On Friday, November 10, 2023 at 10:51:25 AM UTC-8, FredJeffries wrote:
> On Thursday, November 9, 2023 at 9:10:23 PM UTC-8, Adam Polak wrote:
> >
> > The theory of infinite sets by Cantor, aspires to be a correct description of the reality in the realm of infinite sets, the relationships between infinite sets, and their elements. In reality, however, it is simply a FALLACIOUS description. It is a dream of grandeur dreamt by the mind, or perhaps the ego of Cantor (initially), and now by the minds/egos of the overwhelming majority of the scientific community. It is a beautiful dream in which a finite human comprehends infinity, e.g. infinity of natural numbers, and probably due to its beauty, it has been so infectious and widely spread. In the end, it is, unfortunately, only a waking dream with very little in common with reality, especially in terms of the true nature of infinity.
> >
> In fact, Cantor took great pains to disassociate and distinguish the infinities both of theology and metaphysics from his mere mathematical transfinite.
>
> <quote translated>
> The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or order type. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.
> </quote>
>
> <quote translated>
> I have never assumed a “Genus Supremum” of the actual infinite. Quite on the contrary I have proved that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”, which is by many called “God”.
> </quote>
>
> https://en.wikipedia.org/wiki/Absolute_Infinite
>
> https://pointatinfinityblog.wordpress.com/2017/06/12/cantor-and-the-absolute-universal-structures-iv/

Sure, trot out the oldest nag at the working farm.

That, "the infinite", is, "uncountable", is reduced to a footnote of Kant is his last Judgment on the Sublime,
Or his last Critique the Critique on Judgment, what's connected is sublime and it's continuous.

So, all the rest of the canon is kept then this also struck through as "an infinity". I.e., what philosophy
says logic says is only phenomenological and infinity is only noumenal. It is, Kant says Des Cartes says,
or says for him, that Duns Scotus, says. "There is a mathematical infinity.."

Which Platonists say, .... On a good day.

There's still the rest of the Absolute and "G-d's Infinity", universal G-d, monist G-d, "All Infinity".
It's just there's let in the usual notion of sublime as greater that lets in continuity and
infinite-divisibility, all together, "only mathematically".

So, formally that's just a distinction rule with whatever finite, bounded model and all things.
"Infinity: there is one."

Then Kant in his "On the Old Saw", "Ethics", "Government", .... No, this is the "technical",
philosophy, logic, it's technical, humanity is arbitrarily an abstract concept, "philosophy".
So, in Kant's world, it's just pointed down "sublime" and the "subliminal", infinitely-divisible,
continuous, "Kant's DesCartes' Euclid's geometry's points including at infinity".

Then, foundations, I tell you, I'm glad I leafed all through a text of Courant, my differential
results, under "Cantor Space and Square Cantor Space, a non-Cartesian function", put
the most direct sorts applications, for differential analysis, into usual laws of large numbers
in the infinitely-divisible and continuum analysis. That: "the sweep function is its own anti-derivative",
is about the most remarkable fact to function theory, that "the exponential function is its own anti-derivative".

Not a real function, ..., that there is one.

Not-a-real-function, ....

So, for the differential analysis, there is this idee fixe for the fixed point, the identity dimension,
it is two Cartesian dimension, then x = y together, and in the infinite identity dimension x = y= z
and so on, relating two dimensions to indicate the level dimension, and having upper and lower
reflections, in the quadrant in the half-plane.

This being the envelope of a usual ramp, f(x) = x, involves a sort of style of differential analysis,
bounding to the corners under the hyperbola, families of those, that though fill the space and
area, like Fourier-style analysis has the windowing and the boxing as it were or the intervals,
this has results differentially from the origin and identity, out perspective what reflects bounds.

So, that's called "implicits" these days then also Courant introduces quite a few methods,
and justifications including "counterexamples in demonstration of convergence".
(That result convergence, ....) Here there's a completion established under the symmetries
of the two Cartesian dimensions and the identity dimension, either one and both, or
then simply the origin and identity, and envelope, "completions", that result similarly
to a Fourier-style analysis and its justifications windowing and boxing and thus resulting
transforms after the orthogonal and the kernels, "completions", is under hyperbolas,
resulting a sort wave-integration, the coefficients of the elements of the terms of the
infinite and the telescoping, series.

That though is defined algebraically itself, "foundations", yet of course such useful notions
as "the origin" and "angle", define all sorts forms, proportions, and geometry.

Analytically, ....

These days a generally approach of "completions" and "almost everywhere, equal",
(that completions are everywhere, and almost everywhere isn't), is most usual
"approximation, which is close" and "error, which is off".

Then you only need three definitions of continuity, any one of which are neat and small.
Given most objects are properties resulting, you don't have to attach your ontological
commitment to whatever makes it so, while of course there is what it is, as so.

So, renewed observance of what was expected a formal system, helps show not
just the common line, but all the so formal systems, what results a wealth common
results about usually laws of numbers: laws of large numbers and laws of small numbers,
in laws of numbers, here simply the types of infinities in the infinitely-divisible have their
own completions, and as large numbers of those sorts, laws of large numbers, and
"extra", actually "beyond", necessarily, any finite. I.e., together they're as so,
common systems of infinite divisibility.

Any regular language has a model in ordinary set theory.
They can't all use it at once, ..., the model of all ordinary set theory.

It's a set theory, it's a continuum mechanics, "foundations" has to be both..

On Thursday, November 9, 2023 at 9:10:23 PM UTC-8, Adam Polak wrote:
> Thank you for your comment Ross. Very interesting.
> e.g.:
> "So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
> The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity and the ordering infinity, in ordinals.
> The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces, above each constructive, regular, ordinary, ..., theory of words like sets, here elementary objects.
> That's one reason why cardinals and ordinals are different, different infinities."
> "I sort of organize analysis in continuum mechanics."
> I'll start from the end. I can assure you, Ross, and provide numerous independent, but logically related pieces of evidence that everything what quantitatively is represented by any infinite set, including: Alephs, cardinals, and Omegas, fits in, "happens" within the quantity represented by the infinite set of natural numbers.
>
> The theory of infinite sets by Cantor, aspires to be a correct description of the reality in the realm of infinite sets, the relationships between infinite sets, and their elements. In reality, however, it is simply a FALLACIOUS description. It is a dream of grandeur dreamt by the mind, or perhaps the ego of Cantor (initially), and now by the minds/egos of the overwhelming majority of the scientific community. It is a beautiful dream in which a finite human comprehends infinity, e.g. infinity of natural numbers, and probably due to its beauty, it has been so infectious and widely spread. In the end, it is, unfortunately, only a waking dream with very little in common with reality, especially in terms of the true nature of infinity.
>
> From the history of science, we know many sometimes very spectacular cases when "great" theories, accepted by "everyone," turned out to be greatly flawed or at least imperfect at some point. The same is about to happen soon with Cantor's Set Theory. I'm looking for a volunteer, preferably someone who is a "expert" in set theory professionally, someone who professionally deals with it, BUT! who allows for even the slightest possibility that there might be something wrong with set theory, just like with anything created by humans. (a person without this "BUT" will not wake up, he is simply fanatically attached to the illusion he is stuck in)
>
> I will undertake to guide such a person through detox and awaken from the dream of cardinalities greater than N, to the reality, which I assure is no less, but more intriguing..
>
> BR, Adam
> środa, 8 listopada 2023 o 17:55:47 UTC+1 Ross Finlayson napisał(a):
> > On Wednesday, November 8, 2023 at 6:55:13 AM UTC-8, Timothy Golden wrote:
> > > On Monday, November 6, 2023 at 4:09:22 PM UTC-5, Adam Polak wrote:
> > > > FINITE
> > > > A set S is finite when there exists an n ϵ N such that S has exactly n elements.
> > > >
> > > > INFINITE
> > > > Consequently:
> > > > A set S is infinite when there does not exist an n ϵ N such that S has exactly n elements.
> > > It is really modulo behaved elements that satisfy here. All others give me the creeps.
> > > >
> > > > From the above, it follows that:
> > > > Infinity is not a number, particularly not a natural number. It is not the number of elements in an infinite set, nor is it the largest or last element in an infinite set, e.g. such as the infinite set of natural numbers N.
> > > > poniedziałek, 6 listopada 2023 o 17:37:41 UTC+1 Dan Christensen napisał(a):
> > > > > On Monday, November 6, 2023 at 6:01:03 AM UTC-5, Adam Polak wrote:
> > > > > [snip]
> > > > > > 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.
> > > > > >
> > > > > Hilbert's Infinite Hotel should not be taken too literally. It is simply a humorous illustration that an infinite set like the set of natural numbers N = {1, 2, 3, ... } can be mapped bijectively to a proper subset of itself, namely {2, 3, 4, ... }. This property is the defining characteristic othenf ANY infinite set. In this case, the required bijection is f: N --> {2, 3, 4, ... } such that f(x)=x+1.
> > > > >
> > > > > For a somewhat less mind-blowing development, you might consider my alternative approach. I start by defining what we mean by a finite set. Then an infinite set is just one that is not finite.
> > > > >
> > > > > See my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> > > > >
> > > > > There, I present informal and formal set-theoretic developments of a non-numeric definition of a finite set. Refute it if you think you can.,
> > > > >
> > > > > Dan
> > > > >
> > > > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > > > Visit my Math Blog at http://www.dcproof.wordpress.com
> > Hmm, "SET THEORY, is dead", I like it, Adam.
> >
> > It's fair of you to say "here's a theory where I've axiomatized away my model of
> > a trust in set theory, I must make myself another way to trust", the set theory,
> > then you constructively bring up what you want, while then pointing out a
> > crankish argument in set theory, that shows something you've hobbled yourself, from.
> >
> > Powers of 2?
> >
> > What you get is ordering, numbering, and counting, and when numbering invoves counting.
> >
> > Set theory models these sufficiently all their "regular" way, "well-founded", for example,
> > the regular set theory.
> >
> > You can make inconsistent models of set theory and show how they're inconsistent.
> > It's not considered constructivist, say, insofar as formal rigor and "can't not trust it".
> >
> > So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
> > The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity
> > and the ordering infinity, in ordinals.
> >
> > The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces,
> > above each constructive, regular, ordinary, ..., theory of words like sets, here elementary
> > objects.
> >
> > That's one reason why cardinals and ordinals are different, different infinities.
> >
> > Anyways usually insofar as any mistake you write here someone will point it out to you.
> >
> > Anyways what results I enjoyed this for some time, currently looking at my own slates,
> > I sort of organize analysis in continuum mechanics.
> >
> > "Infinitely-many", ....
> >
> > So, what you want to do, I think to really get an understanding of the cardinal and ordinal
> > numbers, and, the cardinal and ordinal infinities, is give yourself axioms for example "inverse",
> > but for example "counting" or whatever other results "infinity" axioms, then figuring out
> > where their sameness and differences, do or strongly do or don't or strongly don't, hold,
> > what do.
> >
> > "It's a continuum mechanics, ...", just saying, Adam, that if you're looking for a theory that
> > really digs up set theory, I made one with both cardinals and ordinals and their infinities
> > what otherwise sometimes aren't "extra" enough to be real.
> >
> > I've even gone so far as to stand up letting a simplest mathematical infinity, back into
> > the philosophy, of the theory, the one that science is missing.
> >
> > So, when I suggest, "Powers of 2?", I suggest that you're thoroughly familiar with them,
> > all the powers of 2, then for the two infinities you call "w", omega, and "2^w", 2 to the omega,
> > which as a number, is an ordinal, but also results when writing ordinals regular-ly, is the
> > space "2^w", each of the infinite sequences of zeros and ones, with a beginning.
> >
> > It's set theory, ....

On 10.11.2023 06:10, Adam Polak wrote:

> From the history of science, we know many sometimes very spectacular
cases when "great" theories, accepted by "everyone," turned out to be
greatly flawed or at least imperfect at some point. The same is about to
happen soon with Cantor's Set Theory.
Cantor's mistake is this dilemma:

(1) He assumes that infinity is actual. This assumption need not be true
but it appears credible, in my opinion, when we look at the natural
numbers or at the fractions or at the points of the real line. However
we can use, define, or manipulate as individuals only a potentially
infinite collection, for instance
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| remains infinite.

(2) Cantor uses only the potentially infinite collections for his
bijections (because more is not possible). His result is that all
collections have the same number of elements that can be paired
(bypassing his mistaken result with real numbers here).

If infinity is actual however, then it is fixed. Then {0, 1, 2, 3, ...}
has exactly one element more than {1, 2, 3, ...}. Hilbert's hotel
suffers from the same mistake.

But since these "bijections" have resulted in so drastically
"counterintuitive", i.e. false results, a religious belief necessarily
has spread among the disciples of this matheology which will be
impossible to exorcise. No argument will have an effect.

Regards WM

It happens that WM formulated :
> On 10.11.2023 06:10, Adam Polak wrote:
>
> > From the history of science, we know many sometimes very spectacular
> cases when "great" theories, accepted by "everyone," turned out to be greatly
> flawed or at least imperfect at some point. The same is about to happen soon
> with Cantor's Set Theory.
> Cantor's mistake is this dilemma:
>
> (1) He assumes that infinity is actual. This assumption need not be true but
> it appears credible, in my opinion, when we look at the natural numbers or at
> the fractions or at the points of the real line. However we can use, define,
> or manipulate as individuals only a potentially infinite collection, for
> instance
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| remains infinite.
>
> (2) Cantor uses only the potentially infinite collections for his bijections
> (because more is not possible). His result is that all collections have the
> same number of elements that can be paired (bypassing his mistaken result
> with real numbers here).
>
> If infinity is actual however, then it is fixed. Then {0, 1, 2, 3, ...} has
> exactly one element more than {1, 2, 3, ...}. Hilbert's hotel suffers from
> the same mistake.
>
> But since these "bijections" have resulted in so drastically
> "counterintuitive", i.e. false results, a religious belief necessarily has
> spread among the disciples of this matheology which will be impossible to
> exorcise. No argument will have an effect.

Then your work is done here, congratulations.

On 11.11.2023 10:10, FromTheRafters wrote:
> It happens that WM formulated :
>
>> But since these "bijections" have resulted in so drastically
>> "counterintuitive", i.e. false results, a religious belief necessarily
>> has spread among the disciples of this matheology which will be
>> impossible to exorcise. No argument will have an effect.
>
> Then your work is done here, congratulations.

Are here only matheologians around? There are certainly many readers who
can become informed about the real situation.

Regards, WM

On Saturday, November 11, 2023 at 9:39:09 AM UTC+1, WM wrote:

> {0, 1, 2, ...} has exactly one element more than {1, 2, 3, ...}.

Look dumbo, I once asked you if {0, 1, 2, ...} and {{0}, {1}, {2}, ...} are equinumerous.

Your answer was YES. After all, {{0}, {1}, {2}, ...} is just {{n} : n e {0, 1, 2, ...}}. In other words, n |-> {n} is a bijection from {0, 1, 2, ...} onto {{0}, {1}, {2}, ...}.

This fact does not depend on any specific definition of the natural numbers..

Now in the context of Zermelo's ORIGINAL set theory (Z). The natural numbers are defined the following way: 0 = {}, 1 = {0}, 2 = {1}, 3 = {2}, in general: {n} = n+1 (for n e IN, n =/= 0).

THIS means that {1, 2, 3, ...} just *is* {{0}, {1}, {2}, ...} in the context of Z.

Hence you AGREE that {0, 1, 2, ...} and {1, 2, 3, ...} are equinumerous.

But this CONTRADICTS your claim: "{0, 1, 2, ...} has exactly one element more than {1, 2, 3, ...}".

Mückenheim, you are just too dumb for any form of mathematics.

MOST of your claims are SELF-CONTRADICTORY.

On Friday, November 10, 2023 at 11:03:42 PM UTC-8, Ross Finlayson wrote:
> On Thursday, November 9, 2023 at 9:10:23 PM UTC-8, Adam Polak wrote:
> > Thank you for your comment Ross. Very interesting.
> > e.g.:
> > "So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
> > The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity and the ordering infinity, in ordinals.
> > The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces, above each constructive, regular, ordinary, ..., theory of words like sets, here elementary objects.
> > That's one reason why cardinals and ordinals are different, different infinities."
> > "I sort of organize analysis in continuum mechanics."
> > I'll start from the end. I can assure you, Ross, and provide numerous independent, but logically related pieces of evidence that everything what quantitatively is represented by any infinite set, including: Alephs, cardinals, and Omegas, fits in, "happens" within the quantity represented by the infinite set of natural numbers.
> >
> > The theory of infinite sets by Cantor, aspires to be a correct description of the reality in the realm of infinite sets, the relationships between infinite sets, and their elements. In reality, however, it is simply a FALLACIOUS description. It is a dream of grandeur dreamt by the mind, or perhaps the ego of Cantor (initially), and now by the minds/egos of the overwhelming majority of the scientific community. It is a beautiful dream in which a finite human comprehends infinity, e.g. infinity of natural numbers, and probably due to its beauty, it has been so infectious and widely spread. In the end, it is, unfortunately, only a waking dream with very little in common with reality, especially in terms of the true nature of infinity.
> >
> > From the history of science, we know many sometimes very spectacular cases when "great" theories, accepted by "everyone," turned out to be greatly flawed or at least imperfect at some point. The same is about to happen soon with Cantor's Set Theory. I'm looking for a volunteer, preferably someone who is a "expert" in set theory professionally, someone who professionally deals with it, BUT! who allows for even the slightest possibility that there might be something wrong with set theory, just like with anything created by humans. (a person without this "BUT" will not wake up, he is simply fanatically attached to the illusion he is stuck in)
> >
> > I will undertake to guide such a person through detox and awaken from the dream of cardinalities greater than N, to the reality, which I assure is no less, but more intriguing..
> >
> > BR, Adam
> > środa, 8 listopada 2023 o 17:55:47 UTC+1 Ross Finlayson napisał(a):
> > > On Wednesday, November 8, 2023 at 6:55:13 AM UTC-8, Timothy Golden wrote:
> > > > On Monday, November 6, 2023 at 4:09:22 PM UTC-5, Adam Polak wrote:
> > > > > FINITE
> > > > > A set S is finite when there exists an n ϵ N such that S has exactly n elements.
> > > > >
> > > > > INFINITE
> > > > > Consequently:
> > > > > A set S is infinite when there does not exist an n ϵ N such that S has exactly n elements.
> > > > It is really modulo behaved elements that satisfy here. All others give me the creeps.
> > > > >
> > > > > From the above, it follows that:
> > > > > Infinity is not a number, particularly not a natural number. It is not the number of elements in an infinite set, nor is it the largest or last element in an infinite set, e.g. such as the infinite set of natural numbers N.
> > > > > poniedziałek, 6 listopada 2023 o 17:37:41 UTC+1 Dan Christensen napisał(a):
> > > > > > On Monday, November 6, 2023 at 6:01:03 AM UTC-5, Adam Polak wrote:
> > > > > > [snip]
> > > > > > > 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.
> > > > > > >
> > > > > > Hilbert's Infinite Hotel should not be taken too literally. It is simply a humorous illustration that an infinite set like the set of natural numbers N = {1, 2, 3, ... } can be mapped bijectively to a proper subset of itself, namely {2, 3, 4, ... }. This property is the defining characteristic othenf ANY infinite set. In this case, the required bijection is f: N --> {2, 3, 4, ... } such that f(x)=x+1.
> > > > > >
> > > > > > For a somewhat less mind-blowing development, you might consider my alternative approach. I start by defining what we mean by a finite set.. Then an infinite set is just one that is not finite.
> > > > > >
> > > > > > See my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> > > > > >
> > > > > > There, I present informal and formal set-theoretic developments of a non-numeric definition of a finite set. Refute it if you think you can.,
> > > > > >
> > > > > > Dan
> > > > > >
> > > > > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > > > > Visit my Math Blog at http://www.dcproof.wordpress.com
> > > Hmm, "SET THEORY, is dead", I like it, Adam.
> > >
> > > It's fair of you to say "here's a theory where I've axiomatized away my model of
> > > a trust in set theory, I must make myself another way to trust", the set theory,
> > > then you constructively bring up what you want, while then pointing out a
> > > crankish argument in set theory, that shows something you've hobbled yourself, from.
> > >
> > > Powers of 2?
> > >
> > > What you get is ordering, numbering, and counting, and when numbering invoves counting.
> > >
> > > Set theory models these sufficiently all their "regular" way, "well-founded", for example,
> > > the regular set theory.
> > >
> > > You can make inconsistent models of set theory and show how they're inconsistent.
> > > It's not considered constructivist, say, insofar as formal rigor and "can't not trust it".
> > >
> > > So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
> > > The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity
> > > and the ordering infinity, in ordinals.
> > >
> > > The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces,
> > > above each constructive, regular, ordinary, ..., theory of words like sets, here elementary
> > > objects.
> > >
> > > That's one reason why cardinals and ordinals are different, different infinities.
> > >
> > > Anyways usually insofar as any mistake you write here someone will point it out to you.
> > >
> > > Anyways what results I enjoyed this for some time, currently looking at my own slates,
> > > I sort of organize analysis in continuum mechanics.
> > >
> > > "Infinitely-many", ....
> > >
> > > So, what you want to do, I think to really get an understanding of the cardinal and ordinal
> > > numbers, and, the cardinal and ordinal infinities, is give yourself axioms for example "inverse",
> > > but for example "counting" or whatever other results "infinity" axioms, then figuring out
> > > where their sameness and differences, do or strongly do or don't or strongly don't, hold,
> > > what do.
> > >
> > > "It's a continuum mechanics, ...", just saying, Adam, that if you're looking for a theory that
> > > really digs up set theory, I made one with both cardinals and ordinals and their infinities
> > > what otherwise sometimes aren't "extra" enough to be real.
> > >
> > > I've even gone so far as to stand up letting a simplest mathematical infinity, back into
> > > the philosophy, of the theory, the one that science is missing.
> > >
> > > So, when I suggest, "Powers of 2?", I suggest that you're thoroughly familiar with them,
> > > all the powers of 2, then for the two infinities you call "w", omega, and "2^w", 2 to the omega,
> > > which as a number, is an ordinal, but also results when writing ordinals regular-ly, is the
> > > space "2^w", each of the infinite sequences of zeros and ones, with a beginning.
> > >
> > > It's set theory, ....
> Well, that should be rather clear in a direct sort of way,
> that ordinary set theory is inconsistent.
>
> It's not inconsistent at all insofar as it's regular, but there
> are theorems resulting from quantification, that result
> extra elements, "extra-ordinary", whether the ordinary is
> extra or it's extra, the ordinary, not inconsistent at all,
> then though that "as a model of a 'closed' regular no-infinite-
> descending-chain well-founded theory, it is inconsistent,
> set theory".
>
> Ordinary set theory is inconsistent about infinity only insofar
> as to be consistent again as it was consistent from and as it
> is consistent, toward, what is ordering or numbering what results
> counting in what's the most usual sort basis of a counting argument,
> assigning counts upon numbers, those being all ordinary and finite
> and usual, then all the ideas about their number and order,
> qualitatively/quantitatively, that way.
>
> So, what comes around from "set theory is inconsistent, because
> infinite is infinite and quantification in the infinite", it only comes around
> again, stronger, that it's "extra-consistent" and "infinity equals infinity plus one"
> still retains "and also is greater than it".
>
> Then, all that's just called "sublime", extra and greater, making all the thinkers
> already set up, have to make it so, also, what they can accept.

Dear Ross,
Many words, for which I thank you very much. Some are quite valuable. I apologize if I'm not interpreting them correctly, but it seems a bit like an (perhaps not entirely conscious) attempt to "talk out/nock out" the problem, blur it in words, drown it in words. The problem EXISTS, it is significant, and it won't sink.

Many statements (aspiring to the role of a proper description of reality) contained in set theory or arising from it are simply UNTRUE, a description that is FLAWED.

Example: The claim that any of infinite sets should be perceived quantitatively as more numerous than the infinite set of natural numbers is a huge mistake. (It is easy to demonstrate, and I will do it in several ways, and in a louder manner - limiting presence here to get some spare time).

Of course, in the working phase, various theories can be formulated and explored as potential alternative descriptions of reality. However, if one of these theories turns out to be erroneous in many places, describing an imaginary reality it generated itself, it ceases to be a scientific theory, ceases to be a part of science. It becomes science-fiction, and in some cases, pure fantasy.

I know that engaging with science-fiction or outright fantasy can be very enjoyable and, in a way, very satisfying. Fiction offers far greater possibilities than science. However, consciously playing with fantasy while pretending to be a scientist is simply extremely immoral; it is deceit. It essentially degrades the function of a scientist to that of a fraud.

On the other hand, unconsciously indulging in fantasy under the guise of running scientific work undermines intellectual competence and essentially reduces such a "scientist" to the role of a fool.

I apologize for these blunt words. I don't mean anyone specific, but many should contemplate this.

The only possible result of mixing milk with a puddle is more puddle.

BR,
Adam

On Sunday, November 12, 2023 at 10:03:52 PM UTC-8, Adam Polak wrote:
> Dear Ross,
> Many words, for which I thank you very much. Some are quite valuable. I apologize if I'm not interpreting them correctly, but it seems a bit like an (perhaps not entirely conscious) attempt to "talk out/nock out" the problem, blur it in words, drown it in words. The problem EXISTS, it is significant, and it won't sink.
>
> Many statements (aspiring to the role of a proper description of reality) contained in set theory or arising from it are simply UNTRUE, a description that is FLAWED.
>
> Example: The claim that any of infinite sets should be perceived quantitatively as more numerous than the infinite set of natural numbers is a huge mistake. (It is easy to demonstrate, and I will do it in several ways, and in a louder manner - limiting presence here to get some spare time).
>
> Of course, in the working phase, various theories can be formulated and explored as potential alternative descriptions of reality. However, if one of these theories turns out to be erroneous in many places, describing an imaginary reality it generated itself, it ceases to be a scientific theory, ceases to be a part of science. It becomes science-fiction, and in some cases, pure fantasy.
>
> I know that engaging with science-fiction or outright fantasy can be very enjoyable and, in a way, very satisfying. Fiction offers far greater possibilities than science. However, consciously playing with fantasy while pretending to be a scientist is simply extremely immoral; it is deceit. It essentially degrades the function of a scientist to that of a fraud.
>
> On the other hand, unconsciously indulging in fantasy under the guise of running scientific work undermines intellectual competence and essentially reduces such a "scientist" to the role of a fool.
>
> I apologize for these blunt words. I don't mean anyone specific, but many should contemplate this.
>
> The only possible result of mixing milk with a puddle is more puddle.
>
> BR,
> Adam

Well, you might learn about Skolem, and about how he shows there's a countably infinite model
of things.

I understand that some people's idea of the integers, zero to infinity, scale exactly, to zero to one.
Of course, not everybody, and indeed that the powerset theorem is a thing gets into how there
can be a space of 0's and 1's, and, what's "ordinary" and "extra-ordinary".

There's a bunch to study to figure out that these days' classical logic is at contest with logic
of refutations and relevance, what most would consider as common sense, because what's
called classical pretty much isn't and de Morgan has more going on for resolving duals, than Boole.

But, it's it's not really so much of a bunch: it's not really that much to get a grasp on what goes
into various logics what result they address all the objects of logical discourse, about same.

Then, whether "there doesn't even exist a _standard_ model of integers, only extensions and fragments,
of models of integers", gets into why you should explored open-mindedly, while still of course
keeping the absolutes apart: empty and infinity. (... and that they spring from the same and
are one.)

"Hope and a bottle of ketchup isn't a hamburger."

Anyways there is _ordinary_ set theory and many are familiar with it and it suffices for their
platforms if not foundations itself, and there is _extra-ordinary_ set theory which is by definition
more of the universe of sets, and as after and about class/set distinction, and the lack thereof.

In a set theory it's a set theory, in a part theory a part theory, ..., it's set theory.

Adam Polak schrieb am Montag, 13. November 2023 um 07:03:52 UTC+1:

> The claim that any of infinite sets should be perceived
quantitatively as more numerous than the infinite set of natural numbers
is a huge mistake.

In every finite section from 0 to n there are far more rational numbers
than natural numbers. Therefore, to infere the infinite from the finite,
there are far more rationals. Equinumerosity can only be accepted if
there is no complete reality of numbers but the potential infinity of
numbers created by us.

Regards, WM

Adam Polak schrieb am Montag, 13. November 2023 um 07:03:52 UTC+1:

> The claim that any of infinite sets should be perceived
quantitatively as more numerous than the infinite set of natural numbers
is a huge mistake.

In every finite section from 0 to n there are far more rational numbers
than natural numbers. Therefore, to infere the infinite from the finite,
there are far more rationals. Equinumerosity can only be accepted if
there is no complete reality of numbers but the potential infinity of
numbers created by us.

Regards, WM

Any "finite section" / finite set is not a representative object for conducting research and adjudicating on the specifics of any infinite set.
Almost without exaggeration, I will say that the similarity between any, even the largest, finite set and an infinite set, on a scale from 1 to 10 is ZERO.

poniedziałek, 13 listopada 2023 o 09:20:18 UTC+1 WM napisał(a):
> Adam Polak schrieb am Montag, 13. November 2023 um 07:03:52 UTC+1:
>
> > The claim that any of infinite sets should be perceived
> quantitatively as more numerous than the infinite set of natural numbers
> is a huge mistake.
> In every finite section from 0 to n there are far more rational numbers
> than natural numbers. Therefore, to infere the infinite from the finite,
> there are far more rationals. Equinumerosity can only be accepted if
> there is no complete reality of numbers but the potential infinity of
> numbers created by us.
>
> Regards, WM

On Monday, November 13, 2023 at 12:38:00 AM UTC-8, Adam Polak wrote:
> Any "finite section" / finite set is not a representative object for conducting research and adjudicating on the specifics of any infinite set.
> Almost without exaggeration, I will say that the similarity between any, even the largest, finite set and an infinite set, on a scale from 1 to 10 is ZERO.
> poniedziałek, 13 listopada 2023 o 09:20:18 UTC+1 WM napisał(a):
> > Adam Polak schrieb am Montag, 13. November 2023 um 07:03:52 UTC+1:
> >
> > > The claim that any of infinite sets should be perceived
> > quantitatively as more numerous than the infinite set of natural numbers
> > is a huge mistake.
> > In every finite section from 0 to n there are far more rational numbers
> > than natural numbers. Therefore, to infere the infinite from the finite,
> > there are far more rationals. Equinumerosity can only be accepted if
> > there is no complete reality of numbers but the potential infinity of
> > numbers created by us.
> >
> > Regards, WM

Ah, you should not ignore models of 'effective infinity', that are basically large finite numbers,
where all the precision of measurements is for example less than the root of it, it suffices
that for any measurement precision, that its increments are arbitrarily small.

For example, modern-day Democritan atomic theory, and Planck length/time and 1 AMU,
has that Democritus or Demokrites, theoretical atoms are after the infinitely-divisible,
but the way it works as for substances the regime of the atomic elements, that Avogadro's
number is arbitrarily large, 10^23, about that "atoms are on the order of 25 orders of magnitude,
smaller, than us".

Then, the idea is that "superstrings" or "superstring theory" are that much arbitrarily smaller
again, or around 50 orders of magnitude, while still, not "infinitely-divisible" or "infinite".

If you wonder, every few years the fundamental small atomic constants their dimensions
are updated or as NIST CODATA, they don't just get more precise, but, smaller.

So, anyways you should be aware that there are lots of notions of the infinitely-divisible
that aren't, but are "effectively" infinitely-divisible, then getting into all the mathematics
of the Euclidean and so on in the Planck regimes, of course they aren't infinite but many
of the usual results associated with properties of the infinitely-divisible hold for it.

(Which is basically a notion that there's an arbitrarily small iota i and arbitrarily large
infinity I that iI = 1, or that 1/oo = iota and the sum or multiples of iota-values,
makes exactly one.)

You can exclude them, "potentially infinite" from the "actually infinite", but the
"effectively infinite" is very usual as "least measurable quantity's inverse".

On 13.11.2023 09:37, Adam Polak wrote:
> Any "finite section" / finite set is not a representative object for conducting research and adjudicating on the specifics of any infinite set.

If you mean actual infinity, that is right. But potential infinity is
only a never ending sequence of finite sets.

> Almost without exaggeration, I will say that the similarity between any, even the largest, finite set and an infinite set, on a scale from 1 to 10 is ZERO.

Yes, this is true for actual infinity because the finite part is
vanishing compared to the infinite part:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = always actually infinite.
Nevertheless even in actually infinite sets basic logic is valid.
Example: If every definable natural number has infinitely many natural
successors, then all definable natural numbers have infinitely many
natural successors.

Regards, WM

On 11/13/2023 2:24 PM, WM wrote:
> On 13.11.2023 09:37, Adam Polak wrote:

>> [...]

> Nevertheless
> even in actually infinite sets
> basic logic is valid.

The problem here is not with the claim itself,
but with what you mean by the claim.

What you (WM) mean by basic logic
isn't basic logic.
It is a quantifier shift.

Even in infinite sets,
a quantifier shift is unreliable (invalid).

> Example:
> If every definable natural number has
> infinitely many natural successors,
> then all definable natural numbers have
> infinitely many natural successors.

No.

Each definable natural has
infinitely-many definable natural successors.
∀n ∈ ℕ, ∃S ᐜ⊆ ℕ: n <ᣔ S

Each definable natural has
infinitely-many definable naturals which
it does not succeed.
∀n ∈ ℕ, ∃S ᐜ⊆ ℕ: S ᣔ≮ n

Each definable natural does not succeed
all definable naturals.
∀n ∈ ℕ: ¬(ℕ ᣔ≤ n)

All definable naturals have
zero-many definable natural successors.
¬∃n ∈ ℕ: ℕ ᣔ≤ n

On 11/13/2023 11:24 AM, WM wrote:
> On 13.11.2023 09:37, Adam Polak wrote:
>> Any "finite section" / finite set is not a representative object for
>> conducting research and adjudicating on the specifics of any infinite
>> set.
>
> If you mean actual infinity, that is right. But potential infinity is
> only a never ending sequence of finite sets.
[...]

MORON!!!!!

Chris M. Thomasson was thinking very hard :
> On 11/13/2023 11:24 AM, WM wrote:
>> On 13.11.2023 09:37, Adam Polak wrote:
>>> Any "finite section" / finite set is not a representative object for
>>> conducting research and adjudicating on the specifics of any infinite set.
>>
>> If you mean actual infinity, that is right. But potential infinity is only
>> a never ending sequence of finite sets.
> [...]
>
> MORON!!!!!

He never gets it. I thought he was going to get it this time, but no.
Well, maybe in another decade or so it will sink in.

Here is a question to ponder:

Consider the set of points in the real interval [pi,pi+1] and how many
irrationals that you can remove from consideration in defining a new
set as follows.

Usually in a real interval like [0,1] you can 'address' one or more
rational points by divide by two (halfway point) and then two more by
dividing by three (one third and two thirds points) and on and on; but
this pi to pi plus one interval won't work that way as the numbers
'addressed' in this manner cannot be in Q.

However, in building your new set, only countably many irrationals will
be excised from consideration this way giving you only rationals for
your new set unless there are more irrationals than there are
rationals.

How can there be only one size of infinte in his world?

On Monday, November 13, 2023 at 11:30:28 PM UTC+1, Jim Burns wrote:
> On 11/13/2023 2:24 PM, WM wrote:
> >
> > If every definable natural number has
> > infinitely many natural successors,
> > then all definable natural numbers have
> > infinitely many natural successors.
> >
> No.

Actually, YES.

Hint: If every x in A has the property P, then all x in A have the property P.

In the context of mathematics we (except of you, it seems) would formalize this claim the following way:

Ax e A: P(x) -> Ax e A: P(x).

This is clearly a substitution instance of the logical tautology

P -> P.

Hence it's certrainly /true/.

On Monday, November 13, 2023 at 11:35:25 PM UTC+1, Chris M. Thomasson wrote:
> On 11/13/2023 11:24 AM, WM wrote:
> > On 13.11.2023 09:37, Adam Polak wrote:
> >>
> >> Any "finite section" / finite set is not a representative object for
> >> conducting research and adjudicating on the specifics of any infinite
> >> set.
> >>
> > If you mean actual infinity, that is right. But potential infinity is
> > only a never ending sequence of finite sets. [...]
> >
> MORON!!!!!

You think?

On Monday, November 13, 2023 at 2:30:28 PM UTC-8, Jim Burns wrote:
> On 11/13/2023 2:24 PM, WM wrote:
> > On 13.11.2023 09:37, Adam Polak wrote:
> >> [...]
> > Nevertheless
> > even in actually infinite sets
> > basic logic is valid.
> The problem here is not with the claim itself,
> but with what you mean by the claim.
>
> What you (WM) mean by basic logic
> isn't basic logic.
> It is a quantifier shift.
>
> Even in infinite sets,
> a quantifier shift is unreliable (invalid).
> > Example:
> > If every definable natural number has
> > infinitely many natural successors,
> > then all definable natural numbers have
> > infinitely many natural successors.
> No.
>
> Each definable natural has
> infinitely-many definable natural successors.
> ∀n ∈ ℕ, ∃S ᐜ⊆ ℕ: n <ᣔ S
>
> Each definable natural has
> infinitely-many definable naturals which
> it does not succeed.
> ∀n ∈ ℕ, ∃S ᐜ⊆ ℕ: S ᣔ≮ n
>
> Each definable natural does not succeed
> all definable naturals.
> ∀n ∈ ℕ: ¬(ℕ ᣔ≤ n)
>
> All definable naturals have
> zero-many definable natural successors.
> ¬∃n ∈ ℕ: ℕ ᣔ≤ n

Ah, that a "quantifer shift" may be valid is called according to the "transfer principle",
that what's true for "each" is true for "all", or as with respect to "heap/sorites" sometimes.
There's even where "quantifier shift" is "anti-transfer", or for variously when it does
or doesn't validly apply, like so.

There are even functions that according to usual convergence tests and laws of large
numbers, for example ranging from zero to one, appear to have a limit that equals one,
but actually go to zero, "functions that belie their finite inputs", if of course, not
necessarily, "standard real functions".

The "Ramsey theory" has a lot going on about various asymptotics.

On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
> On Monday, November 13, 2023
> at 11:30:28 PM UTC+1, Jim Burns wrote:
>> On 11/13/2023 2:24 PM, WM wrote:

>>> If every definable natural number has
>>> infinitely many natural successors,

WM: if ∀n ∈ ℕ, ∃Sₙ ᐜ⊆ ℕ: n <ᣔ Sₙ

>>> then all definable natural numbers have
>>> infinitely many natural successors.

WM: then ∃S ᐜ⊆ ℕ, ∀nₛ ∈ ℕ: nₛ <ᣔ S

>>
>> No.
>
> Actually, YES.
>
> Hint:
> If every x in A has the property P,
> then all x in A have the property P.

Yes.
However,
I think WM means something else.

> In the context of mathematics
> we (except of you, it seems) would formalize
> this claim the following way:
>
> Ax e A: P(x) -> Ax e A: P(x).
>
> This is clearly a substitution instance of
> the logical tautology
>
> P -> P.
>
> Hence it's certrainly /true/.

P -> P
which you (FF) think WM means,
is true.

∀n∈ℕ, ∃Sₙᐜ⊆ℕ: n<ᣔSₙ -> ∃Sᐜ⊆ℕ, ∀nₛ∈ℕ: nₛ<ᣔS
which I (JB) think WM means,
is not true.

On Monday, November 13, 2023 at 3:27:50 PM UTC-8, FromTheRafters wrote:
> Chris M. Thomasson was thinking very hard :
> > On 11/13/2023 11:24 AM, WM wrote:
> >> On 13.11.2023 09:37, Adam Polak wrote:
> >>> Any "finite section" / finite set is not a representative object for
> >>> conducting research and adjudicating on the specifics of any infinite set.
> >>
> >> If you mean actual infinity, that is right. But potential infinity is only
> >> a never ending sequence of finite sets.
> > [...]
> >
> > MORON!!!!!
> He never gets it. I thought he was going to get it this time, but no.
> Well, maybe in another decade or so it will sink in.
>
> Here is a question to ponder:
>
> Consider the set of points in the real interval [pi,pi+1] and how many
> irrationals that you can remove from consideration in defining a new
> set as follows.
>
> Usually in a real interval like [0,1] you can 'address' one or more
> rational points by divide by two (halfway point) and then two more by
> dividing by three (one third and two thirds points) and on and on; but
> this pi to pi plus one interval won't work that way as the numbers
> 'addressed' in this manner cannot be in Q.
>
> However, in building your new set, only countably many irrationals will
> be excised from consideration this way giving you only rationals for
> your new set unless there are more irrationals than there are
> rationals.
>
> How can there be only one size of infinte in his world?

It's pretty simple, you can conscientiously be an ultra-finitist, and indeed for each
there's a certain requirement of finitude and boundedness for matters of comprehension,
but there's also plenty of understanding of large numbers and that there's not just
one "law of large numbers", then besides there's that the "infinite" has its places according
to geometry and perspective, number theory, set theory, and so on, and "infinitesimals"
have their places in topology, analysis, and so on. So, you're welcome to reject the infinite,
and be ultra-finitist, but not refute it, which would be a "retro-finitist crankety troll".

What does set theory mean in math?
A matrix is a set at an angle.
Why would that change anything?

poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):

> If you mean actual infinity, that is right. But potential infinity is
> only a never ending sequence of finite sets.

> Nevertheless even in actually infinite sets basic logic is valid.
> Example: If every definable natural number has infinitely many natural
> successors, then all definable natural numbers have infinitely many
> natural successors.

Regarding logic. Yes, it is crucial and is completely missing in set theory and elsewhere, unfortunately.

"everyone" and "all" have no quantitative context when used to describe an infinite set (infinity), ONLY qualitative! If you interpret it differently, you are making a mistake.
Look,
these:
1, 2, 3, 4, 5, 6, 7, 8...
I, II, III, IV, 5, 6, 7, 8, ...
are QUANTITATIVELY
EXACTLY THE SAME SET/ THE SAME CONSTRUCTION
they differ only in the qualitative parameters of a few elements

consequently, this:
0, 1, 2, 3, 4, 5, 6, 7, ...
is also the same set (from QUANTITATIVE perspective!)
same as:
-2, -1, 0, 1, 2, 3, 4, 5, ...
and so on

It can be said that in a quantitative context there is ONLY ONE infinite set. Or to put it another way, there is only one model of the infinite set, with which - in a quantitative context! - EVERY infinite set, WITHOUT EXCEPTION, is consistent.
The differences between infinite sets come down only to differences in the qualitative parameters of individual objects that are elements of different infinite sets.

It is impossible to understand these issues if you do not understand a number of subtle but key issues, e.g.: as I wrote above: "all" "every" only in the context of qualitative parameters, never in the quantitative context in relation to an infinite set.

Another extremely important issue: Number and numerical value

"numerical value" - is not a mathematical object, it is only a qualitative parameter that an object may have,

"a number" - is a mathematical object with a qualitative parameter that is numerical value, preferably if it is the only qualitative parameter of the number,
if additional ones appear, problems will also arise
e.g.:
4 - "four" (number, mathematical object)
IV - "four" (number, mathematical object)
but!:
IV - "four Roman" (it is an object with an additional quality parameter "Roman", as a result it is no longer a correctly defined number, it is not "a number" anymore)

This may seem like little or nothing important all, but it's an illusion.
This exact error leads to the conclusion that there may be objects other than irrational numbers in the right column of the matrix (Cantor's Diagonal Method). This is simply not true.
This:
0.001
and this:
0.0010000000000000...
are two completely different objects
only the first of which is a properly constructed decimal, "a number".
Yes, both objects have the same parameter that is a numerical value: 1/1000, but what does it matter if they differ in other parameters and in effect constitute two completely different mathematical objects and only one can be called the number: "one thousandth"

BR, Adam

On 14.11.2023 07:09, Adam Polak wrote:
> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you mean actual infinity, that is right. But potential infinity is
>> only a never ending sequence of finite sets.
>
>> Nevertheless even in actually infinite sets basic logic is valid.
>> Example: If every definable natural number has infinitely many natural
>> successors, then all definable natural numbers have infinitely many
>> natural successors.
>
> Regarding logic. Yes, it is crucial and is completely missing in set theory and elsewhere, unfortunately.
>
> "everyone" and "all" have no quantitative context when used to describe an infinite set (infinity), ONLY qualitative! If you interpret it differently, you are making a mistake.
> Look,
> these:
> 1, 2, 3, 4, 5, 6, 7, 8...
> I, II, III, IV, 5, 6, 7, 8, ...
> are QUANTITATIVELY
> EXACTLY THE SAME SET/ THE SAME CONSTRUCTION
> they differ only in the qualitative parameters of a few elements

They differ only in the form of naming the first elements.
>
> consequently, this:
> 0, 1, 2, 3, 4, 5, 6, 7, ...
> is also the same set (from QUANTITATIVE perspective!)
> same as:
> -2, -1, 0, 1, 2, 3, 4, 5, ...
> and so on

If you wish to call 0 now -2, and so on, then both sets are the same. If
you use the names for the well-known numbers, then both sets differ,
namely by two elements of the second set which are missing in the first set.

Regards, WM