You wrote: "At first we might see a problem in the existence of Aspect’s machine, since here we have a system that can be defined and constructed with a relatively minimal number of statements, but yet is capable of producing strings of any possible length which can then be proved to be random through violations of Bell’s inequalities showing their quantum origin."

You are mixing 2 very...

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You wrote: "At first we might see a problem in the existence of Aspect’s machine, since here we have a system that can be defined and constructed with a relatively minimal number of statements, but yet is capable of producing strings of any possible length which can then be proved to be random through violations of Bell’s inequalities showing their quantum origin."

You are mixing 2 very different notions of randomness. The notion of quantum randomness is relative to the time when a device is in a specific state and ready to produce a number which will come at random in the sense that it is not determined yet, as repeating the same experiment with exact copies of the device with the same initial state may give different results. We have a physical indetermination of the choice between many possible values it is still able to take.

On the other hand, you mentioned the notion of whether a given fixed number is a random number or not. This is a completely different question.

Indeed by Chaitin's theorem, it is not possible to prove that any specific big number is a random one, however this does not diminish the fact that anyway most big numbers are random. Concretely, when a quantum device is going to produce a number at random (that quantum paradoxes show to be random in a physical sense, i.e. that its value is not fixed in advance but any number still has actual chances to be produced by the device at this time), there is then a provably high probability for the future yet undetermined output to have the property of "being a random number" in the sense of Kolmogorov complexity.

This "high probability of being a random number" is related to the fact that the average expectable complexity is a high one.

The point is that the provably high value of the average expectable complexity of the future undetermined output, does not change anything to the fact that, according to Chaitin's theorem, it is not possible to formally prove the high value of the complexity of any specific number, among all possibilities.

To explain very simply this "paradox" which I do not even find strange, imagine a function f that will play the role of the Kolmogorov complexity, to be applied to a future output that may turn out to be either a or b with probability 1/2 each.

Then we know that the average expectable level of complexity is high : f(a)+f(b)>2 is provable. However for each specific possibility, there is no way to prove its high complexity : neither f(a)>1 nor f(b)>1 is provable.

Nevertheless, we have a proof of f(a)+f(b)>2 and thus of (f(a)>1 or f(b)>1).

But this provability of (f(a)>1 or f(b)>1) does not anyhow contradict that f(a)>1 is unprovable and f(b)>1 is also unprovable, because the formal proof of (f(a)>1 or f(b)>1) does not provide any way to formally find out which of both numbers f(a) and f(b) is >1.

Still there is a real mathematical truth about it but it is not algorithmically computable. (This non-computability should not be confused with physical indetermination.)

If you are amazed that physical devices may produce random numbers, well, remember that a physical quantum device is NOT a classical deterministic Turing machine, which Chaitin's theorem exclusively refers to. As I explained in my essay, I interpret quantum randomness as having a non-physical source (conciousness). See also my arguments against the hidden variable approach.

You wrote: "As photons are generated by excited atoms, we can only state the exact polarization of photons is indeterminate prior to measurement. Since the polarization states are generated from some quantum source, then Aspect’s machine contains a component that is indescribable in the formal language that can describe its construction and prove its result."

It all depends on the precise experiment. Some kinds of excited atoms may have a spin that determines the polarization of the emitted photon. In practice the result is usually indeterminate because systems are made of many atoms whose spins are randomly distributed, full of entropy.

You wrote: "real numbers can be characterized as being rational functions, which are simply the ratio of polynomials"

Looking at the reference you gave where this amazing result on the simple nature of real numbers comes from, the explanation is this one:

"the real numbers are a subset of the rational functions" where "By a rational function in the variable t, we will mean a function of the form p(t)/q(t), where p(t) and q(t) are polynomials with standard real coefficients", so that "for instance, the number 7 can be viewed as 7/1, where 7 and 1 are both polynomials of degree 0". So more generally in this way, we can get any real number u in the form of the rational function u/1 where u and 1 are seen as polynomials of degree 0 with real coefficients. Wow !

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