Consequence number one: the numerical axis exhibits the properties of fractality not only through the Cantor fractal.

One significant consequence would have been that Baysian methodology of statistics+ probability is the one Mother Nature obeys . . ..

Dear Ed Gerck !

You have been fearful enough to even remove your name-mentioning mechanism from this thread .

You hastily write :

" Nature is ontically digital, quantum. Otherwise, R~Q, and R~Q*, is an approximation, to use R. Every effect in physical reality can be accomplished with Q or Q* only; none requires R, C, or G. Also, infinitesimals do not have to be used at all. "

Thanks for the cooperation. "

Thanks for whose cooperation ?

how many true scientists ( non-charlatans like you ) came over this thread to oppose you ?

How many dozens of them affirmed and confirmed your "Genius" posts after posts after posts for proving "Digitalism" ?? !! In how many posts and by how many mathematical formulae did you possibly manage to prove this ? And in what peer-reviewed journal ?? !!

True scientists here did not even bother to reply back to a writing like :

Quote :

" Nature is ontically digital, quantum. Otherwise, R~Q, and R~Q*, is an approximation, to use R. Every effect in physical reality can be accomplished with Q or Q* only; none requires R, C, or G. Also, infinitesimals do not have to be used at all. "

Ontically digital ?

Or ontically made up of spacetime continuous fabric as proven repeatedly by General Relativity ?

Are you brushing all of that aside by a few words instead of by writing a full tome ?? !!

In connection with the issue under consideration, the following hypothesis arises: corpuscular-wave dualism, quantum entanglement, and even the Heisenbrrg uncertainty relation are consequences of the connectedness of the whole connected system of such regularities as the Cantor-Robinson transfinite ladder, the Levenheim-Skolem theorem, the Whittaker-Nyewist-Kotelnikov-Shannon theorem and fractality numerical axis

Dear Dr. Ed Gerck,

Don't you think that in your works [12] and [13] you have not so much revealed the mistake of your predecessors as formulated a new principle of equivalence?

Ann Gerck claims :

" ... the math currently taught in middle school is now outdated. "

so you may ,please , not use [ in your everyday life ] divisions , multiplications , subtractions , additions , , , , , ,etc . . .. . .etc ; ; ; ; that you learnt from elementary and middle schools ...............

Appreciations !!

Laurent Schwartz also began his approach to distribution theory by developing a concept of generalized solutions of partial differential equations. But he was not entirely satisfied with his approach because it did not make sense of the derivatives on the right hand and left hand side of the wave equation separately. Schwartz wanted a theory where the derivative f made sense also for non differentiable functions. In 1945 he found the answer. First he developed a theory of what he called convolution operators and some months later he simplified the theory and defined a generalized function as a functional, as described above. In fact , Schwartz had been anticipated by Sergei Sobolev in 1936, but Sobolev had not developed the theory as far as Schwartz did.

Work on generalized solutions of partial differential equations began around the turn of the 20th century. One of the earliest explicit discussions of generalized solutions was given in Hilbert's 20th problem which asked whether every regular variation problem had a solution, provided that the notion of solution was suitably extended. Serrin came to credit the beginning of the study of generalized solutions of elliptic boundary-value problems to Hilbert [1900]. This work also gave rise to "direct methods" in the calculus of variations. For the first time , sets did not go one on one .

Mr Ed Gerck gets so many repetitive "private messages" ...............

How private !!

Indeed, if the Hamiltonian is deemed an r/Q Functional , then time is an integral multiple of a fundamental quantum of time ‡ . Consequently, not the hamiltonian but the evolution operator for one time-step, U(∆t) = e −i∆t H , is the most fundamental object describing standing for INFINITESIMALS . The difficulty that we then have to face is that there is no way to define a ground state, or vacuum. Without a ground state we are unable to do thermodynamics, and we are deprived from our stability arguments: the distinction between stable, low mass particles and unstable massive ones can no longer be made, and it is impossible to define low-energy and high-energy domains of the theory. At first sight this may sound as an insignificant complication, but it is fundamental. We are not able to formulate the requirement that our theory should approach a conventional field theory at the large-distance scale by differentials of discontinuous functions, and R~Q.

by quantizing space-time in such a way that can "functionally" calculate all of the constants of physics/economics and derive and explain every equation therein , appears only when the related field seems to change faster than phenomenon of the objective reality itself .

Quote :

" This means, using the Curry-Howard relationship, that structural logic itself must change, and reflect that. "

This is a very deep statement on the part of Mr Gerck , which has to be taken into second and third thought most meticulously . . .. This brings about ( possibly ) a RE_discovery of Gödel , too. . . . .

The undecidability , if algorithmic , takes more than issues of functionality discontinuity to progress forward into true discreteness . The way I think about it is in the context of type theory, in which case a program is a proof that its type is inhabited into discrete steps . The answer to this question depends on how expressive your type system is.

Oh !

My !

One of those non-ending series of totally private massages !!

ALright ,,,,,,,,,,,,,

In Physics a "theory" is a mathematical model based on various assumptions and valid for a limited range of physical conditions. Newton's laws are a mathematical model that is limited to non-relativistic speeds and low gravitational fields, and within those limits it is exceedingly accurate. There is no sense in which Newton was proved wrong by Einstein. There weren't really any hints that Newtonian gravity was limited until special relativity came along and raised problems. In the context of pre-SR mechanics Newtonian gravity was a complete theory. It would have been natural, but not justified, to assume then that Newtonian gravity was exactly correct. This is the sense in which GR "overthrew" Newton.

These days physicists tend to think we know better where the limits of our theories are. Our "effective theories" have built-in breaking points which tell us where to look for new physics. Not so much for Newton pre-special relativity. What relativity did is expand the range of physical conditions over which the theory applied. Special relativity extended the range to include high speeds, and general relativity extended it again to include high gravitational fields. Even GR is not applicable everywhere because it fails at singularities like the centre of black holes. We expect that some future theory (string theory?) will extend GR to describe places that are singular in GR.

The notion of Lebesgue integral still seems absolutely vital with the self-styled R~Q. Illiteracy is rampant here on this thread AND charlatanism goes sky-rocketing .............

The most fearful charlatan Mr Ed Gerck is still in a dither as to activate his Mention system on the nonsense that he writes here on this thread and others . Plzz : At least have the courage to take off those thick dark-smoked glasses and hats ,, and activate your "Mention" name here so that people around can take you a tiny bit seriously .

Quote :

" The conclusion is that any mathematical choice against R~Q has an extremely low chance to succeed."

Right now everything against your self-styled R~Q is good and well and going fine . Calculus of infinitesimals which you refute for no valid reason is doing its job perfect well , too .

Quote :

" Consider 3 states for each result n, n=1,16, where each result can be:

(1) R is equivalent to Q;

(2) R is different from Q; or

(3) something else.

The case (3) can be considered a negation of (1), as a subcase of the state (2), and thus any result can have only 2 possible states, (1) or (2). "

Under no system of logic does such a conclusion pops out . What you are writing is mere Alice-in-wonderland hallucinations.

I have aleady wrrote that "Consequence number one: the numerical axis exhibits the properties of fractality not only through the Cantor fractal". It seems that it was on this path that one of the possible answers to your question was found. Since I am not very fond of probabilistic methods, my version is different from yours.

https://www.researchgate.net/publication/367542059_NATURAL_BOUNDARIES_OF_e-NEIGHBORHOODS

https://www.researchgate.net/publication/368831616_Rational_numbers_in_the_hyperrreal_world

Hierarchy of the ε-boundaries is a fractal system on the numeric axis. This system is related to the Cantor fractal, but does not coincide with it (the Cantor fractal is defined over the field of real numbers). This fractal system of infinitely small but limited intervals is at least one of the possible variants of dualism (complementarity) between discreteness and (hyper)continuum.

Alexey Orlovsky

how happy you and Ed may be to reciprocally write posts and admire one another !!

Quote :

" Hierarchy of the ε-boundaries is a fractal system on the numeric axis. This system is related to the Cantor fractal, but does not coincide with it (the Cantor fractal is defined over the field of real numbers). This fractal system of infinitely small but limited intervals is at least one of the possible variants of dualism (complementarity) between discreteness and (hyper)continuum. "

The term fractal set or fractal coined by MANDELBROT refers to all "non-smooth" sets. In order to make this term productive both in the mathematical sense and in the sense of the applications, he means Objects that are self-similar in one way or another, for which there is - roughly speaking - no characteristic size: Rather, these objects merge into themselves with all enlargements (or reductions), at least in a certain size range and for essential Characteristics only .. .. .. . ..

Quote :

" I wish there would be a funny button at RG ... "

I , too , wish so.........

AND : Wish for more than that :

I wish there was another button for charlatan-detectionism , as well ; ; ; ; ; ; ; ;

Quote :

" This is called the Dharma in Hindu tradition. To everyone, their actions in return. The good ones and the bad ones can evolve, letting actions have consequences -- reducing the space for bad actions, in time.

The end is an evolved space and no conflict or censorship. "

I didn't believe we could discuss religion here on this thread ...........

Alexey Orlovsky

ed Gerck

The program for approximating the square computations is correct with respect to its (mathematical) specification if, for an input 𝑎, it returns an output 𝑥 whose square 𝑥 2 differs by less than 𝜖 from 𝑎 (postcondition). The program returns the value of root asymptomaticity , so we can equate 𝑥 with root and the condition must hold for root. Now the program ends (and thus returns a value) only when root satisfies the condition abs(root * root - a) < e, which corresponds exactly to the above postcondition. If the program ends or, as one says in computer science, terminates, then the result is correct according to specification, and this independently of the input.

As a result , in our eyes any search for once-and-for-all valid genealogies implies an originary distortion, which defines the type as a form of intellectual appropriation of reality. From this point of view , there can only be relative genealogies of root computations even by Q_Computers , ,,, , Such load-bearing fields of commutability can be defined only locally/regionally, and by the automatically simultaneously targeted migratory movements of the type****

REZA

We are correcting a 500 year error -- complex numbers do not exist and are not needed. Who lives will see the results. The Fourier transform and the FFT can use only rational numbers.

The sooner complex numbers are not used, the better. We are suggesting the addition of set Q*.

FFT=FT and can be faster. Galois theory needs to change. Polynomials have coefficients in the set Q. The quintic can be always solved in the set Q. The formula for third-degree polynomials works always in the set Q. The equation for second-degree polynomials change. Discontinuous functions can be differentiated and integrated.

Mathematics evolves.

" Mathematics evolves. "

The only proposition any sentient scientist can non-mechanically agree with .

REZA

I changed the title of this question, to reflect that computers do consider that R=Q, not just as an equivalence (~).

Computers also never use infinitesimals, epsilon-deltas, complex numbers, mathematical real-numbers, or complex Fourier Transforms, as the hardware can only do operations in the Boolean set, where the Boolean set is B={0,1}. Everything else is emulated, even in co-processors, necessarily with errors, needless complications, and loss of time. By the Curry-Howard relationship, computers also refute the Gödel's uncertainties.

And yet, computers can calculate anything. Computers are the actual driving force evolving mathematics and logic through experimentation -- we are just reporting.

To aid us, we find at least 4 quantum properties that are always present in the number systems of sets N, Z, Q, and Q* [1] -- and we can use them for quantum consciousness and quantum computing.

This is explored somewhere else, but the first three quantum properties are discussed in [1], while the fourth quantum property of numbers can be seen in prime numbers, as indistinguishability.

Indistinguishability sets in because one cannot distinguish any number in the prime number 163283381, it is a “blob” and cannot be split in any way. This is valid, using the set N, here on Earth and in the star Betelgeuse. It is a universal quantum property, the fourth -- and adds in the factoring of numbers using quantum mechanics.

[1] DOI: 10.3390/math11010068

There is only one way to avoid criticism: do nothing, say nothing, and be nothing. ~ Aristotle

Albeit, there is another way to avoid criticism -- keep doing it.

If one gets a rational number (which I say happens in any physical measurement), one can get a natural number from it by a known isometric relationship, Q --> N.

One cannot get an irrational number by adding numbers in the set Q -- the set Q is closed for addition, +Q--> Q.

Any number x in the set Q, to add computability, obeys N --> B^n --> B ={0,1}. Computers can calculate anything measurable-- p-adic numbers, and the sets R, C, G, do not appear!

No physical measurement can be in sets R or C (which has two sets R).

No physical measurement can be in set G, either, because a physical path must begin and in the set Q -- and one can always get two set Q numbers between any two set G numbers.

QED.

Is π a rational number?

Quantum computing says: certainly YES.

A stunning solution of the mystery of π was discovered by Indian mathematicians about 1500 CE and rediscovered by our quantum computing in 2023.

The number π can be exactly represented by the infinite, but amazingly simple, series:

π/4 = 1 − 1/3 + 1/5 − 1/7 +⋯.

Since the set Q is closed under addition/subtraction, the result of the series must be rational, hence π is a rational number.

The same applies to e, square-root(2), square-root(n) when n is in the set Q, and other expressions (finite or infinite) using elements of the sets N, Z, Q, and Q*.

The ancient Greeks were right: the square-root(2) is CERTAINLY a rational number, even though it results from an infinite addiction/subtraction.

What could not be calculable is now reachable, and the Hurwitz Theorem can be extended to cover all such cases.

Cryptography cannot be done with mathematical real-numbers, or numbers from other "imagined mathematics".

Cryptography makes R=Q with no room for irrational numbers.

Cryptography has to be done with rational numbers. That affirms the usefulness of real mathematics in physical reality.

Voting or opinions in contrary are not relevant, nor soviet methods.