On 2/20/24 3:12 AM, WM wrote:
> Le 19/02/2024 à 21:21, Jim Burns a écrit :
>> On 2/18/2024 4:00 PM, WM wrote:
>>> I will never give up the following self-evidence:
>>> If
>>> there are ℵo unit fractions in
>>> the interval (0, eps),
>>> then
>>> there is an x with only a finite number of
>>> unit fractions in (0, x).
>>> Why?
>>> Because unit fractions are real points on
>>> the real line.
>>> They cannot appear as
>>> an infinite swarm without a finite start.
>
>> In other words,
>> infinitely.many are leftward
>> finitely.many are rightward
>> for each ⅟j
>
> That correctly describes an evolving infinite collection, i.e., a
> potentially infinite set where more and more elements are created which
> initially have not existed.
> A complete, i.e. actually infinite set of ℵo real fixed points on the
> real axis can be subdivided by any of its elements (since all are
> existing) such that the subsets have cardinalities from 0, ℵo over n, ℵo
> to ℵo, n, and ℵo, 0.

But ℵo / n is still ℵo, so you never get to 0

You still don't understand the mathematics of Trans-finite values,
because you misused your finite principles.

>>
>> No ⅟j is in a finite start
>> That is the reason that
>> an infinite swarm is the only possibility
>
> Then you are not talking about really existing invariable points. But
> that is what I discuss.

But neither are you talking about what really exists, as you are using
invalid logic.

>
> Regards, WM

Le 20/02/2024 à 13:49, Richard Damon a écrit :
> On 2/20/24 3:15 AM, WM wrote:

>> But if all are there existing from the scratch as an actually infinite
>> set, then each one can be addressed as border between two subsets, in
>> principle. Even the smallest one.
>
> But there isn't a "Smallest One".
>
You are obviosuly wrong. If all are there, then all can be used to divide
the set into two parts. None is exempt.

Regards, WM

Le 20/02/2024 à 13:49, Richard Damon a écrit :
> On 2/20/24 3:12 AM, WM wrote:
>> Le 19/02/2024 à 21:21, Jim Burns a écrit :

>>> infinitely.many are leftward
>>> finitely.many are rightward
>>> for each ⅟j
>>
>> That correctly describes an evolving infinite collection, i.e., a
>> potentially infinite set where more and more elements are created which
>> initially have not existed.
>> A complete, i.e. actually infinite set of ℵo real fixed points on the
>> real axis can be subdivided by any of its elements (since all are
>> existing) such that the subsets have cardinalities from 0, ℵo over n, ℵo
>> to ℵo, n, and ℵo, 0.
>
> But ℵo / n is still ℵo,

subsets have cardinalities from (0, ℵo) over (n, ℵo) to (ℵo, n), and
(ℵo, 0).

> so you never get to 0.

Of course not. They are dark. But nevertheless these sets are existing.

>>> No ⅟j is in a finite start
>>> That is the reason that
>>> an infinite swarm is the only possibility
>>
>> Then you are not talking about really existing invariable points. But
>> that is what I discuss.
>
> But neither are you talking about what really exists,

If all unit fractions really exist, then we can talk about each one,
although we cannot find most.

Regards, WM

On 02/20/2024 04:49 AM, Richard Damon wrote:
> On 2/20/24 3:15 AM, WM wrote:
>> Le 19/02/2024 à 21:37, Jim Burns a écrit :
>>
>>> One can avoid using the I.word by saying
>>> | we CAN fit more than any finite number of them in
>>> | a finite space.
>>
>> But if all are there existing from the scratch as an actually infinite
>> set, then each one can be addressed as border between two subsets, in
>> principle. Even the smallest one.
>> Regards, WM
>
> But there isn't a "Smallest One".
>
> You just don't understand that fact, because you mind is just to filled
> with Darkness.

The iota-values of lines reals
that model [0,1] a continuous domain,
they have a smallest positive member, "iota".

Conveniently the word "iota" also
means "a smallest non-zero amount",
so when one says "not one iota",
there are none, while,
"it won't give an iota",
also implies none.

Here of course it's mathematics
what makes it so,
while the natural definitions
of "iota" and "infinity" here
sort of make for "iota's infinity"
is a continuum limit, that the
"iota's infinity's-many iota-values
the iota-multiples" equals 1.

Their iota-sum equals 2, because it's a
doubling-space among the
continuous and discrete,
it's a very special fact
of a very special function
and it's attributed to Vitali,
and measure theory, in accords
how it's so.

That the "sums" and "products" are
different for these infinitesimals
is a simplest sort of way to split them out.

Really, just take pencil to paper,
mark a point 0, mark a point 1,

Then, put the pencil down at zero
connect zero and one with a line-drawing,
without lifting the pencil, of course.
Now, lift the pencil:
congratulations, your pencil just
drew line-reals [0,1].

I understand that from one perspective
there's no first point between 0 and 1,
the line-drawing through must have gone
through them in order.

All the iota-values's, ....

These are about the most _standard_
notion of the _nonstandard_, these
iota-values are about the most and
infact about the only _standard_,
infinitesimals.

This is because they're defined
exactly by 0 and 1 and a _standard_,
exactly, continuum limit, the
iota-value's infinity: _standard_.

On 2/17/2024 3:33 AM, FromTheRafters wrote:
> Chris M. Thomasson expressed precisely :
>> On 2/16/2024 2:33 PM, FromTheRafters wrote:
>>> After serious thinking Chris M. Thomasson wrote :
>>>> Take a number that wants to get close to zero. Say:
>>>>
>>>> [0] = 1
>>>> [1] = .1
>>>> [2] = .01
>>>> [3] = .001
>>>> [...] = [...]
>>>>
>>>> This gets close to zero, yet never will equal zero. Okay so:
>>>
>>> This sequence doesn't reach zero, but this series (1.111...) equals
>>> one and one ninth.
>>>
>>>> arbitrarily close seems to be the accepted term.
>>>
>>> Approaching arbitrarily closely seems right to me.
>>>
>>>> infinitely close is the wrong wording?
>>>
>>> That sort of works too. What you want to avoid is the 'infinite
>>> <something>' which you often say when you mean 'infinitely many
>>> something(s').
>>>
>>>> The function f(n) = 10^(-n) gets "infinitely close" to 0... lol.
>>>> Using the "metaphysical formation" of arbitrarily close... ;^)
>>>
>>> I don't know what you are getting at with this.
>>
>> I was told one time that infinitely closer is in the realm of
>> "metaphysical" because of the word infinite. However, the term
>> arbitrarily close is something others can deal with "better", so to
>> speak. Make any sense?
>
> You could say approaching asymtotically.
>
> https://www.dictionary.com/browse/asymptotically

Fine with me.

On 2/20/2024 3:12 AM, WM wrote:
> Le 19/02/2024 à 21:21, Jim Burns a écrit :

>> In other words,
>> infinitely.many are leftward
>> finitely.many are rightward
>> for each ⅟j
>
> That correctly describes
> an evolving infinite collection, i.e.,
> a potentially infinite set where
> more and more elements are created which
> initially have not existed.

For anything which is a final.ordinal.reciprocal
∃j: ⅟j⋅j = 1 ∧ {<j} ⃒⇇ {<j}⁺ᴮᵒᵇ

For anything which isn't a final.ordinal.reciprocal
¬∃j: ⅟j⋅j = 1 ∧ {<j} ⃒⇇ {<j}⁺ᴮᵒᵇ

Those _statements_ don't evolve,
however it might be with final.ordinal.reciprocals.

We can follow them with a sequence of statements,
statements which also don't evolve.

We are finite beings.
The statements are finitely.many.
Their finitely.many.ness doesn't evolve.

It is a property of
finite sequences of statements that
if any of them is false,
then one of then is first.false.

That is equivalent to:
if each statement is not.first.false,
then each statement is not.false.
It is a property which doesn't evolve.

For some statements in
some sequences of statements,
_we can see_ that they are not.first.false
by examining the sequence they're in.
For example,
Q is not.first.false in ⟨ P P⇒Q Q ⟩
We see that
either Q is true
or Q is preceded by false P or false P⇒Q
We can see that there is no circumstance
in which Q first.false in ⟨ P P⇒Q Q ⟩

Consider
a finite sequence of statements about
final.ordinal.reciprocals
such that
we know that each statement is not.first.false
either because (1)
we know what a final.ordinal.reciprocal is
as for ∃j: ⅟j⋅j = 1 ∧ {<j} ⃒⇇ {<j}⁺ᴮᵒᵇ
or because (2)
we can see that it is not.first.false.
as for Q in ⟨ P P⇒Q Q ⟩

The statements in that sequence do not evolve.
They remain themselves, there in that sequence.
Type (1) statements remain what we mean, not.evolving.
Type (2) statements remain visibly not.first.false,
not.evolving.

Each part of
how we know the truth of those statements
doesn't evolve.

Therefore,
we know that
the truth of those statements
doesn't evolve.

>> In other words,
>> infinitely.many are leftward
>> finitely.many are rightward
>> for each ⅟j
>
> That correctly describes
> an evolving infinite collection, i.e.,
> a potentially infinite set where
> more and more elements are created which
> initially have not existed.

There is a finite sequence of statements
which holds my claim up.post and others
which only holds types (1) and (2) claims

We know that the truth of those claims
doesn't evolve.

>> No ⅟j is in a finite start
>> That is the reason that
>> an infinite swarm is the only possibility
>
> Then you are not talking about
> really existing invariable points.
> But that is what I discuss.

Judging from your reticence about which
of these claim you give up:
| a final.ordinal.reciprocal.free zone (0,δ)
| exists
or
| a skipping.function isn't all.continuous
or
| for final.ordinal.reciprocal ⅟m
| ⅟(4⋅m) is a final.ordinal.reciprocal
| what you are discussing accepts all three.
Then,
what you are discussing not.exists.

Le 20/02/2024 à 21:02, Jim Burns a écrit :
> On 2/20/2024 3:12 AM, WM wrote:
>> Le 19/02/2024 à 21:21, Jim Burns a écrit :
>
>>> In other words,
>>> infinitely.many are leftward
>>> finitely.many are rightward
>>> for each ⅟j
>>
>> That correctly describes
>> an evolving infinite collection, i.e.,
>> a potentially infinite set where
>> more and more elements are created which
>> initially have not existed.

> We are finite beings.
> The statements are finitely.many.
> Their finitely.many.ness doesn't evolve.

We cannot use everything that exists on the real line, because among them
there is the smallest unit fraction, at least the smallest unit fraction
that exists on the real line. Where else should it be? This existence is
static. You seem to deny it. If we could point to it, we caught the
smallest unit fraction. But we cannot point to it although it must be
there. That proves: It is dark.
>
> It is a property of
> finite sequences of statements that
> if any of them is false,
> then one of then is first.false.

Here is one statement that is true: Every unit fraction exists on the real
line. But there are no marks indicating their places. We cannot go to it
without, in principle, passing through all smaller ones. Counting from 1,
2, 3, ... to n.

Regards, WM

On 2/20/2024 1:59 PM, WM wrote:
[...]
> We cannot use everything that exists on the real line, because among
> them there is the smallest unit fraction, at least the smallest unit
> fraction that exists on the real line.

Barf!

[...]

On 2/20/2024 4:59 PM, WM wrote:
> Le 20/02/2024 à 21:02, Jim Burns a écrit :

>> It is a property of
>> finite sequences of statements that
>> if any of them is false,
>> then one of then is first.false.
>
> Here is one statement that is true:
> Every unit fraction exists on the real line.
> But there are no marks indicating their places.

For each final.ordinal.reciprocal ⅟j
a geometric procedure exists which finds it.
It involves constructing similar triangles.

> We cannot go to it without, in principle,
> passing through all smaller ones.

For each final.ordinal k,
there are more.than.k smaller ones.

> Counting from 1, 2, 3, ... to n.

Whatever final.ordinal n is imagined to be,
counting fails.
There are more.than.n to count.

----
It is a boring property of a "normal" finite set S
that, if Bob is inserted in S, giving S⁺ᴮᵒᵇ,
then S⁺ᴮᵒᵇ doesn't fit into S
S ⃒⇇ S⁺ᴮᵒᵇ
No 1.to.1 map exists to S from S⁺ᴮᵒᵇ

It is a boring property of a final ordinal j
that before.j {<j} is a "normal" finite set.
If Bob is inserted in {<j}, giving {<j}⁺ᴮᵒᵇ,
then then {<j}⁺ᴮᵒᵇ doesn't fit in {<j}
{<j} ⃒⇇ {<j}⁺ᴮᵒᵇ

For each boring "normal" finite set S,
a boring "normal" final.ordinal j exists
such that S fits into before.j
S ⃒⇇ S⁺ᴮᵒᵇ
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
∃j: S ⇉ {<j} ⃒⇇ {<j}⁺ᴮᵒᵇ

The technical term for this is "counting".
It is famously boring.
People "count" sheep
in order to bore themselves to sleep.

Define ℕ to be the set of final ordinals.

For each final ordinal j _and successor_ j⁺¹
{<j} and {<j⁺¹} fit in ℕ
But {<j⁺¹} doesn't fit in {<j}; {<j} is final.

Neither does ℕ fit in {<j},
or else {<j⁺¹} (sub ℕ) fits in {<j}

∀j: {<j} ⃒⇇ {<j}⁺ᴮᵒᵇ ⟹ {<j} ⃒⇇ ℕ
¬∃j: ℕ ⇉ {<j} ⃒⇇ {<j}⁺ᴮᵒᵇ
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(ℕ ⃒⇇ ℕ⁺ᴮᵒᵇ)
ℕ⁺ᴮᵒᵇ ⇉ ℕ

ℕ is not a boring "normal" finite set.
De taediosum non taediosum.

It's a miracle!

On 2/20/24 11:47 AM, WM wrote:
> Le 20/02/2024 à 13:49, Richard Damon a écrit :
>> On 2/20/24 3:15 AM, WM wrote:
>
>>> But if all are there existing from the scratch as an actually
>>> infinite set, then each one can be addressed as border between two
>>> subsets, in principle. Even the smallest one.
>>
>> But there isn't a "Smallest One".
>>
> You are obviosuly wrong. If all are there, then all can be used to
> divide the set into two parts. None is exempt.
>
> Regards, WM
>

And we CAN do that, since none of them are "the first", so ALL of them
have an infinite number of point before them.

Nothing wrong with that, at least as long as you understand how
unbounded sets work.

On 2/20/24 11:52 AM, WM wrote:
> Le 20/02/2024 à 13:49, Richard Damon a écrit :
>> On 2/20/24 3:12 AM, WM wrote:
>>> Le 19/02/2024 à 21:21, Jim Burns a écrit :
>
>>>> infinitely.many are leftward
>>>> finitely.many are rightward
>>>> for each ⅟j
>>>
>>> That correctly describes an evolving infinite collection, i.e., a
>>> potentially infinite set where more and more elements are created
>>> which initially have not existed.
>>> A complete, i.e. actually infinite set of ℵo real fixed points on the
>>> real axis can be subdivided by any of its elements (since all are
>>> existing) such that the subsets have cardinalities from 0, ℵo over n,
>>> ℵo to ℵo, n, and ℵo, 0.
>>
>> But ℵo / n is still ℵo,
>
> subsets have cardinalities from (0, ℵo) over (n, ℵo) to (ℵo, n), and
> (ℵo, 0).

Nope, all the subsets have cardinalities of ℵo (assuming we are using
the Rational Line), if we are using the Real line then there are ℵ1
points between each of them.

>
>> so you never get to 0.
>
> Of course not. They are dark. But nevertheless these sets are existing.

But they are NOT "Dark"

Your mind just seems them as dark as it can't handle the truth about them.

>>>> No ⅟j is in a finite start
>>>> That is the reason that
>>>> an infinite swarm is the only possibility
>>>
>>> Then you are not talking about really existing invariable points. But
>>> that is what I discuss.
>>
>> But neither are you talking about what really exists,
>
> If all unit fractions really exist, then we can talk about each one,
> although we cannot find most.

We can find any one that we want.

At least as long as you don't try to qualify it with an impossible
qualifier, like the lowest value one.

>
> Regards, WM
>
>

On 2/20/24 4:59 PM, WM wrote:
> Le 20/02/2024 à 21:02, Jim Burns a écrit :
>> On 2/20/2024 3:12 AM, WM wrote:
>>> Le 19/02/2024 à 21:21, Jim Burns a écrit :
>>
>>>> In other words,
>>>> infinitely.many are leftward
>>>> finitely.many are rightward
>>>> for each ⅟j
>>>
>>> That correctly describes
>>> an evolving infinite collection, i.e.,
>>> a potentially infinite set where
>>> more and more elements are created which initially have not existed.
>
>> We are finite beings.
>> The statements are finitely.many.
>> Their finitely.many.ness doesn't evolve.
>
> We cannot use everything that exists on the real line, because among
> them there is the smallest unit fraction, at least the smallest unit
> fraction that exists on the real line. Where else should it be? This
> existence is static. You seem to deny it. If we could point to it, we
> caught the smallest unit fraction. But we cannot point to it although it
> must be there. That proves: It is dark.
>>

No, there ISN'T a "Smallest Unit Fraction" as has been shown.

Since ALL Unit Fractions "exist" in the mathematical sense,.

You are just stuck in your wrong thinking and it makes your mind dark.

>> It is a property of
>> finite sequences of statements that
>> if any of them is false,
>> then one of then is first.false.
>
> Here is one statement that is true: Every unit fraction exists on the
> real line. But there are no marks indicating their places. We cannot go
> to it without, in principle, passing through all smaller ones. Counting
> from 1, 2, 3, ... to n.
>
"Counting " the unit fractions can only be done from 1/1 to 1/2 to 1`/3
and so on.

You can't start counting from an end that doesn't exist. Trying to do
so, just breaks your system.

> Regards, WM
>

Le 21/02/2024 à 00:55, Jim Burns a écrit :
> On 2/20/2024 4:59 PM, WM wrote:

>> Every unit fraction exists on the real line.
>> But there are no marks indicating their places.
>
> For each final.ordinal.reciprocal ⅟j
> a geometric procedure exists which finds it.

Not for those existing next to zero. Note that if reciprocals are existing
on the real axis and if all points are timeless, then there is a point
next to zero. So your claim involves time. That is not mathematics, which
is time-independent

Regards, WM

Le 21/02/2024 à 03:17, Richard Damon a écrit :
> On 2/20/24 11:47 AM, WM wrote:

>> You are obviosuly wrong. If all are there, then all can be used to
>> divide the set into two parts. None is exempt.
>
> And we CAN do that, since none of them are "the first", so ALL of them
> have an infinite number of point before them.

If all are there, timeless and static, then one of them is the first.
NUF(x) growing from 0 to ℵo immediately would contradict mathematics
according to which after every unit fraction there are points without unit
fraction.

Regards, WM

Le 21/02/2024 à 03:17, Richard Damon a écrit :
> On 2/20/24 11:52 AM, WM wrote:
>> Le 20/02/2024 à 13:49, Richard Damon a écrit :
>>> On 2/20/24 3:12 AM, WM wrote:
>>>> Le 19/02/2024 à 21:21, Jim Burns a écrit :
>>
>>>>> infinitely.many are leftward
>>>>> finitely.many are rightward
>>>>> for each ⅟j
>>>>
>>>> That correctly describes an evolving infinite collection, i.e., a
>>>> potentially infinite set where more and more elements are created
>>>> which initially have not existed.
>>>> A complete, i.e. actually infinite set of ℵo real fixed points on the
>>>> real axis can be subdivided by any of its elements (since all are
>>>> existing) such that the subsets have cardinalities from 0, ℵo over n,
>>>> ℵo to ℵo, n, and ℵo, 0.

>> subsets have cardinalities from (0, ℵo) over (n, ℵo) to (ℵo, n), and
>> (ℵo, 0).
>
>> If all unit fractions really exist, then we can talk about each one,
>> although we cannot find most.
>
> We can find any one that we want.
>
> At least as long as you don't try to qualify it with an impossible
> qualifier, like the lowest value one.

In a static real line obeying mathematics ∀n ∈ ℕ: 1/n - 1/(n+1) =
d_n > 0 there is a smallest unit fraction existing as a point. The only
alternative would be many smallest ones, but that can be excluded.

Regards, WM

Le 21/02/2024 à 03:17, Richard Damon a écrit :
> On 2/20/24 4:59 PM, WM wrote:

>> We cannot use everything that exists on the real line, because among
>> them there is the smallest unit fraction, at least the smallest unit
>> fraction that exists on the real line. Where else should it be? This
>> existence is static. You seem to deny it. If we could point to it, we
>> caught the smallest unit fraction. But we cannot point to it although it
>> must be there. That proves: It is dark.
>
> No, there ISN'T a "Smallest Unit Fraction" as has been shown.
>
> Since ALL Unit Fractions "exist" in the mathematical sense,.

Then take the first one existing there.

> "Counting " the unit fractions can only be done from 1/1 to 1/2 to 1`/3
> and so on.
>
> You can't start counting from an end that doesn't exist. Trying to do
> so, just breaks your system.

The increase from NUF(0) = 0 to NUF(x>0) > 0 is restricted by logic:
Either one or more than one at one point. More than one is excluded by
mathematics,

Regards, WM

On 2/21/24 3:42 AM, WM wrote:
> Le 21/02/2024 à 03:17, Richard Damon a écrit :
>> On 2/20/24 11:47 AM, WM wrote:
>
>>> You are obviosuly wrong. If all are there, then all can be used to
>>> divide the set into two parts. None is exempt.
>>
>> And we CAN do that, since none of them are "the first", so ALL of them
>> have an infinite number of point before them.
>
> If all are there, timeless and static, then one of them is the first.

Nope.

You don't understand the properties of UNBOUNDED sets.

Being "Unbounded" means there isn't a "Bound" (i.e. and end) in that set.

This is just a property of "infinity" that your logic can't handle.

> NUF(x) growing from 0 to ℵo immediately would contradict mathematics
> according to which after every unit fraction there are points without
> unit fraction.

No more than my Qn and Qd which show that the square root of two is
Rational.

Definied in words does not mean defined to exist.

>
> Regards, WM
>

On 2/21/24 3:47 AM, WM wrote:
> Le 21/02/2024 à 03:17, Richard Damon a écrit :
>> On 2/20/24 11:52 AM, WM wrote:
>>> Le 20/02/2024 à 13:49, Richard Damon a écrit :
>>>> On 2/20/24 3:12 AM, WM wrote:
>>>>> Le 19/02/2024 à 21:21, Jim Burns a écrit :
>>>
>>>>>> infinitely.many are leftward
>>>>>> finitely.many are rightward
>>>>>> for each ⅟j
>>>>>
>>>>> That correctly describes an evolving infinite collection, i.e., a
>>>>> potentially infinite set where more and more elements are created
>>>>> which initially have not existed.
>>>>> A complete, i.e. actually infinite set of ℵo real fixed points on
>>>>> the real axis can be subdivided by any of its elements (since all
>>>>> are existing) such that the subsets have cardinalities from 0, ℵo
>>>>> over n, ℵo to ℵo, n, and ℵo, 0.
>
>>> subsets have cardinalities from (0, ℵo) over (n, ℵo) to (ℵo, n), and
>>> (ℵo, 0).
>>
>>> If all unit fractions really exist, then we can talk about each one,
>>> although we cannot find most.
>>
>> We can find any one that we want.
>>
>> At least as long as you don't try to qualify it with an impossible
>> qualifier, like the lowest value one.
>
> In a static real line obeying mathematics ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n >
> 0 there is a smallest unit fraction existing as a point. The only
> alternative would be many smallest ones, but that can be excluded.
>
> Regards, WM
>
>

Nope. You just don't understand the properties of unbounded sets,
because you mind can't actually handle the unbounded.

There doesn't need to be just one or many with a property, there can be
none, and that is how many have the property "smallest".

Your logic system just can't handle unbounded sets.

On 2/21/24 3:39 AM, WM wrote:
> Le 21/02/2024 à 00:55, Jim Burns a écrit :
>> On 2/20/2024 4:59 PM, WM wrote:
>
>>> Every unit fraction exists on the real line.
>>> But there are no marks indicating their places.
>>
>> For each final.ordinal.reciprocal ⅟j
>> a geometric procedure exists which finds it.
>
> Not for those existing next to zero. Note that if reciprocals are
> existing on the real axis and if all points are timeless, then there is
> a point next to zero. So your claim involves time. That is not
> mathematics, which is time-independent
>
> Regards, WM

Nope, no point "next to" zero. Points on the real axis are "dense" and
there is no "Next To" property, as between ANY two points, there is
another one between them.

You need to get into a trans-finite number system to get the "Next To"
property again.

On 2/21/24 3:52 AM, WM wrote:
> Le 21/02/2024 à 03:17, Richard Damon a écrit :
>> On 2/20/24 4:59 PM, WM wrote:
>
>>> We cannot use everything that exists on the real line, because among
>>> them there is the smallest unit fraction, at least the smallest unit
>>> fraction that exists on the real line. Where else should it be? This
>>> existence is static. You seem to deny it. If we could point to it, we
>>> caught the smallest unit fraction. But we cannot point to it although
>>> it must be there. That proves: It is dark.
>>
>> No, there ISN'T a "Smallest Unit Fraction" as has been shown.
>>
>> Since ALL Unit Fractions "exist" in the mathematical sense,.
>
> Then take the first one existing there.

There isn't one, and you are just proving your ignornacd.

>
>> "Counting " the unit fractions can only be done from 1/1 to 1/2 to
>> 1`/3 and so on.
>>
>> You can't start counting from an end that doesn't exist. Trying to do
>> so, just breaks your system.
>
> The increase from NUF(0) = 0 to NUF(x>0) > 0 is restricted by logic:
> Either one or more than one at one point. More than one is excluded by
> mathematics,
> Regards, WM
>
>
>
And that "restriction" makes NUF(x) just disapper as a defined function
as an artifact of a contradiction.

You can't just "assume" the existance of a function. Your doing that
just makes an ASS out of U. (it doesn't add me, because I won't fall for
it).

On 02/17/2024 10:35 AM, Jim Burns wrote:
> On 2/17/2024 4:56 AM, WM wrote:
>> Mike Terry schrieb am Freitag,
>> 16. Februar 2024 um 22:45:48 UTC+1:
>
>>> Yes,
>>> it's not clear what "infinitely close" means
>>
>> It means dark numbers.
>
> Do you (WM) say that
> a point with a final.ordinal.reciprocal
> ⅟n⋅n = 1 ∧ ⟨1,…,n⟩ ⃒⇇ ⟨1,…,n,n⁺¹⟩
> below it is infinitelyᵂᴹ.close to 0?
> That would be an odd use of "infinite".
>
> A positive dark number has
> a final.ordinal.reciprocal below it.
>
> | Assume otherwise.
> | Also, assume
> | a skipping.function isn't all.continuous, and,
> | for final.ordinal.reciprocal ⅟m
> | ⅟(4⋅m) is a final.ordinal.reciprocal.
> |
> | By assumption,
> | positive dark δ is a positive lower bound of
> | final.ordinal.reciprocals ⅟ℕ₁
> | 0 < δ ≤ᣔ ⅟ℕ₁
> |
> | β is the greatest lower bound of
> | final.ordinal.reciprocals ⅟ℕ₁
> | 0 < δ ≤ β ≤ᣔ ⅟ℕ₁
> | 0 < β/2 < β < 2β
> | 2β isn't a lower bound of ⅟ℕ₁
> | β is the greatest lower bound of ⅟ℕ₁
> | β/2 is a lower bound of ⅟ℕ₁
> |
> | β < 2β
> | 2β isn't a lower bound of ⅟ℕ₁
> | final.ordinal.reciprocal ⅟m₂ᵦ < 2β exists.
> | final.ordinal.reciprocal ⅟(4⋅m₂ᵦ) < β/2 exists.
> | β/2 isn't a lower bound of ⅟ℕ₁
> |
> | However,
> | β/2 < β
> | β/2 is a lower bound of ⅟ℕ₁
> | Contradiction.
>
> Therefore,
> a positive dark number has
> a final.ordinal.reciprocal below it.
>
>> The function
>> Number of Unit Fractions between (0, and x)
>> has the following properties:
>> (1) An increase from NUF(0) = 0 to NUF(x>0) > 0
>> cannot happen unless NUF(x) increases at some x.
>
> NUF(x) increases at 0
>
>> (2) NUF(x) cannot increase other than
>> when passing unit fractions at some x = 1/n.
>
> NUF(x) cannot increase other than
> when ∀β > 0: NUF(x-β) < NUF(x+β)
>
>> (3) NUF(x) cannot pass more than one
>> unit fraction at a single point x because
>> ∀n ∈ ℕ: 1/n =/= 1/(n-1).
>
> ∀n ∈ ℕ: 1/n =/= 0
>
> ∀β > 0: ∀n ∈ ℕ: NUF(0-β) + n < NUF(0+β)
>
> β > ⅟1⁺ᵐᵝ > ... > ⅟n⁺ᵐᵝ > ⅟(n+1)⁺ᵐᵝ > 0
> for
> 0 =< mᵦ =< ⅟β < mᵦ+1 = 1⁺ᵐᵝ
>
>> (4) This requires a first unit fraction,
>> if all are there in actual infinity.
>
> Each final.ordinal.reciprocal
> is preceded by
> another final.ordinal.reciprocal.
>
> The first final.ordinal.reciprocal not.exists.
>
>

So, there's no first example where
"the equivalency function" isn't a model
of "not-a-real-function"
with "real analytical character",

considering that
the infinite and continuum limit
was already run out one-way.

I mean if you want it back,
here's the argument that made it, ....
there's transparency here,
it's clear that it's related its
resources already, ....

It's not.ultimately.untrue, ....

On 2/21/2024 3:39 AM, WM wrote:
> Le 21/02/2024 à 00:55, Jim Burns a écrit :
>> On 2/20/2024 4:59 PM, WM wrote:

>>> Every unit fraction exists on the real line.
>>> But there are no marks indicating their places.
>>
>> For each final.ordinal.reciprocal ⅟j
>> a geometric procedure exists which finds it.
>
> Not for those existing next to zero.

None exist which are next to zero.

β is the greatest.lower.bound of
final.ordinal.reciprocals findable by
geometric procedure.
Point β exists because
all skipping.functions are discontinuous.somewhere.

¬(0 < β)

| Assume otherwise.
| Assume 0 < β
| | 0 < β/2 < β < 2β
| | β is the greatest.lower.bound of
| final.ordinal.reciprocals findable by
| geometric procedure.
| | β/2 is a lower.bound of
| final.ordinal.reciprocals findable by
| geometric procedure.
| | 2β isn't a lower.bound of
| final.ordinal.reciprocals findable by
| geometric procedure.
| | 2β isn't a lower.bound
| Exists ⅟m < 2β which is
| a final.ordinal.reciprocal findable by
| geometric procedure.
| | Exists ⅟(4⋅m) < β/2 which is
| a final.ordinal.reciprocal findable by
| geometric procedure.
| β/2 isn't a lower.bound
| | However,
| β/2 is a lower.bound
| Contradiction.

Therefore,
¬(0 < β)

0 is a lower bound of
final.ordinal.reciprocals findable by
geometric procedure.
0 ≤ β

Thus, 0 = β
β the greatest.lower.bound of
final.ordinal.reciprocals findable by
geometric procedure.

>> For each final.ordinal.reciprocal ⅟j
>> a geometric procedure exists which finds it.
>
> Not for those existing next to zero.

No positive point is next to zero, meaning,
no positive point lacks
some final.ordinal.reciprocal findable by
geometric procedure
between that positive point and zero,
because 0 = β

> Note that
> if reciprocals are existing on the real axis and
> if all points are timeless,
> then there is a point next to zero.

Elaborate.

Do you reject
all skipping.functions being discontinuous.somewhere?

Do you reject
only both or neither ⅟m ⅟(4⋅m) being
final.ordinal.reciprocals findable by
geometric procedure?

> So your claim involves time. That is not
> mathematics, which is time-independent

Describe points, final.ordinal.reciprocals,
skipping.functions, and so on.

Augment the description with
not.first.false claims about points,
final.ordinal.reciprocals, skipping.functions,
and so on.

Know that the augmenting claims are true
because
the finite not.first.false claim.sequence exists,
wherever and whenever it exists.
It is time.independent knowledge.

On 2/21/2024 1:27 PM, Ross Finlayson wrote:
> On 02/17/2024 10:35 AM, Jim Burns wrote:

>> ...]
>
> So, there's no first example where
> "the equivalency function" isn't a model
> of "not-a-real-function"
> with "real analytical character",
> considering that
> the infinite and continuum limit
> was already run out one-way.

The iota.value limit which you describe
is not the real interval [0,1]

Consider (following your lead)
a range of constantly-different monotone
strictly increasing values between zero and one,
an infinitude of them.
I abbreviate that to [0,1]\ι

Define a plus.iota next.operator x⁺ᶥ
∀x ∈ [0,1)\ι: ∃x⁺ᶥ ∈ (0,1]\ι:
x < x⁺ᶥ ∧ ¬∃xₓ ∈ [0,1]\ι: x < xₓ < x⁺ᶥ

It is a constantly.different strictly.increasing
next.operator.
∀x,y ∈ [0,1)\ι: x⁺ᶥ-x = y⁺ᶥ-y = ι > 0

Are there an infinitude of values in [0,1]\ι?
No.

ι > 0
Therefore, there is
a finitely.denominated unit.fraction ⅟n
between ι and 0
and |[0,1]\ι| ≤ n+1

| Assume otherwise.
| Assume ι is a positive lower bound of
| ⅟ℕ₁ the finitely.denominated unit.fractions.
| 0 < ι ≤ᣔ ⅟ℕ₁
| | β is the greatest.lower.bound of ⅟ℕ₁
| | β exists, or else
| a function exists which is
| continuous everywhere and
| skips over some points between.
| | 0 < ι ≤ β
| 0 < β/2 < β < 2β
| β is the greatest lower bound of ⅟ℕ₁
| 2β isn't a lower bound of ⅟ℕ₁
| β/2 is a lower bound of ⅟ℕ₁
| | β < 2β
| 2β isn't a lower bound of ⅟ℕ₁
| finitely.denominated ⅟m < 2β exists.
| finitely.denominated ⅟(4⋅m) < β/2 exists.
| β/2 isn't a lower bound of ⅟ℕ₁
| | However,
| β/2 < β
| β/2 is a lower bound of ⅟ℕ₁
| Contradiction.

Therefore,
there is a finitely.denominated unit.fraction ⅟n
between ι and 0
and |[0,1]\ι| ≤ n+1

For a range of constantly-different monotone
strictly increasing values between zero and one,
there aren't an infinitude of them.

You (RF) can define whatever you choose to define.
That's a matter of letting us know
how you are using language.
But
not everything you are able to define
includes the rationals and excludes
skipping.functions not discontinuous.somewhere.

On 02/21/2024 12:02 PM, Jim Burns wrote:
> On 2/21/2024 1:27 PM, Ross Finlayson wrote:
>> On 02/17/2024 10:35 AM, Jim Burns wrote:
>
>>> ...]
>>
>> So, there's no first example where
>> "the equivalency function" isn't a model
>> of "not-a-real-function"
>> with "real analytical character",
>> considering that
>> the infinite and continuum limit
>> was already run out one-way.
>
> The iota.value limit which you describe
> is not the real interval [0,1]
>
> Consider (following your lead)
> a range of constantly-different monotone
> strictly increasing values between zero and one,
> an infinitude of them.
> I abbreviate that to [0,1]\ι
>
> Define a plus.iota next.operator x⁺ᶥ
> ∀x ∈ [0,1)\ι: ∃x⁺ᶥ ∈ (0,1]\ι:
> x < x⁺ᶥ ∧ ¬∃xₓ ∈ [0,1]\ι: x < xₓ < x⁺ᶥ
>
> It is a constantly.different strictly.increasing
> next.operator.
> ∀x,y ∈ [0,1)\ι: x⁺ᶥ-x = y⁺ᶥ-y = ι > 0
>
> Are there an infinitude of values in [0,1]\ι?
> No.
>
> ι > 0
> Therefore, there is
> a finitely.denominated unit.fraction ⅟n
> between ι and 0
> and |[0,1]\ι| ≤ n+1
>
> | Assume otherwise.
> | Assume ι is a positive lower bound of
> | ⅟ℕ₁ the finitely.denominated unit.fractions.
> | 0 < ι ≤ᣔ ⅟ℕ₁
> |
> | β is the greatest.lower.bound of ⅟ℕ₁
> |
> | β exists, or else
> | a function exists which is
> | continuous everywhere and
> | skips over some points between.
> |
> | 0 < ι ≤ β
> | 0 < β/2 < β < 2β
> | β is the greatest lower bound of ⅟ℕ₁
> | 2β isn't a lower bound of ⅟ℕ₁
> | β/2 is a lower bound of ⅟ℕ₁
> |
> | β < 2β
> | 2β isn't a lower bound of ⅟ℕ₁
> | finitely.denominated ⅟m < 2β exists.
> | finitely.denominated ⅟(4⋅m) < β/2 exists.
> | β/2 isn't a lower bound of ⅟ℕ₁
> |
> | However,
> | β/2 < β
> | β/2 is a lower bound of ⅟ℕ₁
> | Contradiction.
>
> Therefore,
> there is a finitely.denominated unit.fraction ⅟n
> between ι and 0
> and |[0,1]\ι| ≤ n+1
>
> For a range of constantly-different monotone
> strictly increasing values between zero and one,
> there aren't an infinitude of them.
>
> You (RF) can define whatever you choose to define.
> That's a matter of letting us know
> how you are using language.
> But
> not everything you are able to define
> includes the rationals and excludes
> skipping.functions not discontinuous.somewhere.
>
>

This isn't the complete ordered field,
it's iota-values,
the constant monotone strictly increasing property,
is their ordering,
but says nothing about their arithmetic,
at all.

What defines their arithmetic,
is that iota-products
put them back together, length 1,
and iota-sums make
for re-Vitali-izing measure theory, length 2.

I.e. they're only put together altogether.

It's not a Cartesian function either,
this equivalency function, and
it falls out of all the results
otherwise about Cartesian functions
and Cantorian uncountability,
which otherwise of course applies.

It's special, this way, and, it's very special.

It's line-drawing, and it's the function
between discrete and continuous.

That's what it is.

Thank you for your attention to this matter.

On 2/21/2024 3:59 PM, Ross Finlayson wrote:
> On 02/21/2024 12:02 PM, Jim Burns wrote:

>> [...]
>
> This isn't the complete ordered field,

> It's line-drawing, and it's the function
> between discrete and continuous.
>
> That's what it is.

Lines which cross intersect.
It is the points of intersection which
complete the line.

The classical geometers had lines,
and described them as an ordered field.
(× ÷ by similar triangles.)
They maybe didn't state it, but,
when those lines crossed, they intersected.

I think that there are very good reasons for
describing the line as the complete ordered field:
_at least_ the (classical) rational points but also
lines.which.cross intersect.somewhere.
(AKA functions.which.skip jump.somewhere.)

> It's line-drawing, and it's the function
> between discrete and continuous.

There aren't enough discrete points
to fill a continuous (Dedekind.complete) segment.

There are some very basic properties
in conflict with your claims which
I have yet to see you (RF) work around

For a set S and a function f: S → T
if f is 1.to.1
then the image f(S) is the same 'size' as S
For any not.1.to.1 function,
the image f(S) is no 'larger' than S

The discrete is the 'size' of ℕ, of ℤ
The continuous is the 'size' of ℝ, the 'size' of [0,1]
ℝ is 'larger' than ℕ

You express confidence that, in effect,
the equivalency function EF maps ℕ onto ℝ
For that to be true,
the image EF(ℕ) needs to be 'larger' than ℕ
Basic properties say that's not a thing.

How do you (RF) explain that?