, as we can name them.
>
> You cannot, remember the missing eps for almost all of them.

What's missig about them?

For every one of them, I can find an eps that is smaller, and for every
eps I can find a unit fraction that is smaller.

They keep getting smaller all the way down with no end.

>>
>> Your logic just doesn't works for this case, an infinite unbounded set.
>>
>> What is the first point on a circle? or do those not exist?
>
> Here we have no circle but a linear problem containing points with large
> distances.

No, we have a linear problem with boundlessly small distances, so there
is no smallest.

>
> Regards, WM
>
>

Le 08/02/2024 à 19:53, Jim Burns a écrit :
> On 2/8/2024 6:39 AM, WM wrote:

>> NUF(x) cannot increase by
>> more than 1 per point
>> because
>> all unit fractions are
>> points which have giant distances, namely
>> uncountably many points.

Can you understand this?
>>
>> This is disproving your choice.
>
>> However, also your choice implies
>> infinitely many unit fractions which
>> cannot be identified by any eps.
>
> Each unit.fraction ⅟i′ is preceded

Stop to waffle. I know your argument.
Can you understand the above?

Regards, WM

Le 08/02/2024 à 19:53, Jim Burns a écrit :
> On 2/8/2024 6:39 AM, WM wrote:

>> Either NUF increases in one point to 1
>> or to more.
>> The latter is excluded by
>> ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
> Elaborate on why

Not before you have understood the above argument.

Regards, WM

Le 09/02/2024 à 04:40, Richard Damon a écrit :
> On 2/8/24 7:52 AM, WM wrote:

>>
>>> Nope, we KNOW there are points, as we can name them.
>>
>> You cannot, remember the missing eps for almost all of them.
>
> What's missig about them?

You cannot define ℵ smallest existing unit fractions.
>
> For every one of them, I can find an eps that is smaller, and for every
> eps I can find a unit fraction that is smaller.

Bigmout. Liar.

> No, we have a linear problem with boundlessly small distances, so there
> is no smallest.

Impossible for NUF(x). NUF(0) = 0 and NUF(x>0) > 0 cannot exist without
NUF(x) = 1 because between all points 1/n there are uncountably many
points x with NUF(x) constant.

Regards, WM

On 2/9/24 4:59 AM, WM wrote:
> Le 09/02/2024 à 04:40, Richard Damon a écrit :
>> On 2/8/24 7:52 AM, WM wrote:
>
>>>
>>>> Nope, we KNOW there are points, as we can name them.
>>>
>>> You cannot, remember the missing eps for almost all of them.
>>
>> What's missig about them?
>
> You cannot define ℵ smallest existing unit fractions.

Well, there is no number ℵ (that would be a name for a set of values)

And I can define the infinite set of the smallest unit fractions:

1/n, 1/(n+1), 1/(n+2), 1/(n+3), ... 1/(n+k), ...

>>
>> For every one of them, I can find an eps that is smaller, and for
>> every eps I can find a unit fraction that is smaller.
>
> Bigmout. Liar.

Good self-discription.

Give me a number it can't be done with.

>
>> No, we have a linear problem with boundlessly small distances, so
>> there is no smallest.
>
> Impossible for NUF(x). NUF(0) = 0 and NUF(x>0) > 0 cannot exist without
> NUF(x) = 1 because between all points 1/n there are uncountably many
> points x with NUF(x) constant.

Bcause your definition of NUF(x) is broken.

Note, you said BETWEEN points, and that only applies to points between
unit fractions, not below all of them.

Since there isn't a point below all the unit fractions and above 0,
there is no constant point for NUF(x) to be 1.

Your space has a discontinuity in definition, where NUF(x) jumps from
its defined value of 0, to its poorly defined value of infinity. (since
you claim to be working in the finite space, it can't have the value of
ℵ, since that isn't a finite value)

>
> Regards, WM
>
>
>

On 2/9/2024 4:44 AM, WM wrote:
> Le 08/02/2024 à 19:53, Jim Burns a écrit :
>> On 2/8/2024 6:39 AM, WM wrote:

>>> Either NUF increases in one point to 1
>>> or to more.
>>> The latter is excluded by
>>> ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
>
>> Elaborate on why
>
> Not before you have understood
> the above argument.

Until you elaborate,
you do not have an argument.

https://sneltraining.nl/then-a-miracle-occurs/
| | "then a miracle occurs"
| | I think you should be more explicit
| here in step two"

You have _at best_
in what's far too _generous_ a framing
a demand that _someone else_ make your argument
_for you_

I _understand_ that the argument you owe
can't be made.
Your argument _not.exists_ Wolfgang Mückenheim of
Hochschule für angewandte Wissenschaften Augsburg,

>>> Either NUF increases in one point to 1
>>> or to more.
>>> The latter is excluded by
>>> ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.

At each unit.fraction ⅟n
yes,
there is one other[1]
unit.fraction.to.the.left
from left.of.⅟n to right.of.⅟n

However,
0 is not a unit.fraction, and
0 is where NUF(x) jumps from 0 to ℵ₀
| ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
doesn't even begin to talk about
the ℵ₀.jump.

The difference between ⅟n and 0 is that
around ⅟n is a small open interval,
(left.of.⅟n,⅟n,right.of.⅟n)
such that, in that small open interval,
no unit.fractions are left.of.⅟n
no unit.fractions are right.of.⅟n

Thus,
for each _unit.fraction_ ⅟n
there is one other[1]
unit.fraction.to.the.left
from left.of.⅟n to right.of.⅟n

In contrast,
around _non.unit.fraction_ 0,
no small open interval like that exists.

For each right.of.0 (0,β)
there is a unit.fraction ⅟1⁺ᵐᵝ in (0,β)
mᵦ ≤ ⅟β < 1⁺ᵐᵝ

For each final ordinal k
there are more than k in (0,β),
⅟1⁺ᵐᵝ ⅟2⁺ᵐᵝ ... ⅟k⁺ᵐᵝ ⅟(k+1)⁺ᵐᵝ

[1]
At each unit.fraction ⅟n
yes,
there is one other
unit.fraction.to.the.left
from left.of.⅟n to right.of.⅟n

I say "other" instead of "more"
because,
for unit.fractions.to.the.left,
including one other _isn't_
including one more.

For a final.ordinal,
including one other _is_
including one more.

However,
unit.fractions.to.the.left
_aren't_ any final ordinal k
⅟1⁺ᵐᵝ ⅟2⁺ᵐᵝ ... ⅟k⁺ᵐᵝ ⅟(k+1)⁺ᵐᵝ
and
one other isn't one more.

Le 09/02/2024 à 13:19, Richard Damon a écrit :
> On 2/9/24 4:59 AM, WM wrote:
>> Le 09/02/2024 à 04:40, Richard Damon a écrit :
>>> On 2/8/24 7:52 AM, WM wrote:
>>
>>>>
>>>>> Nope, we KNOW there are points, as we can name them.
>>>>
>>>> You cannot, remember the missing eps for almost all of them.
>>>
>>> What's missig about them?
>>
>> You cannot define ℵ smallest existing unit fractions.
>
> Well, there is no number ℵ (that would be a name for a set of values)

I use it for infinitely many.
>
> And I can define the infinite set of the smallest unit fractions:
>
> 1/n, 1/(n+1), 1/(n+2), 1/(n+3), ... 1/(n+k), ...

ℵ are missing.
>>
>> Impossible for NUF(x). NUF(0) = 0 and NUF(x>0) > 0 cannot exist without
>> NUF(x) = 1 because between all points 1/n there are uncountably many
>> points x with NUF(x) constant.

> Your space has a discontinuity in definition, where NUF(x) jumps from
> its defined value of 0, to its poorly defined value of infinity. (since
> you claim to be working in the finite space, it can't have the value of
> ℵ, since that isn't a finite value)

It can't jump from 0 to more than 1.

Regards, WM

Le 09/02/2024 à 17:06, Jim Burns a écrit :
> On 2/9/2024 4:44 AM, WM wrote:

>>>> Either NUF increases in one point to 1
>>>> or to more.
>>>> The latter is excluded by
>>>> ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
>
> At each unit.fraction ⅟n
> yes,

Only they are counted.

> However,
> 0 is not a unit.fraction,

Therefore NUF(0) = 0

> and
> 0 is where NUF(x) jumps from 0 to ℵ₀

Impossible because NUF jumps only at unit fractions, by definition.

Regards, WM

Fritz Feldhase schrieb am Freitag, 9. Februar 2024 um 19:54:29 UTC+1:

> His "argument" is somehow like "the finite distances (>0) between any 2
> 'successive' unit fractions do not allow for infinitely many unit fractions in (0,
> x] [for all x > 0]", or so.

Nonsense. Correct is: The finite distances force NUF(x) to pause after
every increase by 1.

Regards, WM

On 2/10/2024 1:18 PM, WM wrote:
> Le 09/02/2024 à 17:06, Jim Burns a écrit :

>> and
>> 0 is where NUF(x) jumps from 0 to ℵ₀
>
> Impossible because
> NUF jumps only at unit fractions,
> by definition.

Probably what you (WM) mean by
| ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0
is
| For each unit.fraction ⅟k
| exists small.enough β > 0 such that
| there is only one other unit.fraction < ⅟k+β
| than there is ⅟k-β

Yes, but
for point 0 (not a unit.fraction)
for each β > 0 (none are too small)
unit.fractions < 0-β are 0.many
and
unit.fractions < 0-β are more than any final k

0 < mᵦ ≤ ⅟β < mᵦ+1 = 1⁺ᵐᵝ
β > ⅟1⁺ᵐᵝ > ... > ⅟k⁺ᵐᵝ > ⅟(k+1)⁺ᵐᵝ > 0

Not( exists small.enough β > 0 such that
there is only one other unit.fraction < 0+β
than there is 0-β )

Moreover,
not( exists small.enough β > 0 such that
there are only _two_ other unit.fraction < 0+β
than there is 0-β )

Not( exists small.enough β > 0 such that
there are only k.many other unit.fraction < 0+β
than there is 0-β )
for any final ordinal k

The jump NUF(x) makes at 0 is more than 1, more than 2,
more than any final ordinal k.

The jump NUF(x) makes at 0 isn't a final ordinal.

>> and
>> 0 is where NUF(x) jumps from 0 to ℵ₀
>
> Impossible because
> NUF jumps only at unit fractions,
> by definition.

Your definition is incorrect.

What is correct is that
NUF(x) jumps _near_ unit fractions.

0 is _near_ unit fractions.

On 2/10/2024 12:35 PM, WM wrote:
> Le 09/02/2024 à 13:19, Richard Damon a écrit :
>> On 2/9/24 4:59 AM, WM wrote:

>>> You cannot define
>>> ℵ smallest existing unit fractions.
>>
>> Well, there is no number ℵ
>> (that would be a name for a set of values)
>
> I use it for infinitely many.

Unicode U+221E INFINITY ∞

On 2/10/2024 12:35 PM, WM wrote:
> Le 09/02/2024 à 13:19, Richard Damon a écrit :

>> And I can define
>> the infinite set of the smallest unit fractions:
>> 1/n, 1/(n+1), 1/(n+2), 1/(n+3), ... 1/(n+k), ...
>
> ℵ are missing.

A set from which any unit fraction ≤ ⅟n is missing
isn't the set of unit.fractions ≤ ⅟n

> It can't jump from 0 to more than 1.

0 < mᵦ ≤ ⅟β < mᵦ+1 = 1⁺ᵐᵝ
β > ⅟1⁺ᵐᵝ > ... > ⅟k⁺ᵐᵝ > ⅟(k+1)⁺ᵐᵝ > 0

On 2/10/24 12:35 PM, WM wrote:
> Le 09/02/2024 à 13:19, Richard Damon a écrit :
>> On 2/9/24 4:59 AM, WM wrote:
>>> Le 09/02/2024 à 04:40, Richard Damon a écrit :
>>>> On 2/8/24 7:52 AM, WM wrote:
>>>
>>>>>
>>>>>> Nope, we KNOW there are points, as we can name them.
>>>>>
>>>>> You cannot, remember the missing eps for almost all of them.
>>>>
>>>> What's missig about them?
>>>
>>> You cannot define ℵ smallest existing unit fractions.
>>
>> Well, there is no number ℵ (that would be a name for a set of values)
>
> I use it for infinitely many.

So, not a "number", since you reject "Mathelogical" definitions.

"Natural Mathematics" doesn't have "infinite" as a "Value" only a
concept that you can approach (because we are finite, and thus can't
"see" infinity).

Normally indicated with the symbol ∞

>>
>> And I can define the infinite set of the smallest unit fractions:
>>
>> 1/n, 1/(n+1), 1/(n+2), 1/(n+3), ... 1/(n+k), ...
>
> ℵ are missing.

Which ones?

since n+k gets unboundedly large, 1/(n+k) gets unboundedly small, so no
unit fraction gets missed.

>>>
>>> Impossible for NUF(x). NUF(0) = 0 and NUF(x>0) > 0 cannot exist
>>> without NUF(x) = 1 because between all points 1/n there are
>>> uncountably many points x with NUF(x) constant.
>
>> Your space has a discontinuity in definition, where NUF(x) jumps from
>> its defined value of 0, to its poorly defined value of infinity.
>> (since you claim to be working in the finite space, it can't have the
>> value of ℵ, since that isn't a finite value)
>
> It can't jump from 0 to more than 1.

Why not?

The other question, can it actually exist, since ℵ is not a "number" in
the finite number systems (Natural Numbers, Rationals or Reals).

To have the value of ℵ as an actual number, you need to accept some
"mathelogical" framework that defines it, as "infinity" isn't a natural
mathematical value.

> Regards, WM

On 2/10/24 1:18 PM, WM wrote:
> Le 09/02/2024 à 17:06, Jim Burns a écrit :
>> On 2/9/2024 4:44 AM, WM wrote:
>
>>>>> Either NUF increases in one point to 1
>>>>> or to more.
>>>>> The latter is excluded by
>>>>> ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0.
>>
>> At each unit.fraction ⅟n
>> yes,
>
> Only they are counted.
>
>> However,
>> 0 is not a unit.fraction,
>
> Therefore NUF(0) = 0
>
>> and
>> 0 is where NUF(x) jumps from 0 to ℵ₀
>
> Impossible because NUF jumps only at unit fractions, by definition.
>
> Regards, WM

You only have an "Natural Language" definition of NUF(x). You haven't
shown that it can actually exist.

Your definition creates a contradiction, that NUF(x) will have the value
1 at the smallest unit fraction, but there isn't a smallest unit
fraction, so you don't have a workable definition.

That is sort of like asking, what is the truth value for the following
statement:

This statement is false.

Assuming you have NUF(x), breaks your mathematics.