On 1/19/2024 5:49 PM, WM wrote:
> Le 19/01/2024 à 21:33, Jim Burns a écrit :

>> "They have no FISON" is true the same way
>> "The present king of France is bald" is true.
>> In the least useful way possible.
>
> Wrong.
> There is no present king but,
> assuming actual infinity,
> there is ω.

ω is defined to be the set of final.ordinals
ω isn't a final.ordinal.
No final.ordinal has.no.FISON.
No element.of.ω has.no.FISON.
No final.ordinal is the last final.ordinal.
No element.of.ω is the last element.of.ω

ω isn't a final ordinal
| | Assume otherwise.
| Assume ω is a final ordinal
| | ω is the set of final.ordinals.
| ω+1 = {0,1,2,…;ω} ⊆ ω
| ≡: ω+1 ⇉ ω 1.to.1
| |ω| = |ω+1|
| ω isn't a final ordinal.
| Contradiction.

No final.ordinal is the last final.ordinal.
| | Assume otherwise.
| Assume λ is the last final.ordinal.
| |λ| < |λ+1| = |λ+2|
| | |λ+1| = |λ+2|
| exists g: λ+2 ⇉ λ+1 1.to.1
| | Define f: λ+1 ⇉ λ
| such that
| if g(α) ≠ λ+1
| then f(α) = g(α)
| else f(α) = g(λ+1)
| | exists f: λ+1 ⇉ λ 1.to.1
| |λ| = |λ+1|
| | However,
| |λ| < |λ+1|
| not.exists f: λ+1 ⇉ λ 1.to.1
| Contradiction.

On Saturday, January 20, 2024 at 4:47:48 AM UTC-8, Jim Burns wrote:
> On 1/19/2024 6:29 PM, Ross Finlayson wrote:
> > On Friday, January 19, 2024
> > at 12:51:15 PM UTC-8, Jim Burns wrote:
> >> On 1/18/2024 10:09 PM, Ross Finlayson wrote:
> >>> On Thursday, January 18, 2024
> >>> at 12:28:54 PM UTC-8, Jim Burns wrote:
>
> >>>> Our current theory of the line,
> >>>> a theory stress.tested beyond the dreams of Zeno,
> >>>> is that, in topological terms,
> >>>> the line is _one component_ that it can't be
> >>>> partitioned into two non.empty open sets.
> >>>
> >>> Well there are lots of topologies.
> >>
> >> There is one meaning I intend for "component",
> >> as used in topology.area.of.study.
> >>
> >> One meaning across all
> >> topologies.set.collections.designated.the.opens.
> > I'm unfamiliar with this usage of, "component".
> > Can you detail its definition?
> We need "topological space" "disconnected" and
> "connected component"
>
> | A topological space is the most general type of
> | a mathematical space that allows for the definition of
> | limits, continuity, and connectedness. [...]
> |
> | A topology on a set X may be defined as
> | a collection τ of subsets of X, called open sets
> | and satisfying the following axioms:
> |
> | 1. The empty set and X itself belong to τ
> | 2. Any arbitrary (finite or infinite) union of
> | members of τ belongs to τ
> | 3. The intersection of any finite number of
> | members of τ belongs to τ
> [1]
>
> | A topological space X is said to be disconnected if
> | it is the union of two disjoint non-empty open sets.
> | Otherwise, X is said to be connected. [...]
> |
> | The connected component of a point x in X is
> | the union of all connected subsets of X that contain x
> | it is the unique largest (with respect to ⊆)
> | connected subset of X that contains x
> [2]
>
> I said: ℝ has one component.
> Better: ℝ is connected.
>
> Delete ℼ from ℝ and ℝ\{ℼ} is disconnected..
> ℝ\{ℼ} = (-∞,ℼ)∪(ℼ,+∞)
>
> It's possible to map
> (-∞,ℼ) ⟶ 0 and (ℼ,+∞) ⟶ 1
> and be continuous at each point in ℝ\{ℼ}
> But that's not what we want
> when we ask for a continuous function.
>
> In order to get what we want,
> we must include ℼ in our discussion,
> even if we never use ℼ
> We must include uncountably.many points,
> which even an endless process of naming can't use.
>
> They must be there, used or unused, in order to
> be able talk about continuous functions
> with it meaning what we want.
>
> [1]
> https://en.wikipedia.org/wiki/Topological_space
>
> [2]
> https://en.wikipedia.org/wiki/Connected_space
> > Of course you know that "open" and "closed" aren't
> > together a binary property, in topology.
> Point x is in the interior of set S if
> some open set Oₓ which holds x is entirely in S
>
> Point x is in the exterior of set S if
> some open set O′ₓ which holds x is entirely not-in S
>
> Point x is in the boundary of set S if
> if it isn't in its interior or its exterior.
>
> A closed set contains all of its boundary.
> An open set contains none of its boundary.
>
> And then, there are the sets containing
> neither all nor none.
>
>
> For some sets, their boundary is the empty set.
> They contain all and none of the empty set.
> They are closed and open.
> They are called clopen sets,
> which proves mathematicians have a sense of humor.

Ah. Connectedness is a familiar property.

Thanks for this write-up, I'll look to a fuller reply.

We're conscientious logicians,
and have a tower of giants the canon to uphold.

So, line-reals follow the integer continuum,
a unit magnitude of an integer,
for the integer part, and non-integer part,
of a real number, of the linear continuum.

Of course, their establishment as continuous domains,
involves their relations, in function theory, and topology.

On Saturday, January 20, 2024 at 10:05:42 AM UTC-8, Ross Finlayson wrote:
> On Saturday, January 20, 2024 at 4:47:48 AM UTC-8, Jim Burns wrote:
> > On 1/19/2024 6:29 PM, Ross Finlayson wrote:
> > > On Friday, January 19, 2024
> > > at 12:51:15 PM UTC-8, Jim Burns wrote:
> > >> On 1/18/2024 10:09 PM, Ross Finlayson wrote:
> > >>> On Thursday, January 18, 2024
> > >>> at 12:28:54 PM UTC-8, Jim Burns wrote:
> >
> > >>>> Our current theory of the line,
> > >>>> a theory stress.tested beyond the dreams of Zeno,
> > >>>> is that, in topological terms,
> > >>>> the line is _one component_ that it can't be
> > >>>> partitioned into two non.empty open sets.
> > >>>
> > >>> Well there are lots of topologies.
> > >>
> > >> There is one meaning I intend for "component",
> > >> as used in topology.area.of.study.
> > >>
> > >> One meaning across all
> > >> topologies.set.collections.designated.the.opens.
> > > I'm unfamiliar with this usage of, "component".
> > > Can you detail its definition?
> > We need "topological space" "disconnected" and
> > "connected component"
> >
> > | A topological space is the most general type of
> > | a mathematical space that allows for the definition of
> > | limits, continuity, and connectedness. [...]
> > |
> > | A topology on a set X may be defined as
> > | a collection τ of subsets of X, called open sets
> > | and satisfying the following axioms:
> > |
> > | 1. The empty set and X itself belong to τ
> > | 2. Any arbitrary (finite or infinite) union of
> > | members of τ belongs to τ
> > | 3. The intersection of any finite number of
> > | members of τ belongs to τ
> > [1]
> >
> > | A topological space X is said to be disconnected if
> > | it is the union of two disjoint non-empty open sets.
> > | Otherwise, X is said to be connected. [...]
> > |
> > | The connected component of a point x in X is
> > | the union of all connected subsets of X that contain x
> > | it is the unique largest (with respect to ⊆)
> > | connected subset of X that contains x
> > [2]
> >
> > I said: ℝ has one component.
> > Better: ℝ is connected.
> >
> > Delete ℼ from ℝ and ℝ\{ℼ} is disconnected.
> > ℝ\{ℼ} = (-∞,ℼ)∪(ℼ,+∞)
> >
> > It's possible to map
> > (-∞,ℼ) ⟶ 0 and (ℼ,+∞) ⟶ 1
> > and be continuous at each point in ℝ\{ℼ}
> > But that's not what we want
> > when we ask for a continuous function.
> >
> > In order to get what we want,
> > we must include ℼ in our discussion,
> > even if we never use ℼ
> > We must include uncountably.many points,
> > which even an endless process of naming can't use.
> >
> > They must be there, used or unused, in order to
> > be able talk about continuous functions
> > with it meaning what we want.
> >
> > [1]
> > https://en.wikipedia.org/wiki/Topological_space
> >
> > [2]
> > https://en.wikipedia.org/wiki/Connected_space
> > > Of course you know that "open" and "closed" aren't
> > > together a binary property, in topology.
> > Point x is in the interior of set S if
> > some open set Oₓ which holds x is entirely in S
> >
> > Point x is in the exterior of set S if
> > some open set O′ₓ which holds x is entirely not-in S
> >
> > Point x is in the boundary of set S if
> > if it isn't in its interior or its exterior.
> >
> > A closed set contains all of its boundary.
> > An open set contains none of its boundary.
> >
> > And then, there are the sets containing
> > neither all nor none.
> >
> >
> > For some sets, their boundary is the empty set.
> > They contain all and none of the empty set.
> > They are closed and open.
> > They are called clopen sets,
> > which proves mathematicians have a sense of humor.
> Ah. Connectedness is a familiar property.
>
> Thanks for this write-up, I'll look to a fuller reply.
>
> We're conscientious logicians,
> and have a tower of giants the canon to uphold.
>
> So, line-reals follow the integer continuum,
> a unit magnitude of an integer,
> for the integer part, and non-integer part,
> of a real number, of the linear continuum.
>
> Of course, their establishment as continuous domains,
> involves their relations, in function theory, and topology.

Here of course is already quite a bit of background reading.

https://groups.google.com/g/sci.logic/search?q=Finlayson%20author%3ABurns
https://groups.google.com/g/sci.math/search?q=Finlayson%20author%3ABurns

An interview with Robert Kuhn and Roger Penrose:
https://www.youtube.com/watch?v=ujvS2K06dg4
https://www.youtube.com/watch?v=H9Q6SWcTA9w&lc=Ugwv2tuaKKbGwtMCYHp4AaABAg&t=529s

"What are Breakthroughs in Mathematics?"
https://www.youtube.com/watch?v=hpkf6-C9E-E&t=970s

On 1/20/2024 1:05 PM, Ross Finlayson wrote:
> On Saturday, January 20, 2024
> at 4:47:48 AM UTC-8, Jim Burns wrote:

>> [...]
>
> So, line-reals follow the integer continuum,
> a unit magnitude of an integer,
> for the integer part, and non-integer part,
> of a real number, of the linear continuum.
>
> Of course,
> their establishment as continuous domains, involves
> their relations, in function theory,
> and topology.

What I consider essential:

Natural numbers are not too big to fail at
matching nearby natural numbers.
There are no 1.to.1 maps from 1.inserted.
¬(⟨0,…,k⟩ ⇇ ⟨0,…,k,k+1⟩) ⟺ k ∈ ℕ

Rational numbers do not have an inherent scale.
∀p ∈ ℚ: ∀n ∈ ℕ₁: p/n ∈ ℚ ∧ n⋅p ∈ ℚ

Real numbers do not have functions.which.jump
and are continuous everywhere.
For each non.empty split F,H of ℝ
exists x ∈ ℝ between F and H

These claims are sufficient to begin,
for our purposes, for _those_ purposes:
not.matching, scaling, not.jumping.

They can be augmented by
claims which are not.first.false

On Saturday, January 20, 2024 at 4:49:42 PM UTC-8, Jim Burns wrote:
> On 1/20/2024 1:05 PM, Ross Finlayson wrote:
> > On Saturday, January 20, 2024
> > at 4:47:48 AM UTC-8, Jim Burns wrote:
> >> [...]
> >
> > So, line-reals follow the integer continuum,
> > a unit magnitude of an integer,
> > for the integer part, and non-integer part,
> > of a real number, of the linear continuum.
> >
> > Of course,
> > their establishment as continuous domains, involves
> > their relations, in function theory,
> > and topology.
> What I consider essential:
>
> Natural numbers are not too big to fail at
> matching nearby natural numbers.
> There are no 1.to.1 maps from 1.inserted.
> ¬(⟨0,…,k⟩ ⇇ ⟨0,…,k,k+1⟩) ⟺ k ∈ ℕ
>
> Rational numbers do not have an inherent scale.
> ∀p ∈ ℚ: ∀n ∈ ℕ₁: p/n ∈ ℚ ∧ n⋅p ∈ ℚ
>
> Real numbers do not have functions.which.jump
> and are continuous everywhere.
> For each non.empty split F,H of ℝ
> exists x ∈ ℝ between F and H
>
>
> These claims are sufficient to begin,
> for our purposes, for _those_ purposes:
> not.matching, scaling, not.jumping.
>
> They can be augmented by
> claims which are not.first.false

Also not.ultimately.untrue.

Here a continuous domain, as connecting integers,
has

extent (for all neighborhoods, recursively)
density (for all neighborhoods, recursively)
completeness (gaplessness, LUB)
measure (a metric)

with regards to that thusly the IVT and thusly the FTC's arise.

About free comprehension and the sputnik of quantification,
or "Russell's element" or "Burali-Forti's ordinal", your usual
re-writing of a combination of "there exists a well-founded
inductive set, restricting free comprehension, combined
with a model of ordinals", well besides that that's always
a fragment and an axiom schema i.e. never realized,
that way, also it doesn't really declare modularity
of integral moduli, i.e., even multiples of integers,
their integer factors.

So, that helps detail at least these "axioms"
are usually added to free comprehension of setes, logical objects,
which sort of inarguably already include "expansion of comprehension".

1) well-foundedness
2) doubly-well-foundedness, an ordinary inductive set
3) usually it's underdefined the integral moduli but that's more form not an axiom
4) Least Upper Bound is an Axiom
5) Measure 1.0 is an Axiom

Then we've been getting into usual facets of relations of the spaces, topology.

6) Derivations follow only the usual open topology, restriction of comprehension

Line continuity fills Least Upper bound and Measure 1.0 for itself with
the trivial Least Upper Bound and length assignment as extent.

The notions of _minimality_ of axioms and _independence_ of axioms,
are key requirements and desiderata of the elementarily fundamental.

Le 20/01/2024 à 13:29, Richard Damon a écrit :
> On 1/20/24 6:00 AM, WM wrote:
>> Le 20/01/2024 à 00:18, Richard Damon a écrit :
>>> On 1/19/24 5:54 PM, WM wrote:
>>>> Le 19/01/2024 à 22:48, Richard Damon a écrit :
>>>>> On 1/19/24 2:26 PM, WM wrote:
>>>>>> Le 19/01/2024 à 19:22, Richard Damon a écrit :
>>>>>>> On 1/19/24 1:02 PM, WM wrote:
>>>>>>>>
>>>>>>>> Unit fractions sit only at finite points. Only there NUF can grow.
>>>>>>>
>>>>>>> Nothing in its definition says that.
>>>>>>
>>>>>> Try again.
>>>>>
>>>>> You try again. Where does the definiton of NUF(x) say what you are
>>>>> claiming.
>>>>
>>>> NUF(x) counts the unit fractions between 0 and x.
>>>
>>> And why does that say it can have the value of 1?
>>>
>>> That would require there to be a last unit fraction, and thus a
>>> highest Natural Number, which doesn't exist.
>>
>> The sequence of unit fractions ends before zero. Therefore there is a
>> last one or more last ones. Since all have gaps, there is one last one.
>
> Nope.
>
> Note "More last ones" just means they continue.

They cannot continue beyond zero. More last ones would mean many between 0
and (0, 1]. Your infinitesimals.

>> The first number has no direction, the last number has no direction, no
>> index has a direction. They are simply there or are not there. Actual
>> infinity says that all are there. That is presumed.
>
> No, the first has a direction, the dirrection of the "next" operator.

The operator does not belong to the number.
>
> First has a next but no previous, so it establishes a direction.

The set has an order, may be called a direction, but not any number.
>
Regards, WM

Le 20/01/2024 à 17:24, Jim Burns a écrit :
> On 1/19/2024 5:49 PM, WM wrote:
>
> ω is defined to be the set of final.ordinals
> ω isn't a final.ordinal.

And many finite ordinals before ω are no final ordinals too. They have no
FISONs. They are dark.

> No final.ordinal has.no.FISON.

By definition.

> No element.of.ω has.no.FISON.

Wrong. This would imply that no element of (0, 1] has less than ℵ LHS
unit fractions, but the interval has no LHS unit fractions. A beautiful
contradiction in geometry.

Regards, WM

On 1/21/24 3:25 AM, WM wrote:
> Le 20/01/2024 à 13:29, Richard Damon a écrit :
>> On 1/20/24 6:00 AM, WM wrote:
>>> Le 20/01/2024 à 00:18, Richard Damon a écrit :
>>>> On 1/19/24 5:54 PM, WM wrote:
>>>>> Le 19/01/2024 à 22:48, Richard Damon a écrit :
>>>>>> On 1/19/24 2:26 PM, WM wrote:
>>>>>>> Le 19/01/2024 à 19:22, Richard Damon a écrit :
>>>>>>>> On 1/19/24 1:02 PM, WM wrote:
>>>>>>>>>
>>>>>>>>> Unit fractions sit only at finite points. Only there NUF can grow.
>>>>>>>>
>>>>>>>> Nothing in its definition says that.
>>>>>>>
>>>>>>> Try again.
>>>>>>
>>>>>> You try again. Where does the definiton of NUF(x) say what you are
>>>>>> claiming.
>>>>>
>>>>> NUF(x) counts the unit fractions between 0 and x.
>>>>
>>>> And why does that say it can have the value of 1?
>>>>
>>>> That would require there to be a last unit fraction, and thus a
>>>> highest Natural Number, which doesn't exist.
>>>
>>> The sequence of unit fractions ends before zero. Therefore there is a
>>> last one or more last ones. Since all have gaps, there is one last one.
>>
>> Nope.
>>
>> Note "More last ones" just means they continue.
>
> They cannot continue beyond zero. More last ones would mean many between
> 0 and (0, 1]. Your infinitesimals.

But they do extend from what ever point you are at towards zero, and
there is an unbounded number of them, so you never get to the last one,
as such a thing doesn't exist.

>
>>> The first number has no direction, the last number has no direction,
>>> no index has a direction. They are simply there or are not there.
>>> Actual infinity says that all are there. That is presumed.
>>
>> No, the first has a direction, the dirrection of the "next" operator.
>
> The operator does not belong to the number.

No, but it creates numbers, and establishes a direction of the number
sequence.

>>
>> First has a next but no previous, so it establishes a direction.
>
> The set has an order, may be called a direction, but not any number.

Then why did YOU try to establish a "forward" direction from the "last"
(you call it lowest) number?

NUF(x) == 1 requires that you can find a lowest x that is a unit
fraction that only has a forward towards larger unit fractions, (lower
natural numbers), but we know that every natural number has a higher
natural number, and thus every unit fraction has a lower unit fraction,
and thus no unit fraction is at the point where NUF(x) == 1.

Your whole logic is based on the idea that there must be a "last"
natural number, (to make a smallest unit fraction) but ZFC says such a
thing does not exist.

>>
> Regards, WM
>
>
>

On 1/21/2024 3:34 AM, WM wrote:
> Le 20/01/2024 à 17:24, Jim Burns a écrit :
>> On 1/19/2024 5:49 PM, WM wrote:

>> ω is defined to be the set of final.ordinals
>> ω isn't a final.ordinal.
>
> And many finite ordinals before ω are
> no final ordinals too.

Each ordinal is represented as
the set of ordinals preceding it.
α ∈ β ⟺ α < β

ω is defined to be the set of final.ordinals.
Each ordinal is represented as
the set of ordinals preceding it.

| ∀α:( α ∈ ω ⟺ ¬(α ⇇ α∪{α}) )
is the same as
| ∀α:( α < ω ⟺ ¬(α ⇇ α∪{α}) )

| α is in ω iff α is not.final
is the same as
| α is before ω iff α is not.final

¬∃α < ω: ¬¬(α ⇇ α∪{α})
No not.final.ordinal is before ω

> They have no FISONs.
> They are dark.

They are not in/before ω

I think that,
when you (WM) see "infinite"
you think "ridiculously.big",
a thought I think at least partly because,
when you (WM) try to give a counter.example to Cantor,
you describe some ridiculously.big
(final) number.which.inserting.another.does.not.fit.

That's not what "infinite" means.
Infinitely.many is more than each
(final) number.which.inserting.another.does.not.fit,
even ridiculously.big ones.

>> No final.ordinal has.no.FISON.
>
> By definition.
>
>> No element.of.ω has.no.FISON.
>
> Wrong.

And yet,
final.ordinal == element.of.ω

How can one be true and the other false?

> This would imply that
> no element of (0, 1] has less than
> ℵ LHS unit fractions, but
> the interval has no LHS unit fractions.

In (0,x]
for each
(final) number.which.inserting.another.does.not.fit,
there are more unit fractions than that.
|⅟ℕ₁∩(0,x]| ∉ ω

In (x,1]
there is some
(final) number.which.inserting.another.does.not.fit
number of unit fractions.
|⅟ℕ₁∩(x,1]| ∈ ω

In (-∞,0]
there are no unit fractions.
|⅟ℕ₁∩(-∞,0]| = 0

> A beautiful contradiction in geometry.

Yes,
the set ω of final.ordinals is not a final.ordinal.

However,
that isn't a contradiction.
We get that by augmenting what we mean
with only not.first.false claims.

Le 21/01/2024 à 14:03, Richard Damon a écrit :
> On 1/21/24 3:25 AM, WM wrote:

>>>> The sequence of unit fractions ends before zero. Therefore there is a
>>>> last one or more last ones. Since all have gaps, there is one last one.
>>>
>>> Nope.
>>>
>>> Note "More last ones" just means they continue.
>>
>> They cannot continue beyond zero. More last ones would mean many between
>> 0 and (0, 1]. Your infinitesimals.
>
> But they do extend from what ever point you are at towards zero, and
> there is an unbounded number of them, so you never get to the last one,
> as such a thing doesn't exist.

Anyhow you cannot name all, because "all" implies that none is missing.
>
>>>> The first number has no direction, the last number has no direction,
>>>> no index has a direction. They are simply there or are not there.
>>>> Actual infinity says that all are there. That is presumed.
>>>
>>> No, the first has a direction, the dirrection of the "next" operator.
>>
>> The operator does not belong to the number.
>
> No, but it creates numbers, and establishes a direction of the number
> sequence.

Yes, a direction of the sequence but not of any number.
>
>>>
>>> First has a next but no previous, so it establishes a direction.
>>
>> The set has an order, may be called a direction, but not any number.
>
> Then why did YOU try to establish a "forward" direction from the "last"
> (you call it lowest) number?

Because the sequence has an order.
>
> Your whole logic is based on the idea that there must be a "last"
> natural number, (to make a smallest unit fraction) but ZFC says such a
> thing does not exist.

My logic is based on mathematics:
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
∀1/n ∈ ℝ: (∀x ∈ (0,1]: 1/n < x) ⇒ 1/n ≤ 0.
∀x ∈ (0, 1] ∃^ℵo 1/n < x

Regards, WM

Le 21/01/2024 à 20:58, Jim Burns a écrit :
>
> Infinitely.many is more than each
> (final) number.

Infinity is more than can be expressed by finite numbers. But all visible
numbers are measured by the last temporarily last final or visible number,
and therefore are far less than ω or ℵ.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.

Regards, WM

On 1/22/2024 3:16 AM, WM wrote:
> Le 21/01/2024 à 20:58, Jim Burns a écrit :

>> Infinitely.many is more than each
>> (final) number.
>
> Infinity is more than
> can be expressed by finite numbers.

ω is the set of finite (final) ordinals.
ω is an infinite (not.final) ordinal.
α ∈ ω ⟺ ¬(a ⇇ a+1)
ω ⇇ ω+1

ordinal β is the set of preceding ordinals
α ∈ β ⟺ α < β

α+1 means α∪{α}

α ⇇ β
means
exists 1.to.1 map to α from β
means
¬(|α| < |β|)

α ∈ ω ⟺ |α| < |α+1|
¬(|ω| < |ω+1|)

|α| < |β| ⊻ α ⇇ β

non.empty ordinal.set S holds a first
S ⊆ Ord ∧ S≠{} :⇒
∃α∈S ∀β∈S α≤β

> But all visible numbers are measured by
> the last temporarily last
> final or visible number,

No final number n measures all final numbers.
final: |n| < |n+1| < |n+2|

> and therefore are far less than ω or ℵ.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.

∀n ∈ ℕ:
ℕ\{1,…,n} ⇇ ℕ
ℕ\{1,…,n} ⇉ ℕ

∀n ∈ ℕ:
¬(|ℕ\{1….,n}| < |ℕ|)
¬(|ℕ\{1,…,n}| > |ℕ|)
|ℕ\{1,…,n}| = |ℕ|

On 1/22/24 3:02 AM, WM wrote:
> Le 21/01/2024 à 14:03, Richard Damon a écrit :
>> On 1/21/24 3:25 AM, WM wrote:
>
>>>>> The sequence of unit fractions ends before zero. Therefore there is
>>>>> a last one or more last ones. Since all have gaps, there is one
>>>>> last one.
>>>>
>>>> Nope.
>>>>
>>>> Note "More last ones" just means they continue.
>>>
>>> They cannot continue beyond zero. More last ones would mean many
>>> between 0 and (0, 1]. Your infinitesimals.
>>
>> But they do extend from what ever point you are at towards zero, and
>> there is an unbounded number of them, so you never get to the last
>> one, as such a thing doesn't exist.
>
> Anyhow you cannot name all, because "all" implies that none is missing.

And none are from the set that I can give a name to.

Of course I can't name an infinite number of them at once, but that
doesn't mean there are some that I can not name individually.

You are just using a wrong tool to define definable.

>>
>>>>> The first number has no direction, the last number has no
>>>>> direction, no index has a direction. They are simply there or are
>>>>> not there. Actual infinity says that all are there. That is presumed.
>>>>
>>>> No, the first has a direction, the dirrection of the "next" operator.
>>>
>>> The operator does not belong to the number.
>>
>> No, but it creates numbers, and establishes a direction of the number
>> sequence.
>
> Yes, a direction of the sequence but not of any number.

Which still says you can't "start" at the end that has no end.

>>
>>>>
>>>> First has a next but no previous, so it establishes a direction.
>>>
>>> The set has an order, may be called a direction, but not any number.
>>
>> Then why did YOU try to establish a "forward" direction from the
>> "last" (you call it lowest) number?
>
> Because the sequence has an order.

Right from 1/1 and up.

It doesn't have a "last" to start going the other way.

>>
>> Your whole logic is based on the idea that there must be a "last"
>> natural number, (to make a smallest unit fraction) but ZFC says such a
>> thing does not exist.
>
> My logic is based on mathematics:
> ∀n ∈ ℕ: 1/n - 1/(n+1) > 0
> ∀1/n ∈ ℝ: (∀x ∈ (0,1]: 1/n < x) ⇒ 1/n ≤ 0.
> ∀x ∈ (0, 1] ∃^ℵo 1/n < x
>
> Regards, WM
>

Which no where says there is a smallest number, in fact, we can prove
that there can't be.

In fact, your middle line proves that there is no 1/n that is less than
all numbers (0,1] as there is no 1/n that is <= 0. Which of course
doesn't actually prove anything as we knew that 1/n will be > 0 for all n.

Not sure what you are trying to say with ∃^ℵo 1/n < x but if you are
trying to say there are always an ℵo number of unit fraction below any
positive number, yes there are, which means there can not be a smallest
unit fraction.

On 1/22/24 3:16 AM, WM wrote:
> Le 21/01/2024 à 20:58, Jim Burns a écrit :
>>
>> Infinitely.many is more than each
>> (final) number.
>
> Infinity is more than can be expressed by finite numbers. But all
> visible numbers are measured by the last temporarily last final or
> visible number, and therefore are far less than ω or ℵ.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>
> Regards, WM

But there is no "last" even temporarily.

All you are saying really is that any chosen finite initial sequence of
the Natural Numbers leaves most of the Natural Numbers out.

So?

Le 22/01/2024 à 13:52, Richard Damon a écrit :
> ℵo number of unit fraction below any
> positive number, yes there are

Even below all of your infinitesimals?

Regards, WM

Le 22/01/2024 à 13:54, Richard Damon a écrit :
> On 1/22/24 3:16 AM, WM wrote:
>> Le 21/01/2024 à 20:58, Jim Burns a écrit :
>>>
>>> Infinitely.many is more than each
>>> (final) number.
>>
>> Infinity is more than can be expressed by finite numbers. But all
>> visible numbers are measured by the last temporarily last final or
>> visible number, and therefore are far less than ω or ℵ.
>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>
> But there is no "last" even temporarily.

There is a last known prime number temporarily.
Same with the last visible natnumbers.

Regards, WM

On 1/22/24 5:29 PM, WM wrote:
> Le 22/01/2024 à 13:52, Richard Damon a écrit :
>> ℵo number of unit fraction below any positive number, yes there are
>
> Even below all of your infinitesimals?
>
> Regards, WM
>
>

No, that should below any positive finite number, but then you keep on
switching between excluding the transfinite and allowing them, because
you don't know what you are talking about.

Once we move from finite to transfinite there are no unit fractions.

On 1/22/24 5:32 PM, WM wrote:
> Le 22/01/2024 à 13:54, Richard Damon a écrit :
>> On 1/22/24 3:16 AM, WM wrote:
>>> Le 21/01/2024 à 20:58, Jim Burns a écrit :
>>>>
>>>> Infinitely.many is more than each
>>>> (final) number.
>>>
>>> Infinity is more than can be expressed by finite numbers. But all
>>> visible numbers are measured by the last temporarily last final or
>>> visible number, and therefore are far less than ω or ℵ.
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>>
>> But there is no "last" even temporarily.
>
> There is a last known prime number temporarily.
> Same with the last visible natnumbers.
>
> Regards, WM

Again with "known".

There may be a last expressed d Natural Number, you can't even get away
with "Known" there, as we know all of the numbers exist, even if we
don't know yet which are prime.

Knowledge of the properties of specific numbers is different then them
having the properites.

All the Prime Numbers are Prime Nubers, we just can't name all of them
since we don't know which ones they are out of the set of Natural Numbers.

We do know which of the Natural Numbers are Natural Numbers, that is all
of them, and thus all are visible.

On Monday, January 22, 2024 at 8:00:24 PM UTC-8, Richard Damon wrote:
> On 1/22/24 5:29 PM, WM wrote:
> > Le 22/01/2024 à 13:52, Richard Damon a écrit :
> >> ℵo number of unit fraction below any positive number, yes there are
> >
> > Even below all of your infinitesimals?
> >
> > Regards, WM
> >
> >
> No, that should below any positive finite number, but then you keep on
> switching between excluding the transfinite and allowing them, because
> you don't know what you are talking about.
>
> Once we move from finite to transfinite there are no unit fractions.

It's Aristotle's usual notion, yes we know you're reflecting upon
a particularly salient standard development most easily relatable
for its employment of inequalities in arithmetic and the delta-epsilonics.

Then though you might want to know about Jordan measure which is
adding up like a line integral what aren't not infinitesimals, though
it's sort of named Jordan content to not confuse people that don't
want that though it means exactly measure, would sort of immediately
present a counterexample, to a counterexample to this sort bleating of MW.

Or, the ancient methods of exhaustion are quite thorough.

"You're an ancient Pythagorean, in a maze of twisty tunnels all alike, ...."

Le 23/01/2024 à 05:00, Richard Damon a écrit :
> On 1/22/24 5:29 PM, WM wrote:
>> Le 22/01/2024 à 13:52, Richard Damon a écrit :
>>> ℵo number of unit fraction below any positive number, yes there are
>>
>> Even below all of your infinitesimals?
>>
> No, that should below any positive finite number, but then you keep on
> switching between excluding the transfinite and allowing them,

I do not switch, but you agree to the existence of ℵo unit fractions
which cannot be found. You call them infinitesimal. That is nonsense. But
you have recognized at least that invisible unit fractions exist.

Regards, WM

Le 23/01/2024 à 05:00, Richard Damon a écrit :
> On 1/22/24 5:32 PM, WM wrote:

>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>>>
>>> But there is no "last" even temporarily.
>>
>> There is a last known prime number temporarily.
>> Same with the last visible natnumbers.

> Again with "known".
>
> There may be a last expressed d Natural Number, you can't even get away
> with "Known" there, as we know all of the numbers exist, even if we
> don't know yet which are prime.

You don't know all of the numbers individually.
>
> Knowledge of the properties of specific numbers is different then them
> having the properites.

So it is.
>
> All the Prime Numbers are Prime Nubers, we just can't name all of them
> since we don't know which ones they are out of the set of Natural Numbers.

So it is.
>
> We do know which of the Natural Numbers are Natural Numbers, that is all
> of them, and thus all are visible.

No. You know about the set, not about the individuals.

Regards, WM

WM explained on 1/23/2024 :
> Le 23/01/2024 à 05:00, Richard Damon a écrit :
>> On 1/22/24 5:32 PM, WM wrote:
>
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>>>>
>>>> But there is no "last" even temporarily.
>>>
>>> There is a last known prime number temporarily.
>>> Same with the last visible natnumbers.
>
>> Again with "known".
>>
>> There may be a last expressed d Natural Number, you can't even get away
>> with "Known" there, as we know all of the numbers exist, even if we don't
>> know yet which are prime.
>
> You don't know all of the numbers individually.
>>
>> Knowledge of the properties of specific numbers is different then them
>> having the properites.
>
> So it is.
>>
>> All the Prime Numbers are Prime Nubers, we just can't name all of them
>> since we don't know which ones they are out of the set of Natural Numbers.
>
> So it is.
>>
>> We do know which of the Natural Numbers are Natural Numbers, that is all of
>> them, and thus all are visible.
>
> No. You know about the set, not about the individuals.

It doesn't matter, when pairing them. Pairing is not like matching.

On 1/23/24 6:29 AM, WM wrote:
> Le 23/01/2024 à 05:00, Richard Damon a écrit :
>> On 1/22/24 5:29 PM, WM wrote:
>>> Le 22/01/2024 à 13:52, Richard Damon a écrit :
>>>> ℵo number of unit fraction below any positive number, yes there are
>>>
>>> Even below all of your infinitesimals?
>>>
>> No, that should below any positive finite number, but then you keep on
>> switching between excluding the transfinite and allowing them,
>
> I do not switch, but you agree to the existence of ℵo unit fractions
> which cannot be found. You call them infinitesimal. That is nonsense.
> But you have recognized at least that invisible unit fractions exist.
>
> Regards, WM
>

I didn't say they could not be found, you just think that is what I said
as you don't seem to understand logical statement.

There is nothing "nonsense" about infinitesimals, they are a fully
developed number system. Now, your logic system and brain may not
comprehend them, but that is YOUR problem, not theirs,

Your definition of NUF(x) does not imply that NUF(x) needs to have any
value other than 0 or Aleph0, and claiming that some unit fractions are
"missing" where it has a finite, non-zero falue is just incorrect.

There is no "smallest" unit fraction, but they are unboundedly small

All Unit Fractions are "Defined" and "Visible" since that holds for all
Natural Numbers too.

Your logic is presuming the existance of something that doesn't exist,
and thus has exploded in a sea of contradictions, and creates incorrect
results.

PERIOD.

On 1/23/24 6:32 AM, WM wrote:
> Le 23/01/2024 à 05:00, Richard Damon a écrit :
>> On 1/22/24 5:32 PM, WM wrote:
>
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>>>>
>>>> But there is no "last" even temporarily.
>>>
>>> There is a last known prime number temporarily.
>>> Same with the last visible natnumbers.
>
>> Again with "known".
>>
>> There may be a last expressed d Natural Number, you can't even get
>> away with "Known" there, as we know all of the numbers exist, even if
>> we don't know yet which are prime.
>
> You don't know all of the numbers individually.

Maybe not, since it is an infinite set, but I can know any of them, and
know of all of them.

>>
>> Knowledge of the properties of specific numbers is different then them
>> having the properites.
>
> So it is.

And thus, arguments about known numbers can' be used about the actual
properties of numbers.

>>
>> All the Prime Numbers are Prime Nubers, we just can't name all of them
>> since we don't know which ones they are out of the set of Natural
>> Numbers.
>
> So it is.
>>
>> We do know which of the Natural Numbers are Natural Numbers, that is
>> all of them, and thus all are visible.
>
> No. You know about the set, not about the individuals.

I know they all exist and that I can know any one of them. That is enough.

If you logic says you need to individually "know" every member of a set,
then your logic system can not handle unbouded sets.

And that seems to be where you are, using an inadiquite logic system on
an unbounded set.

>
> Regards, WM
>
>
>

On 1/23/24 6:29 AM, WM wrote:
> Le 23/01/2024 à 05:00, Richard Damon a écrit :
>> On 1/22/24 5:29 PM, WM wrote:
>>> Le 22/01/2024 à 13:52, Richard Damon a écrit :
>>>> ℵo number of unit fraction below any positive number, yes there are
>>>
>>> Even below all of your infinitesimals?
>>>
>> No, that should below any positive finite number, but then you keep on
>> switching between excluding the transfinite and allowing them,
>
> I do not switch, but you agree to the existence of ℵo unit fractions
> which cannot be found. You call them infinitesimal. That is nonsense.
> But you have recognized at least that invisible unit fractions exist.
>
> Regards, WM
>

I didn't say they could not be found, you just think that is what I said
as you don't seem to understand logical statement.

Your definition of NUF(x) does not imply that NUF(x) needs to have any
value other than 0 or Aleph0, and claiming that some unit fractions are
"missing" where it has a finite, non-zero falue is just incorrect.

There is no "smallest" unit fraction, but they are unboundedly small

All Unit Fractions are "Defined" and "Visible" since that holds for all
Natural Numbers too.

Your logic is presuming the existance of something that doesn't exist,
and thus has exploded in a sea of contradictions, and creates incorrect
results.

PERIOD.