On 12/13/2023 5:28 AM, Adam Polak wrote:
> poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
>> Dear Friends,
>>
>> The Set Theory, creator of which is considered to be Professor Georg Cantor, currently adhered to by the vast majority of scientists, is an undoubtedly flawed theory, based on erroneous assumptions and, as a result, filled with errors and internal contradictions.
>>
>> The wide "Analysis of mistakes in infinity study attempts" within set theory can be found here on YouTube:
>> https://www.youtube.com/watch?v=s23Cz8A0BKs
>>
>> In the upcoming presentations, we will together take a colser look on numerous errors in set theory, we will identify Hilbert's Grand Hotel Paradox errors, easily solve the Continuum Hypothesis (allegedly undecidable), Russell's Paradox, the Paradox of the set of all sets, and we will confirm even more emphatically that the set theory can be seen only as erroneous and disproven.
>>
>> A small sample below. A comparison that decisively, in an unquestionable manner, refutes the Cantor's Diagonal Argument as evidence of the inequality of the infinite set of real numbers relative to the infinite set of natural numbers.
>>
>> A hotel with an infinite number of rooms.
>> There is a guest in each room.
>> As a result, you have two infinite sets:
>>
>> An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
>>
>> An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
>>
>> A new guest appears: NG1
>> The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
>>
>> If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
>> you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
>>
>> You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
>>
>> It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
>>
>> The set theory is clearly contradictory in many places.
>>
>> Best Regards,
>> Adam Polak
>
> A week ago, a month passed since the first post in this thread was published.
> I have not noticed any substantive counter-argument that would question the correctness of the presented method of pairing any quantity of "new" real numbers with natural numbers.
>
> For the sake of order, it should be stated that:
>
> !!!
> Cantor's Diagonal Method IS NOT a tool that in any way demonstrates or confirms that the quantity of elements of an infinite set of real numbers should be viewed as greater than the quantity of elements of an infinite set of natural numbers.
>
> From this perspective:
> The infinite set of natural numbers and the infinite set of real numbers remain always quantitatively equal.
> !!!
>
> Thank you for interesting discussions.

Real numbers are not countable. Think of how you can progress from, say
one to two when there are infinite reals between one and two?

1, 2

1.1, 1.01, 1.001, 1.0001, ect.

That right there is infinite but it does not allow us to go from one to
two. In the naturals we go 1 + 1 = 2, simple. The reals are not countable.

On Wednesday, December 13, 2023 at 4:49:02 AM UTC-8, Adam Polak wrote:
> środa, 13 grudnia 2023 o 08:18:23 UTC+1 Fritz Feldhase napisał(a):
> > On Wednesday, December 13, 2023 at 7:07:59 AM UTC+1, Adam Polak wrote:
> > > wtorek, 12 grudnia 2023 o 18:11:25 UTC+1 FromTheRafters napisał(a):
> > > > >
> > > > > Wasn't it Georg Cantor who ended his life in a nursing home with serious mental problems? [Adam Polak]
> > > > >
> > > > More cranksign, as if we needed any.
> > > >
> > > Do you deny the quite widely known fact that Prof. Georg Cantor finally was in a very bad mental state [...]?
> >
> > No, we don't deny that, but we are not interested in that fact when talking about /axiomatic set theories/ (and their origins as mathematical theories).
> >
> > See: https://en.wikipedia.org/wiki/Ad_hominem
> I hope you can see that a lot of your responses, which are actually ridiculous personal attacks,
> are funny for this very reason and often makes me smile :)

That seems the sign of a sociopath who gets off on sowing confusion. (And agitation.)

Whereas, we are not, so, ..., so we more sort of like-minds
keep our canon and our conscience and our conduct.

.... Especially in an unmoderated sort of realm where it can't be
said that alternative opinions are excluded, on their merits.

Anyways, here we get our kicks above the waistline, sunshine.
Drawing flack for its own sake isn't considered de rigeur.

On Wednesday, December 13, 2023 at 5:29:02 AM UTC-8, Adam Polak wrote:
> poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
> > Dear Friends,
> >
> > The Set Theory, creator of which is considered to be Professor Georg Cantor, currently adhered to by the vast majority of scientists, is an undoubtedly flawed theory, based on erroneous assumptions and, as a result, filled with errors and internal contradictions.
> >
> > The wide "Analysis of mistakes in infinity study attempts" within set theory can be found here on YouTube:
> > https://www.youtube.com/watch?v=s23Cz8A0BKs
> >
> > In the upcoming presentations, we will together take a colser look on numerous errors in set theory, we will identify Hilbert's Grand Hotel Paradox errors, easily solve the Continuum Hypothesis (allegedly undecidable), Russell's Paradox, the Paradox of the set of all sets, and we will confirm even more emphatically that the set theory can be seen only as erroneous and disproven.
> >
> > A small sample below. A comparison that decisively, in an unquestionable manner, refutes the Cantor's Diagonal Argument as evidence of the inequality of the infinite set of real numbers relative to the infinite set of natural numbers.
> >
> > A hotel with an infinite number of rooms.
> > There is a guest in each room.
> > As a result, you have two infinite sets:
> >
> > An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
> >
> > An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
> >
> > A new guest appears: NG1
> > The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
> >
> > If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
> > you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
> >
> > You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
> >
> > It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
> >
> > The set theory is clearly contradictory in many places.
> >
> > Best Regards,
> > Adam Polak
>
> A week ago, a month passed since the first post in this thread was published.
> I have not noticed any substantive counter-argument that would question the correctness of the presented method of pairing any quantity of "new" real numbers with natural numbers.
>
> For the sake of order, it should be stated that:
>
> !!!
> Cantor's Diagonal Method IS NOT a tool that in any way demonstrates or confirms that the quantity of elements of an infinite set of real numbers should be viewed as greater than the quantity of elements of an infinite set of natural numbers.
>
> From this perspective:
> The infinite set of natural numbers and the infinite set of real numbers remain always quantitatively equal.
> !!!
>
> Thank you for interesting discussions.
> Adam Polak
>
> t.b.c.

Note exactly where that's so stated in the video it seems you stole,
as time t_Hilbert in seconds after the start, and add &t=t_Hilbert
to the link.

That infinite sets are infinite is accorded to Galileo,
and of course was well-known before, indeed it may be
so that Cantor assembled his results from like notions,

the m-w rpoof,
nested intervals,
the antidiagonal,
continued fractions,
....,
the powerset theorem,
....

so that if there is to be shewn a function counterexample, then, it must
be so built to pass all of those, "alive".

Here it's "the natural/unit equivalency function", for cardinal equivalency,
then "the successor is ordertype is powerset in ubiquitous ordinals",
then there was also shewn a result after the rationals being huge,
that there are at least three "definitions", of continuity, completeness,
gaplessness, topologically, for the world of continuous functions of
continuous domains to continuous domains.

line reals
field reals
signal reals
long-line reals

Then it sort of results all you're saying is "infinite sets are infinite"
is "I can increment". Which, would be true.

On Wednesday, December 13, 2023 at 8:01:42 AM UTC-8, Fritz Feldhase wrote:
> On Wednesday, December 13, 2023 at 2:29:02 PM UTC+1, Adam Polak wrote:
> >
> > set theory is clearly contradictory in many places.
> Though you aren't able to point out such a place. Surprise!

Here it's "all functions are Cartesian".

This is that for some domains, their Cartesian product,
only has one function in it, not Cartesian in the Cartesian
product of the set of all ordered pairs of elements from domain and range.

I.e. what results re-ordering it, as Cartesian functions may be so re-ordered,
though it doesn't necessarily take continuous functions to continuous functions,
this example, is not a function. ("N/U EF", "sweep".) Similarly the signal reals
only connect to their neighbors.

So, there's simply a counterexample to "all functions are Cartesian",
but, that's a definition anyways, the definition of function.

Anyways, ZF set theory, if you ignore the Axiom of Infinity for a moment,
and, quantify over the finite sets, then, you get the Russell set of the antinomy,
that's not ordinary any more. If you don't think so just ask Frege.

Sweep: big winner.

On Wednesday, December 13, 2023 at 12:45:02 PM UTC-8, Chris M. Thomasson wrote:
> On 12/13/2023 5:28 AM, Adam Polak wrote:
> > poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
> >> Dear Friends,
> >>
> >> The Set Theory, creator of which is considered to be Professor Georg Cantor, currently adhered to by the vast majority of scientists, is an undoubtedly flawed theory, based on erroneous assumptions and, as a result, filled with errors and internal contradictions.
> >>
> >> The wide "Analysis of mistakes in infinity study attempts" within set theory can be found here on YouTube:
> >> https://www.youtube.com/watch?v=s23Cz8A0BKs
> >>
> >> In the upcoming presentations, we will together take a colser look on numerous errors in set theory, we will identify Hilbert's Grand Hotel Paradox errors, easily solve the Continuum Hypothesis (allegedly undecidable), Russell's Paradox, the Paradox of the set of all sets, and we will confirm even more emphatically that the set theory can be seen only as erroneous and disproven.
> >>
> >> A small sample below. A comparison that decisively, in an unquestionable manner, refutes the Cantor's Diagonal Argument as evidence of the inequality of the infinite set of real numbers relative to the infinite set of natural numbers.
> >>
> >> A hotel with an infinite number of rooms.
> >> There is a guest in each room.
> >> As a result, you have two infinite sets:
> >>
> >> An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
> >>
> >> An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
> >>
> >> A new guest appears: NG1
> >> The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
> >>
> >> If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
> >> you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
> >>
> >> You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
> >>
> >> It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
> >>
> >> The set theory is clearly contradictory in many places.
> >>
> >> Best Regards,
> >> Adam Polak
> >
> > A week ago, a month passed since the first post in this thread was published.
> > I have not noticed any substantive counter-argument that would question the correctness of the presented method of pairing any quantity of "new" real numbers with natural numbers.
> >
> > For the sake of order, it should be stated that:
> >
> > !!!
> > Cantor's Diagonal Method IS NOT a tool that in any way demonstrates or confirms that the quantity of elements of an infinite set of real numbers should be viewed as greater than the quantity of elements of an infinite set of natural numbers.
> >
> > From this perspective:
> > The infinite set of natural numbers and the infinite set of real numbers remain always quantitatively equal.
> > !!!
> >
> > Thank you for interesting discussions.
> Real numbers are not countable. Think of how you can progress from, say
> one to two when there are infinite reals between one and two?
>
> 1, 2
>
> 1.1, 1.01, 1.001, 1.0001, ect.
>
>
> That right there is infinite but it does not allow us to go from one to
> two. In the naturals we go 1 + 1 = 2, simple. The reals are not countable.

If you know uncountability you might like outpacing,
which helps define sizes of sets including infinite ones,
about the grids they fill.

(Outpacing is that a subset's smaller than a set if a proper subset.)

Also there's "complexities", complexity classes,
means to quantify and make an ordering of sets's, "sizes".
These are the often found in the fractal world or "Sierpinski's",
and related to fractal "dimension" and so on.

That "half of the integers are even" is called "density"
or "asymptotic density" or "Schnirelmann density".

All perfectly compatible with "set theory".

On 12/13/2023 7:23 PM, Ross Finlayson wrote:
> On Wednesday, December 13, 2023 at 12:45:02 PM UTC-8, Chris M. Thomasson wrote:
>> On 12/13/2023 5:28 AM, Adam Polak wrote:
>>> poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
>>>> Dear Friends,
>>>>
>>>> The Set Theory, creator of which is considered to be Professor Georg Cantor, currently adhered to by the vast majority of scientists, is an undoubtedly flawed theory, based on erroneous assumptions and, as a result, filled with errors and internal contradictions.
>>>>
>>>> The wide "Analysis of mistakes in infinity study attempts" within set theory can be found here on YouTube:
>>>> https://www.youtube.com/watch?v=s23Cz8A0BKs
>>>>
>>>> In the upcoming presentations, we will together take a colser look on numerous errors in set theory, we will identify Hilbert's Grand Hotel Paradox errors, easily solve the Continuum Hypothesis (allegedly undecidable), Russell's Paradox, the Paradox of the set of all sets, and we will confirm even more emphatically that the set theory can be seen only as erroneous and disproven.
>>>>
>>>> A small sample below. A comparison that decisively, in an unquestionable manner, refutes the Cantor's Diagonal Argument as evidence of the inequality of the infinite set of real numbers relative to the infinite set of natural numbers.
>>>>
>>>> A hotel with an infinite number of rooms.
>>>> There is a guest in each room.
>>>> As a result, you have two infinite sets:
>>>>
>>>> An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
>>>>
>>>> An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
>>>>
>>>> A new guest appears: NG1
>>>> The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
>>>>
>>>> If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
>>>> you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
>>>>
>>>> You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
>>>>
>>>> It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
>>>>
>>>> The set theory is clearly contradictory in many places.
>>>>
>>>> Best Regards,
>>>> Adam Polak
>>>
>>> A week ago, a month passed since the first post in this thread was published.
>>> I have not noticed any substantive counter-argument that would question the correctness of the presented method of pairing any quantity of "new" real numbers with natural numbers.
>>>
>>> For the sake of order, it should be stated that:
>>>
>>> !!!
>>> Cantor's Diagonal Method IS NOT a tool that in any way demonstrates or confirms that the quantity of elements of an infinite set of real numbers should be viewed as greater than the quantity of elements of an infinite set of natural numbers.
>>>
>>> From this perspective:
>>> The infinite set of natural numbers and the infinite set of real numbers remain always quantitatively equal.
>>> !!!
>>>
>>> Thank you for interesting discussions.
>> Real numbers are not countable. Think of how you can progress from, say
>> one to two when there are infinite reals between one and two?
>>
>> 1, 2
>>
>> 1.1, 1.01, 1.001, 1.0001, ect.
>>
>>
>> That right there is infinite but it does not allow us to go from one to
>> two. In the naturals we go 1 + 1 = 2, simple. The reals are not countable.
>
> If you know uncountability you might like outpacing,
> which helps define sizes of sets including infinite ones,
> about the grids they fill.
>
> (Outpacing is that a subset's smaller than a set if a proper subset.)
>
> Also there's "complexities", complexity classes,
> means to quantify and make an ordering of sets's, "sizes".
> These are the often found in the fractal world or "Sierpinski's",
> and related to fractal "dimension" and so on.
>
> That "half of the integers are even" is called "density"
> or "asymptotic density" or "Schnirelmann density".
>
> All perfectly compatible with "set theory".
>
>

1 + 1 = 2 is countable. However,

1.1, 1.01, 1.011, .10110, 1.01101, 1.011011, ...

Is an infinite process that will never reach 2. So, afaict its fractal.
infinities within infinities ala reals?

On 12/14/2023 7:13 PM, Chris M. Thomasson wrote:
> On 12/13/2023 7:23 PM, Ross Finlayson wrote:
>> On Wednesday, December 13, 2023 at 12:45:02 PM UTC-8, Chris M.
>> Thomasson wrote:
>>> On 12/13/2023 5:28 AM, Adam Polak wrote:
>>>> poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
>>>>> Dear Friends,
>>>>>
>>>>> The Set Theory, creator of which is considered to be Professor
>>>>> Georg Cantor, currently adhered to by the vast majority of
>>>>> scientists, is an undoubtedly flawed theory, based on erroneous
>>>>> assumptions and, as a result, filled with errors and internal
>>>>> contradictions.
>>>>>
>>>>> The wide "Analysis of mistakes in infinity study attempts" within
>>>>> set theory can be found here on YouTube:
>>>>> https://www.youtube.com/watch?v=s23Cz8A0BKs
>>>>>
>>>>> In the upcoming presentations, we will together take a colser look
>>>>> on numerous errors in set theory, we will identify Hilbert's Grand
>>>>> Hotel Paradox errors, easily solve the Continuum Hypothesis
>>>>> (allegedly undecidable), Russell's Paradox, the Paradox of the set
>>>>> of all sets, and we will confirm even more emphatically that the
>>>>> set theory can be seen only as erroneous and disproven.
>>>>>
>>>>> A small sample below. A comparison that decisively, in an
>>>>> unquestionable manner, refutes the Cantor's Diagonal Argument as
>>>>> evidence of the inequality of the infinite set of real numbers
>>>>> relative to the infinite set of natural numbers.
>>>>>
>>>>> A hotel with an infinite number of rooms.
>>>>> There is a guest in each room.
>>>>> As a result, you have two infinite sets:
>>>>>
>>>>> An infinite SET OF ROOMS containing elements with the following
>>>>> symbols: R1, R2, R3, ...
>>>>>
>>>>> An infinite SET OF GUEST containing elements with the following
>>>>> symbols: G1, G2, G3...
>>>>>
>>>>> A new guest appears: NG1
>>>>> The new guest is definitely not among the guests that are already
>>>>> in the hotel because he is different from them, his name is: ("NG"
>>>>> + its individual number ) , everyone present in the hotel is: ("G"+
>>>>> individual number of each ).
>>>>>
>>>>> If you claim that you can accommodate a new guest in room 1 and
>>>>> move everyone currently present in the hotel to rooms n+1
>>>>> you can do exactly the same thing with a "new" real number
>>>>> supposedly created by diagonal method.
>>>>>
>>>>> You assign "new" real numb to 1, and you shift all the real numbers
>>>>> previously in the right column of the diagonal matrix down by one:
>>>>> the one that was assigned to 1 is now assigned to 2, the one
>>>>> assigned to 2 is now assigned to 3, etc.
>>>>>
>>>>> It is mutually contradictory to say that you can accommodate a new
>>>>> guest in Hilbert's hotel and at the same time to say that you
>>>>> cannot find a natural number as a pair for a "new" real number
>>>>> "created" by the diagonal method.
>>>>>
>>>>> The set theory is clearly contradictory in many places.
>>>>>
>>>>> Best Regards,
>>>>> Adam Polak
>>>>
>>>> A week ago, a month passed since the first post in this thread was
>>>> published.
>>>> I have not noticed any substantive counter-argument that would
>>>> question the correctness of the presented method of pairing any
>>>> quantity of "new" real numbers with natural numbers.
>>>>
>>>> For the sake of order, it should be stated that:
>>>>
>>>> !!!
>>>> Cantor's Diagonal Method IS NOT a tool that in any way demonstrates
>>>> or confirms that the quantity of elements of an infinite set of real
>>>> numbers should be viewed as greater than the quantity of elements of
>>>> an infinite set of natural numbers.
>>>>
>>>>  From this perspective:
>>>> The infinite set of natural numbers and the infinite set of real
>>>> numbers remain always quantitatively equal.
>>>> !!!
>>>>
>>>> Thank you for interesting discussions.
>>> Real numbers are not countable. Think of how you can progress from, say
>>> one to two when there are infinite reals between one and two?
>>>
>>> 1, 2
>>>
>>> 1.1, 1.01, 1.001, 1.0001, ect.
>>>
>>>
>>> That right there is infinite but it does not allow us to go from one to
>>> two. In the naturals we go 1 + 1 = 2, simple. The reals are not
>>> countable.
>>
>> If you know uncountability you might like outpacing,
>> which helps define sizes of sets including infinite ones,
>> about the grids they fill.
>>
>> (Outpacing is that a subset's smaller than a set if a proper subset.)
>>
>> Also there's "complexities", complexity classes,
>> means to quantify and make an ordering of sets's, "sizes".
>> These are the often found in the fractal world or "Sierpinski's",
>> and related to fractal "dimension" and so on.
>>
>> That "half of the integers are even" is called "density"
>> or "asymptotic density" or "Schnirelmann density".
>>
>> All perfectly compatible with "set theory".
>>
>>
>
> 1 + 1 = 2 is countable. However,
>
> 1.1, 1.01, 1.011, .10110, 1.01101, 1.011011, ...
>
> Is an infinite process that will never reach 2. So, afaict its fractal.
> infinities within infinities ala reals?

The limit of the std Mandelbrot set is 2, however, the border is
infinitely complex...

On Friday, December 15, 2023 at 2:11:54 AM UTC-8, Fritz Feldhase wrote:
> On Friday, December 15, 2023 at 4:13:20 AM UTC+1, Chris M. Thomasson wrote:
> >
> > 1.1, 1.01, 1.011, .10110, 1.01101, 1.011011, ...
> >
> > Is an infinite process that will never reach 2. So, afaict its fractal.
> Even a binary tree is a fractal.
>
> .
> /\
> 0 1
> /\ /\
> 0 1 0 1
> ...
>
> We can identify the nodes of this tree with the finite paths which ends in this nodes: .0, 0.1, .00, .01, .10, .11, ...
>
> The nodes in this tree are countable. (Just use this schema: count the nodes on level 0, then the nodes on level 1 and so on... There is no node in tree, you won't reach this way - sooner or later.)
>
> On the other hand, if we try to count the nodes following an infinite path, say, .000... we will not succeed. :-P
>
> ______________________________
>
> In addition: the set of (infinite) paths is not countable, while the set of nodes (or finite paths) is.

In fact, according to a reading of Zeno, you never either arrive, depart, start, nor stop.

But, we use deduction to arrive at that there exists a continuous course-of-passage
in a strictly-monotonically-increasing order, either beginning to end or about and around.

We also call that Zeno but it's called "resolving Zeno's paradox" not "failing to resolve Zeno's paradox".

That it involves the limit from both sides and also meeting in the middle, is also a result.