On Sunday, August 29, 2021 at 1:06:42 PM UTC-7, Python wrote:
> Crank Wolfgang Mueckenheim, aka WM wrote:
> > Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
> >> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
> >
> >> There is only one Grossone it's Yaroslav Sergeyev's.
> >>
> > Yes.
> Reminder:
>
> ---
> Following the publication of an article by Sergeyev in EMS Surveys in
> Mathematical Sciences, the editors published the following
> clarification:
>
> Statement of the editorial board
>
> We deeply regret that this article appears in this issue of the EMS
> Surveys in Mathematical Sciences.
>
> It was a serious mistake to accept it for publication. Owing to an
> unfortunate error, the entire processing of the paper, including the
> decision to accept it, took place without the editorial board being
> aware of what was happening. The editorial board unanimously
> dissociates itself from this decision. It is not representative of the
> very high level that we expect to see in our journal, which can be
> assessed from all other papers that we have published.
>
> Both editors-in-chief have assumed responsibility for these
> mistakes and resigned from their position. Having said that, we add
> that this journal would not exist without their dedication and years of
> hard work, and we wish to register our thanks to them.
> ---

Yet, he continues to be quoted and cited in further works....

Do you happen to know much of the substance of his works?

What all, perhaps, was found so detestable?

On 8/29/2021 8:37 PM, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 1:06:42 PM UTC-7, Python wrote:
>> Crank Wolfgang Mueckenheim, aka WM wrote:
>>> Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
>>>> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
>>>
>>>> There is only one Grossone it's Yaroslav Sergeyev's.
>>>>
>>> Yes.
>> Reminder:
>>
>> ---
>> Following the publication of an article by Sergeyev in EMS Surveys in
>> Mathematical Sciences, the editors published the following
>> clarification:
>>
>> Statement of the editorial board
>>
>> We deeply regret that this article appears in this issue of the EMS
>> Surveys in Mathematical Sciences.
>>
>> It was a serious mistake to accept it for publication. Owing to an
>> unfortunate error, the entire processing of the paper, including the
>> decision to accept it, took place without the editorial board being
>> aware of what was happening. The editorial board unanimously
>> dissociates itself from this decision. It is not representative of the
>> very high level that we expect to see in our journal, which can be
>> assessed from all other papers that we have published.
>>
>> Both editors-in-chief have assumed responsibility for these
>> mistakes and resigned from their position. Having said that, we add
>> that this journal would not exist without their dedication and years of
>> hard work, and we wish to register our thanks to them.
>> ---
>
> Yet, he continues to be quoted and cited in further works....
>
> Do you happen to know much of the substance of his works?
>
> What all, perhaps, was found so detestable?
>

bad math

On Sunday, August 29, 2021 at 1:01:36 PM UTC-7, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 12:39:51 PM UTC-7, Ross A. Finlayson wrote:
> > On Sunday, August 29, 2021 at 12:10:22 PM UTC-7, Ross A. Finlayson wrote:
> > > On Sunday, August 29, 2021 at 11:24:51 AM UTC-7, Jim Burns wrote:
> > > > On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> > > > > On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> > > > > Jim Burns wrote:
> > > >
> > > > >> To borrow a term from art, a gap is _negative space_
> > > > >> It is an interval in which points are NOT.
> > > > >> https://en.wikipedia.org/wiki/Negative_space
> > > > >>
> > > > >> In order for there to be a point next to zero,
> > > > >> there must be negative space next to zero, and then the point.
> > > > >> If there is no negative space, there is no next point.
> > > > >> If there is no negative space, there is no gap.
> > > > >
> > > > > Least Upper Bound is the usual axiom attached to the
> > > > > ordered field to result the gapless complete ordered field.
> > > > Here, I'm using "gap" in a more restricted sense than I have
> > > > elsewhere. Call them "multi-point gaps".
> > > >
> > > > The rationals are already sufficient for a multi-point-gapless
> > > > ordered field. No multi-point interval (a,b) with a ~= b
> > > > is without rationals. No (a,b) is a gap.
> > > >
> > > > The Least Upper Bound property eliminates single-point gaps.
> > > >
> > > > One important consequence of the elimination of single-point
> > > > gaps is that we can formally describe and reason about
> > > > Achilles chasing the tortoise along a _continuous_ path.
> > > > Well, we could use the same definitions _without_ LUB, but they
> > > > won't mean what we want them to mean. The path Achilles takes
> > > > can be continuous _at every point_ but, if there are
> > > > single-point gaps, he can execute arbitrarily large teleports
> > > > across that gap. Sure, that wouldn't be continuous _if there_
> > > > _was a point there_ but there isn't.
> > > >
> > > > That is the reason (one, at least) that we need the LUB
> > > > property: so that "continuous at all the points"
> > > > works out to "no teleporting".
> > > > > (That the definition of continuous function applies to
> > > > > domains that preserve a metric.)
> > > > Continuous functions don't need to preserve a metric.
> > > >
> > > > Rotations and translations preserve the metric.
> > > >
> > > > Something along those lines that is true:
> > > > The inverse f^-1 of a continuous function f maps open sets
> > > > to open sets. We could say f^-1 preserves the topology.
> > > Such a topology as about this points-in-a-line would
> > > have two-point neighborhoods at endpoints and
> > > three-point neighborhoods throughout the line.
> > >
> > > Not sure "continuous functions don't preserve a metric".
> > > I'll agree they're perhaps not affine, where they're also
> > > transformations and make a mapping.
> > >
> > > I.e. they define the reversible/invertible mapping the metric.
> > >
> > > I thank you though, about the affine.
> > >
> > > The continuous functions don't so much care that the image
> > > is the rationals or complete ordered field or as about pretty
> > > much any other topology with a metric.
> > https://en.wikipedia.org/wiki/Natural_topology
> My point here is that though the usual "must be no smallest"
> and "must be next largest" are at odds, that, when there are
> both structures then each has all the usual features of what
> makes for modeling each other's natural topologies.
>
> That a vector space makes a basis for a topology,
> and that ran(EF) is a vector space, with its range of iota-values.
>
> "The natural/unit equivalency function that best relates
> integers zero to infinity and reals zero through one is a
> drawing of a line."
>
> Once a model of real numbers that is iota-values is equipped
> with all the other usual features what are handles for analysis,
> there is implemented a neat extensionality what all the results
> go both ways for either.
>
> I.e. the length assignment for ran(f) the iota-values, pretty much
> stands up as a default metric along with all the other usual naive
> properties why this object is more primitive, and primary, in theory,
> than needs require after all of topology the ultrafilter, or a countable
> basis in a space for a characterization of real regions, that it's the
> default natural model as for example by time of constant motion or change.
>
>
> Or, besides being fundamentally at odds,
> models of continuity are reflective and replete.
>
> Then I have written this up and about its topologies
> the contiguous points "in" a line vis-a-vis the
> ordered field's points "on" a line, why each are
> models of continuity, or here continuous domains.

About infinitesimals and their modern rehabilitations,
perhaps you'll find interesting Katz' collected studies.

https://u.math.biu.ac.il/~katzmik/infinitesimals.html

On 8/29/2021 1:24 PM, Jim Burns wrote:
> On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
>> On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
>> Jim Burns wrote:
>
>>> To borrow a term from art, a gap is _negative space_
>>> It is an interval in which points are NOT.
>>> https://en.wikipedia.org/wiki/Negative_space
>>>
>>> In order for there to be a point next to zero,
>>> there must be negative space next to zero, and then the point.
>>> If there is no negative space, there is no next point.
>>> If there is no negative space, there is no gap.
>>
>> Least Upper Bound is the usual axiom attached to the
>> ordered field to result the gapless complete ordered field.
>
> Here, I'm using "gap" in a more restricted sense than I have
> elsewhere. Call them "multi-point gaps".
>
> The rationals are already sufficient for a multi-point-gapless
> ordered field. No multi-point interval (a,b) with a ~= b
> is without rationals. No (a,b) is a gap.
>
> The Least Upper Bound property eliminates single-point gaps.
>
> One important consequence of the elimination of single-point
> gaps is that we can formally describe and reason about
> Achilles chasing the tortoise along a _continuous_ path.
> Well, we could use the same definitions _without_ LUB, but they
> won't mean what we want them to mean. The path Achilles takes
> can be continuous _at every point_ but, if there are
> single-point gaps, he can execute arbitrarily large teleports
> across that gap. Sure, that wouldn't be continuous _if there_
> _was a point there_ but there isn't.
>
> That is the reason (one, at least) that we need the LUB
> property: so that "continuous at all the points"
> works out to "no teleporting".
>
>> (That the definition of continuous function applies to
>> domains that preserve a metric.)
>
> Continuous functions don't need to preserve a metric.
>
> Rotations and translations preserve the metric.
>
> Something along those lines that is true:
> The inverse f^-1 of a continuous function f maps open sets
> to open sets. We could say f^-1 preserves the topology.
>

so, Achilles with a spear is running after this poor scared turtle on a stone path, where the ground is continuous, but each stone (numbered in
ascending natural numbers) had GAPS between all the stones. And WM says each gap is brimming with, and infested with, Dark Numbers.

Question: How does the turtle get across all these gaps packed full of Dark Numbers ? Does he grease the bottom of his shell ?

On Sunday, August 29, 2021 at 6:48:02 PM UTC-7, Serg io wrote:
> On 8/29/2021 8:37 PM, Ross A. Finlayson wrote:
> > On Sunday, August 29, 2021 at 1:06:42 PM UTC-7, Python wrote:
> >> Crank Wolfgang Mueckenheim, aka WM wrote:
> >>> Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
> >>>> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
> >>>
> >>>> There is only one Grossone it's Yaroslav Sergeyev's.
> >>>>
> >>> Yes.
> >> Reminder:
> >>
> >> ---
> >> Following the publication of an article by Sergeyev in EMS Surveys in
> >> Mathematical Sciences, the editors published the following
> >> clarification:
> >>
> >> Statement of the editorial board
> >>
> >> We deeply regret that this article appears in this issue of the EMS
> >> Surveys in Mathematical Sciences.
> >>
> >> It was a serious mistake to accept it for publication. Owing to an
> >> unfortunate error, the entire processing of the paper, including the
> >> decision to accept it, took place without the editorial board being
> >> aware of what was happening. The editorial board unanimously
> >> dissociates itself from this decision. It is not representative of the
> >> very high level that we expect to see in our journal, which can be
> >> assessed from all other papers that we have published.
> >>
> >> Both editors-in-chief have assumed responsibility for these
> >> mistakes and resigned from their position. Having said that, we add
> >> that this journal would not exist without their dedication and years of
> >> hard work, and we wish to register our thanks to them.
> >> ---
> >
> > Yet, he continues to be quoted and cited in further works....
> >
> > Do you happen to know much of the substance of his works?
> >
> > What all, perhaps, was found so detestable?
> >
> bad math

Yeah, it's fair that there's the edifice of formalism to satisfy,
also with respect to underdefined terms, and otherwise as about
validity and soundness and all.

Still, though, in scalar terms, and about something like the
long line, which as the space of functions R to R is bigger than
R but parameterized by values in R, sorts in trichotomy,
there's something to be said for the terms of his theory all
by themselves, tools or platforms or a plank, if not, of course,
foundations of mathematics writ large and "true" definition.

I just find it much, much simpler to have a primitive for time,
central in the theory, instead of building a model of one very,
very high above in ultrafilters, the parametrizing, and as where
the model of countable additivity and well-ordering across the
range uses much, much more numerical resources. (Than sweep.)

I.e., you should know they have one just like that, but,
can't admit its properties back to set theory.

While, revising set theory with the ubiquitous ordinals,
for the powerset theorem, then having sweep as a
non-Cartesian function or even as simply defined,
has a formalism for continuity that can be relayed,
among formalisms of continuity (and not just umpteen
"conservative" extensions of each other).

Or, those two will never stop arguing.

On Sunday, August 29, 2021 at 6:53:20 PM UTC-7, Serg io wrote:
> On 8/29/2021 1:24 PM, Jim Burns wrote:
> > On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> >> On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> >> Jim Burns wrote:
> >
> >>> To borrow a term from art, a gap is _negative space_
> >>> It is an interval in which points are NOT.
> >>> https://en.wikipedia.org/wiki/Negative_space
> >>>
> >>> In order for there to be a point next to zero,
> >>> there must be negative space next to zero, and then the point.
> >>> If there is no negative space, there is no next point.
> >>> If there is no negative space, there is no gap.
> >>
> >> Least Upper Bound is the usual axiom attached to the
> >> ordered field to result the gapless complete ordered field.
> >
> > Here, I'm using "gap" in a more restricted sense than I have
> > elsewhere. Call them "multi-point gaps".
> >
> > The rationals are already sufficient for a multi-point-gapless
> > ordered field. No multi-point interval (a,b) with a ~= b
> > is without rationals. No (a,b) is a gap.
> >
> > The Least Upper Bound property eliminates single-point gaps.
> >
> > One important consequence of the elimination of single-point
> > gaps is that we can formally describe and reason about
> > Achilles chasing the tortoise along a _continuous_ path.
> > Well, we could use the same definitions _without_ LUB, but they
> > won't mean what we want them to mean. The path Achilles takes
> > can be continuous _at every point_ but, if there are
> > single-point gaps, he can execute arbitrarily large teleports
> > across that gap. Sure, that wouldn't be continuous _if there_
> > _was a point there_ but there isn't.
> >
> > That is the reason (one, at least) that we need the LUB
> > property: so that "continuous at all the points"
> > works out to "no teleporting".
> >
> >> (That the definition of continuous function applies to
> >> domains that preserve a metric.)
> >
> > Continuous functions don't need to preserve a metric.
> >
> > Rotations and translations preserve the metric.
> >
> > Something along those lines that is true:
> > The inverse f^-1 of a continuous function f maps open sets
> > to open sets. We could say f^-1 preserves the topology.
> >
> so, Achilles with a spear is running after this poor scared turtle on a stone path, where the ground is continuous, but each stone (numbered in
> ascending natural numbers) had GAPS between all the stones. And WM says each gap is brimming with, and infested with, Dark Numbers.
>
> Question: How does the turtle get across all these gaps packed full of Dark Numbers ? Does he grease the bottom of his shell ?

"Paradoxes of Zeno _again_? Don't we have that at the _beginning_?"

On 8/29/2021 8:37 PM, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 1:06:42 PM UTC-7, Python wrote:
>> Crank Wolfgang Mueckenheim, aka WM wrote:
>>> Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
>>>> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
>>>
>>>> There is only one Grossone it's Yaroslav Sergeyev's.
>>>>
>>> Yes.
>> Reminder:
>>
>> ---
>> Following the publication of an article by Sergeyev in EMS Surveys in
>> Mathematical Sciences, the editors published the following
>> clarification:
>>
>> Statement of the editorial board
>>
>> We deeply regret that this article appears in this issue of the EMS
>> Surveys in Mathematical Sciences.
>>
>> It was a serious mistake to accept it for publication. Owing to an
>> unfortunate error, the entire processing of the paper, including the
>> decision to accept it, took place without the editorial board being
>> aware of what was happening. The editorial board unanimously
>> dissociates itself from this decision. It is not representative of the
>> very high level that we expect to see in our journal, which can be
>> assessed from all other papers that we have published.
>>
>> Both editors-in-chief have assumed responsibility for these
>> mistakes and resigned from their position. Having said that, we add
>> that this journal would not exist without their dedication and years of
>> hard work, and we wish to register our thanks to them.
>> ---
>
> Yet, he continues to be quoted and cited in further works....
>
> Do you happen to know much of the substance of his works?
>
> What all, perhaps, was found so detestable?
>

One critic, Alexander Shen, director of research at the National Center for Scientific Research in Montpellier, France wrote to the editors on Nov. 29,
expressing his concern:

Assuming that this is not a fake or April Fool Day’s joke and the paper was indeed published, I would say that the reputation damage to the Journal,
to EMS and to mathematical community as a whole cannot be fully undone, but a clear apology/retraction seems to be needed.

However, Shen and two mathematicians from Novosibirsk State University in Russia—S. Kutateladze and Alexander Gutman—have published critiques of
Sergeyev’s work. Gutman told us that he thinks the current paper, along with other articles by Sergeyev, are disrespectful to modern mathematics:

[Sergeyev] simply does not call things by their proper names. In his games with “grossone”, the fundamental concepts are redefined, and famous
complex or unresolved problems are replaced by their primitive counterparts. Of course, it is very unpleasant to observe such substitutions, especially
with respect to real mathematical problems, which are complex and beautiful.

Kutateladze added:

The recent publication of the paper by Sergeyev in European Mathematical Society Surveys was a scandalous blunder.

.....................................................................

Sergeyev’s idea is to introduce into arithmetic some infinitely large number— grossone, consider only the numbers that are less than the grossone, and
operate exclusively on these numbers. Sergeyev embellishes his idea with shallow arguments, emphasizing that he does not use Cantor’s approach and
returns to Ancient Greeks. He formulates the three postulates of his own:

Postulate 1. Existence of infinite and infinitesimal objects is postulated but it is also accepted that human beings and machines are able to
execute only a finite number of operations.
Postulate 2. It is not discussed what are the mathematical objects we deal with; we just construct more powerful tools that allow us to improve our
capacities to observe and to describe properties of mathematical objects.

Sergeyev confronts his ideas with the famous nonstandard analysis of Abraham Robinson. Robinsonian infinitesimal analysis is rightfully considered as
one of the most brilliant achievements of mathematics in the twentieth century. Using the sophisticated methods of the recently-invented model theory in
the beginning of the 1960s, Robinson synthesized the approaches by Newton and Leibniz in a new mathematical language and technique. Nonstandard analysis
not only inherited all technical tools of Newton’s method of prime and ultimate ratios and Leibniz’s monads but also explained and justified the
ingenious tricks of Euler who freely dealt with actual infinites and infinitesimals.

Sergeyev has poor knowledge of these classical scientific achievements, counterposing his bizarre surmisals to the modern analysis. But all linguistic
and mathematical tools that are needed to Sergeyev are readily available within nonstandard analysis.

Sergeyev defines his grossone as the “number of elements of the set of natural numbers.” In fact, the role of this would-be mysterious entity can
happily be performed by the factorial of an arbitrary infinite number which are galore in nonstandard analysis. This circumstance is absolutely evident
to specialists but nonetheless it was meticulously elaborated in Siberian Mathematical Journal, 49:5, 835–841 (2008) in order to demonstrate the humble
place of Sergeyev’s speculations. This article also revealed the principal shortcomings of Sergeyev’s attempts at implementing calculations with a
grossone on a computer. Unfortunately, it turned out impossible to stop the flood of Sergeyev’s publications in the variety of the international
journals having little if any in common with foundations of analysis. Miraculously, there are no Sergeyev’s publications on his grossone in Russian in
the Russian mathematical database Math-Net.Ru.

Sergeyev’s writings about invention of new numbers and the “Infinity Computer” are speculations of negligible interest to mathematics but exuberant with
pretensions, which might be perilous to science. The fact that these speculations underlie the Pytharogas Award by the City of Crotone on behalf of the
University of Calabria having Sergeyev on staff cannot improve neither the content of the writings of Sergeyev nor his attitude to the precious
knowledge of the treasure-trove of mathematics.

Ancient Italian grossones are linguistically close to Sergeyev’s grossone but differ in value.
Postulate 3. The principle formulated by Ancient Greeks ‘The part is less than the whole’ is applied to all numbers (finite, infinite, and
infinitesimal) and to all sets and processes (finite and infinite).

On Sunday, August 29, 2021 at 7:10:50 PM UTC-7, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 6:53:20 PM UTC-7, Serg io wrote:
> > On 8/29/2021 1:24 PM, Jim Burns wrote:
> > > On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> > >> On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> > >> Jim Burns wrote:
> > >
> > >>> To borrow a term from art, a gap is _negative space_
> > >>> It is an interval in which points are NOT.
> > >>> https://en.wikipedia.org/wiki/Negative_space
> > >>>
> > >>> In order for there to be a point next to zero,
> > >>> there must be negative space next to zero, and then the point.
> > >>> If there is no negative space, there is no next point.
> > >>> If there is no negative space, there is no gap.
> > >>
> > >> Least Upper Bound is the usual axiom attached to the
> > >> ordered field to result the gapless complete ordered field.
> > >
> > > Here, I'm using "gap" in a more restricted sense than I have
> > > elsewhere. Call them "multi-point gaps".
> > >
> > > The rationals are already sufficient for a multi-point-gapless
> > > ordered field. No multi-point interval (a,b) with a ~= b
> > > is without rationals. No (a,b) is a gap.
> > >
> > > The Least Upper Bound property eliminates single-point gaps.
> > >
> > > One important consequence of the elimination of single-point
> > > gaps is that we can formally describe and reason about
> > > Achilles chasing the tortoise along a _continuous_ path.
> > > Well, we could use the same definitions _without_ LUB, but they
> > > won't mean what we want them to mean. The path Achilles takes
> > > can be continuous _at every point_ but, if there are
> > > single-point gaps, he can execute arbitrarily large teleports
> > > across that gap. Sure, that wouldn't be continuous _if there_
> > > _was a point there_ but there isn't.
> > >
> > > That is the reason (one, at least) that we need the LUB
> > > property: so that "continuous at all the points"
> > > works out to "no teleporting".
> > >
> > >> (That the definition of continuous function applies to
> > >> domains that preserve a metric.)
> > >
> > > Continuous functions don't need to preserve a metric.
> > >
> > > Rotations and translations preserve the metric.
> > >
> > > Something along those lines that is true:
> > > The inverse f^-1 of a continuous function f maps open sets
> > > to open sets. We could say f^-1 preserves the topology.
> > >
> > so, Achilles with a spear is running after this poor scared turtle on a stone path, where the ground is continuous, but each stone (numbered in
> > ascending natural numbers) had GAPS between all the stones. And WM says each gap is brimming with, and infested with, Dark Numbers.
> >
> > Question: How does the turtle get across all these gaps packed full of Dark Numbers ? Does he grease the bottom of his shell ?
> "Paradoxes of Zeno _again_? Don't we have that at the _beginning_?"

"... which is among reasons why there is the Finlayson slates for uncountability and paradox."

torsdag 26 augusti 2021 kl. 14:38:12 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 26. August 2021 um 07:15:08 UTC+2:
> > onsdag 25 augusti 2021 kl. 14:29:18 UTC+2 skrev WM:
>
> > > > ALL rational are indiced. It is trivial to show there is a bijection betwee N and Q+
> > > It is trivial to show the contrary:
> > > ∀ q ∈ (0, 1] ∃ q+n ∈ (n, n+1] und ∀ q+n ∈ (n, n+1] ∃ q ∈ (0, 1]
> > > 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...
> > That does not go against my statement that there is a bijection between N and Q+
> Not against a bijection between the fractions enumerated in (0, 1] and (1000, 1001]. But it goes against the assumption, that Cantor enumerates all fractions. There are less enumerated in (1000, 1001] than in (0, 1] but there are not less in (1000, 1001]. This excludes a bijection between |N and Q+ by lack of surjectivity.
>
> Regards, WM

None of that follows from anything you said. It is surjective and gets all of the fractions.

It is trivial to show that they are of equal cardinality.

Serg io schrieb am Montag, 30. August 2021 um 04:17:05 UTC+2:

> One critic, Alexander Shen, director of research at the National Center for Scientific Research in Montpellier, France wrote to the editors on Nov. 29,
> expressing his concern:
>
> Assuming that this is not a fake or April Fool Day’s joke and the paper was indeed published, I would say that the reputation damage to the Journal,
> to EMS and to mathematical community as a whole cannot be fully undone, but a clear apology/retraction seems to be needed.

There we see that many people are too dense to comprehend new ideas. In particual in France there live many handicapped mathematicians as Arnold reported.
>
> Sergeyev’s work. Gutman told us that he thinks the current paper, along with other articles by Sergeyev, are disrespectful to modern mathematics:

That is the proper position. Modern mathematics is primarily Cantor's nonsense and does not deserve respect but contempt.
>
> Kutateladze added:
>
> The recent publication of the paper by Sergeyev in European Mathematical Society Surveys was a scandalous blunder.

"This is an instance of the amazing power of desire in blinding even very able men to fallacies which would otherwise be obvious at once." [Bertrand Russell]

> Robinsonian infinitesimal analysis is rightfully considered as
> one of the most brilliant achievements of mathematics in the twentieth century.

It had lasted a good while until this was fully recognized. But the fools of matheology will forever constitute the mayjority. Good taste is not the taste of the majority.

Regards, WM

Jim Burns schrieb am Sonntag, 29. August 2021 um 23:04:02 UTC+2:
> On 8/27/2021 10:23 AM, WM wrote:
> > Serg io schrieb
> > am Freitag, 27. August 2021 um 16:18:14 UTC+2:
>
> >> Cantor enumerates all rationals.
> >
> > Why doesn't he start with all fractions n/1?
>
> There are infinitely-many fractions n/1.
> Starting with all n/1 first would make all the other
> fractions inaccessible to one-by-one stepping.

But when hidden "in the infinite", this does not happen? No, it does, but you cannot see it.
>
> Since the claim that that _at least one_ way to order
> the rationals has them all (like the naturals) accessible
> to one-by-one stepping (in that non-standard order)
> from any other rationals,
> starting with all n/1 is not a counter-example.
> It is merely not an example.

If surjectivity would prevail then the order was irrelevant. All is all, whatever you start with.
>
> But there are plenty of examples.
> We didn't need yours, anyway.

You would need plenty of brain to understand that a bijection does not depend on the ordering. Even Cantor made the mistake to admit this: "Those and only those transformations leave the ordinal number unchanged which can be put down to a finite or infinite set of transpositions, i.e., exchanges of two elements." [Cantor, p. 214] Of course this is false for Cantor's fake bijections.

Regards, WM

Jim Burns schrieb am Sonntag, 29. August 2021 um 18:43:59 UTC+2:

> > Arithmetic is consistent.
>
> Arithmetic says
> for each positive rational p, p > 0 ,
> a unique positive rational q, q > 1,
> exists such that q = p + 1.

Arithmetic would remain consistent in potential infinity without any dark numbers. This proves that arithmetic concerns only definable numbers. Although we cannot know about the properties of dark individuals, q = p + 1 is correct for definable and dark numbers, at least collectively, because we have the geometric argument: Translation invariance. The interval (0, 1] is identical to the interval (1, 2] except the added 1.
>
> Arithmetic says
> for each positive rational q, q > 1 ,
> a unique positive rational r, 1 > r > 0,
> exists such that r = 1/q.

Yes, but this can hold only for the definable rationals. Otherwise half of all rationals were in (0, 1], but by adding 1 to every fractions we would find half of all rationals in (1, 2]. This would show arithmetic inconsistent.
>
> Arithmetic says
> for each positive rational r, 1 > r > 0,
> for any positive rationals s,t,
> a unique positive rational u, s < u < s+t,
> exists such that u = s + t*r.

Again, arithmetic can handle only definable numbers, rationals as well as irrationals.
Your further similar arguments are therefore deleted. But my compliments to this very good and helpful contribution!

> > Set theory is inconsistent because
> > if we have an infinite matrix or plane,
> > then its entries or points above the diagonal
> > are precisely as many as below.
>
> You're claiming that set theory is inconsistent
> despite it *agreeing* with arithmetic.
> Arithmetic, though, is consistent.

See above. In order to handle dark numbers or elements which cannot be chosen as individuals we can only use tools which can be applied collectively. That is translation invariance and halving the infinite square by the diagonal.

Again thanks for your contribution. It has cleared a lot in my model.

Regards, WM

Jim Burns schrieb am Sonntag, 29. August 2021 um 19:06:47 UTC+2:
> On 8/29/2021 12:08 PM, WM wrote:
> > William schrieb
> > am Samstag, 28. August 2021 um 22:01:23 UTC+2:
> >> On Saturday, August 28, 2021 at 3:47:34 PM UTC-3,
> >> WM wrote:
>
> >>>> It is not true that there must be
> >>>> a gap next to zero, or a point next to zero.
> >>>
> >>> Either there is a definable point next to zero or
> >>> there is no definable point next to zero.
> >>> Do you agree?
> >>
> >> There is neither a point ("definable" or not)
> >> nor a gap next to zero.
> >
> > What is next to zero?
> > Note that "nothing" is synonymous to "gap".
> Note that "nothing" is NOT synonymous with "gap".

By my definition it is. But you can also say: Either there is a point next to zero or there is no point next to zero.

Regards, WM

Greg Cunt schrieb am Sonntag, 29. August 2021 um 19:26:21 UTC+2:
> On Sunday, August 29, 2021 at 6:35:52 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
> > >
> > > There is no "point next to zero",
> > >
> > Then there is a gap.
> No, there is no gap.

So it is. There are dark numbers, none of which can be identified, let alone be identified as the last one.
>
> Hint: ~Ex e IR: Ay e (0, 1]: 0 < x < y .

Of course not. What you call existing means it can be chosen.

Regards, WM

Jim Burns schrieb am Sonntag, 29. August 2021 um 19:31:15 UTC+2:
> On 8/29/2021 12:31 PM, WM wrote:
> > Jim Burns schrieb
> > am Sonntag, 29. August 2021 um 06:58:57 UTC+2:
> >> In order for x to be next to 0,
> >> there must be no point y between 0 and x,
> >> there must be no point y in (0,x).
> >> In other words, (0,x) must be a gap.
> >
> > You forget that dark points cannot be analyzed
> > as individuals or ordered.
> A gap with points in it is not a gap.
> Please do not forget this.

No, if there are point, there is no gap.

> > But either there is something next to 0 or nothing.
> >
> > This are the alternatives:
> > A defined point or not a defined point.
> For each interval (0, d), there is a unit fraction 1/k
> such that a steppable 0,...,k exists in which,
> for adjacent i,j, j = i+1.

A defined point or a definable point cannot be next to zero.

> What is there that can be next to zero
> without there being a gap between it and zero?

Dark points.

Regards, WM

William schrieb am Sonntag, 29. August 2021 um 22:35:47 UTC+2:
> On Sunday, August 29, 2021 at 4:43:07 PM UTC-3, WM wrote:
> > William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
> > > On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
> > >
> > > > What is next to zero? Note that "nothing" is synonymous to "gap".
> > > Nope
> > > A gap next to 0 is an interval [0,x] (note x must be a real number
> > The end of the gap is a real number x
> and a "gap", like a set, is not something that changes.

The collection of definable points is the complement. Therefore the size of the gap changes.

Regards, WM

Jim Burns schrieb am Sonntag, 29. August 2021 um 22:50:17 UTC+2:
> On 8/27/2021 7:27 AM, WM wrote:
> > Jim Burns schrieb
> > am Freitag, 27. August 2021 um 00:20:04 UTC+2:
>
> >> Each unit fraction 1/k is not _next to_ 0
> >> because 0 < 1/(k+1) < 1/k
> >
> > The cursor moving from 0 to 1 will encounter
> > some unit fraction first.
>
> The cursor can only encounter a non-unit-fraction unit fraction
> first. Any unit-fraction unit fraction will not be first.

One of them must be first. If the cursor is at 1/17, then this will not be the first unit fraction encoutered.

Regards, WM

On Monday, August 30, 2021 at 12:56:15 PM UTC+2, WM wrote:
> Jim Burns schrieb am Sonntag, 29. August 2021 um 22:50:17 UTC+2:
> >
> > The cursor can only encounter a non-unit-fraction unit fraction
> > first. Any unit-fraction unit fraction will not be first.

For simplicity let's call a "unit-fraction unit fraction" "unit-fraction".

> One of them must be first.

No. (Your claim is just one of the dogmas of your matheology.)

> If the cursor is at 1/17, then this will not be the first unit fraction encoutered.

Well, this claim can be compared with Russell's "The present king of France is bald."

Or, say, "1/17 is not the present king of France."

The problem with these sentences/claims is that THERE IS NO present king of France. (Concerning your orignal claim: There is no first unit fraction [encoutered].)

Go figure.

Greg Cunt schrieb am Montag, 30. August 2021 um 14:07:35 UTC+2:

> > If the cursor is at 1/17, then this will not be the first unit fraction encoutered.
> Well, this claim can be compared with Russell's "The present king of France is bald."

Wrong. 1/17 does exist.

> The problem with these sentences/claims is that THERE IS NO present king of France. (Concerning your orignal claim: There is no first unit fraction [encoutered].)

There is a first unit fraction encounterd, depending on the scale, the velocity, the attentiveness, etc. But if having encounterede it, it will be possible to find even smaller unit fractions. Nevertheless ℵo unit fractions will remain dark.

Regards, WM

On Monday, August 30, 2021 at 2:30:49 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Montag, 30. August 2021 um 14:07:35 UTC+2:
> >
> > The problem with these sentences/claims is that THERE IS NO present king of France. (Concerning your orignal claim: There is no first unit fraction [encoutered].)
> >
> There is a first unit fraction encounterd, depending on the scale, the velocity, the attentiveness, etc.

What you mean is "realized" not "encounterd". So your "cursor" is a compexe structure with inner workings? (How do unit fractions and/or rational numbers affect this mechanism, btw.? I mean, how does it recognize unit fractions?)

Still there's no first unit fraction. THAT was the original topic, you silly crank. Not if there's a "first unit fraction realized by a/the cursor".

Holy shit!

Langsam habe ich Ihren Schachsinn dicke, Mückenheim. Soviel Scheiße wie Sie daherlaber, das kann kein Mensch aushalten.

On Monday, August 30, 2021 at 7:52:58 AM UTC-3, WM wrote:
> William schrieb am Sonntag, 29. August 2021 um 22:35:47 UTC+2:
> > On Sunday, August 29, 2021 at 4:43:07 PM UTC-3, WM wrote:
> > > William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
> > > > On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
> > > >
> > > > > What is next to zero? Note that "nothing" is synonymous to "gap".
> > > > Nope
> > > > A gap next to 0 is an interval [0,x] (note x must be a real number
> > > The end of the gap is a real number x
> > and a "gap", like a set, is not something that changes.
> The collection of definable points is the complement.

Nope, the set of points that you can write down plus the points that you cannot write down is the complement. The gap does not change.

--
William Hughes

On 8/30/2021 5:19 AM, WM wrote:
> Greg Cunt schrieb am Sonntag, 29. August 2021 um 19:26:21 UTC+2:
>> On Sunday, August 29, 2021 at 6:35:52 PM UTC+2, WM wrote:
>>> Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
>>>>
>>>> There is no "point next to zero",
>>>>
>>> Then there is a gap.
>> No, there is no gap.
>
> So it is. There are dark numbers, none of which can be identified, let alone be identified as the last one.

nope.
We can prove there is no gap, no number next to zero, this includes your dark numbers,

we can prove unnamed numbers are not next to zero.

On 8/30/2021 3:44 AM, WM wrote:
> Serg io schrieb am Montag, 30. August 2021 um 04:17:05 UTC+2:
>
>> One critic, Alexander Shen, director of research at the National Center for Scientific Research in Montpellier, France wrote to the editors on Nov. 29,
>> expressing his concern:
>>
>> Assuming that this is not a fake or April Fool Day’s joke and the paper was indeed published, I would say that the reputation damage to the Journal,
>> to EMS and to mathematical community as a whole cannot be fully undone, but a clear apology/retraction seems to be needed.
>
> There we see that many people are too dense to comprehend new ideas. In particual in France there live many handicapped mathematicians as Arnold reported.
>>
>> Sergeyev’s work. Gutman told us that he thinks the current paper, along with other articles by Sergeyev, are disrespectful to modern mathematics:
>
> That is the proper position. Modern mathematics is primarily Cantor's nonsense and does not deserve respect but contempt.
>>
>> Kutateladze added:
>>
>> The recent publication of the paper by Sergeyev in European Mathematical Society Surveys was a scandalous blunder.
>
> "This is an instance of the amazing power of desire in blinding even very able men to fallacies which would otherwise be obvious at once." [Bertrand Russell]
>
>> Robinsonian infinitesimal analysis is rightfully considered as
>> one of the most brilliant achievements of mathematics in the twentieth century.
>
> It had lasted a good while until this was fully recognized. But the fools of matheology will forever constitute the mayjority. Good taste is not the taste of the majority.
>
> Regards, WM
>

blaming the rest of the world for your mistakes, is what Joe Biden does.

you snipped out Sergeyev's April Fool Day’s joke,

Sergeyev embellishes his idea with shallow arguments, emphasizing that he does not use Cantor’s approach and
returns to Ancient Greeks. He formulates the three postulates of his own:

Postulate 1. Existence of infinite and infinitesimal objects is postulated but it is also accepted that human beings and machines are able to
execute only a finite number of operations.
Postulate 2. It is not discussed what are the mathematical objects we deal with; we just construct more powerful tools that allow us to improve our
capacities to observe and to describe properties of mathematical objects.

Sergeyev has poor knowledge of these classical scientific achievements, counterposing his bizarre surmisals to the modern analysis. But all linguistic
and mathematical tools that are needed to Sergeyev are readily available within nonstandard analysis.

On 8/30/2021 5:17 AM, WM wrote:
> Jim Burns schrieb am Sonntag, 29. August 2021 um 19:06:47 UTC+2:
>> On 8/29/2021 12:08 PM, WM wrote:
>>> William schrieb
>>> am Samstag, 28. August 2021 um 22:01:23 UTC+2:
>>>> On Saturday, August 28, 2021 at 3:47:34 PM UTC-3,
>>>> WM wrote:
>>
>>>>>> It is not true that there must be
>>>>>> a gap next to zero, or a point next to zero.
>>>>>
>>>>> Either there is a definable point next to zero or
>>>>> there is no definable point next to zero.
>>>>> Do you agree?
>>>>
>>>> There is neither a point ("definable" or not)
>>>> nor a gap next to zero.
>>>
>>> What is next to zero?
>>> Note that "nothing" is synonymous to "gap".
>> Note that "nothing" is NOT synonymous with "gap".
>
> By my definition it is. But you can also say: Either there is a point next to zero or there is no point next to zero.
>
> Regards, WM
>

no, nothing means 0.

A gap is an interval and is specified by two endpoints. What are the endpoint of your assumed "gap" ?

there is no "next to", we have posted the proof using math that there is no "next to". Deal with it.

On 8/30/2021 5:52 AM, WM wrote:
> William schrieb am Sonntag, 29. August 2021 um 22:35:47 UTC+2:
>> On Sunday, August 29, 2021 at 4:43:07 PM UTC-3, WM wrote:
>>> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
>>>> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>>>>
>>>>> What is next to zero? Note that "nothing" is synonymous to "gap".
>>>> Nope
>>>> A gap next to 0 is an interval [0,x] (note x must be a real number
>>> The end of the gap is a real number x
>> and a "gap", like a set, is not something that changes.
>
> The collection of definable points is the complement. Therefore the size of the gap changes.
>
> Regards, WM
>

there is no collection of "definable" points.