Our answer has been YES. Gödel's uncertainty is valid for the B set. The LEM is also valid in the B set. In the B set, numbers are either 0 or 1. And 0^n=0, while 1^n=1, so arithmetic is fast and easy. Digital computers only use the B set, and yet can calculate everything. Gödel's uncertainty is valid.
We, humans, can use the Q* set for fast and easy mental calculations. A negative times a negative is a positive. Gödel's uncertainty is not valid.
Quantum computing uses the set Q, to allow calculus with discontinuous functions--as functions must be in the digital world. We see that world in the XXI century. Gödel's uncertainty is not valid.
By the Curry-Howard relationship, this deprecates Gödel's uncertainties.
So must be finally accepted, under experiments -- not theory or opinions.
No longer individually distinguishable, the digits in each prime number is a "lump" and belong together, in a collective effect beyond digits or names. Peter Shor said this first, in 1994. This is important for quantum computing.
What is your qualified opinion?
(We are not attempting to define nature or evolution. We are just pointing out an illusion. Mathematical results can be absolutely exact.)
This may be useful here. The set ℂ (mathematical complex numbers) is condemned because it uses the set ℝ (mathematical real-numbers), which is condemned (see Gerck 2022, Gisin 2022).
One could think to still use the set G (the Gauss set, with rational imaginary numbers), but not only the other inconsistencies apply, but every path must begin and end in a rational number, with no room for imaginary numbers.
In summary, it is time to give up any investigation. The set ℂ is now an illusion.
But, I don't define nature nor evolution. I am just pointing out "not this way!".
We are correcting a 500 year error -- complex numbers do not exist and are not needed.
The sooner they are not used, the better. 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.
The Gödel uncertainties depend on Boolean logic being applicable. That is not the case in reality, see below.
It was one of the most surprising "discoveries" of the Pythagorean School of Greek mathematicians that would be irrational numbers. According to well-known Courant and Robbins, quite possibly it marked the origin of what one can consider the specifically Greek contribution to "rigorous procedure" in mathematics. Certainly it has profoundly affected mathematics and philosophy from the time of the Greeks to the present day, though it is false -- there are no irrational numbers!
Specifically, the Greeks "discovered" that the diagonal of a square whose sides are 1 unit long has a diagonal whose length "cannot" be rational. By the Pythagorean Theorem, the length of the diagonal equals the square root of 2, where 2 must be equal to p^2/q^2 -- but p and q must be even, which is a false contradiction! In Section 1.1 in the preprint, we use the Hurwitz theorem to find infinitely telescoping approximations in rational numbers of the square root of 2, with ever smaller errors to any arbitrary-length, but one cannot calculate the 0-D point where one would have the square root of 2 -- which remains always undefined.
The error of the traditional Greek "proof" is not to consider "undefined" as a possible logical state, in addition to "yes" and "no". The absence of a number is not "yes" or"no" to that number, but "undefined". The "present King of France" is not 1 neither 0. It is undefined.
This is contrary to the well-known assumption that considers π to be an irrational number, yet no one can numerically evaluate it -- all we can calculate, find, and make, are rational numbers (Section 1.1).
See mnanuscript on pi in my RG profile.
While plenty of us are capable of considering 1 with no reference to physical reality, including experiments with fish and invetebrates (that have no digits), that is not efficient, according to Dedekind, Frege, and others in mathematical semiotics. We are able to do more, when we can use archetypes (cf. Jung).
Now, with quantum mechanics, logic has at least 3 states.
It is a loss if someone does not consider these 2 facts -- what is right may seem downright incomprehensible, like the two-slit experiment. How to divide a photon? Answer: one does not divide, one uses stimulated emission and multiplies.
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