Dear Professor Calude and Professor Svozil,
I like your essay, which is novel and creative, and on topic. I consider Professor's Calude book on constructive mathematics very good. I have some comments and questions about your essay.
The speed of light-based Lorentz invariance seems to be more than simply a convention suggested by Maxwell's equations. It is also related to the proper time of particles, and the experiments confirmed this. But I do not reject the possibility of superluminal speeds. Even a continuously accelerated penetration of light barrier may be possible, if the particles have somehow variable proper mass, going towards zero as the speed approaches that of light.
Assume that we have an accelerated Zeno-Turing machine, asymptotically approaching a time T.
- If the algorithm just switches a bool variable at each step, what value will the variable have after T?
It seems to me that, in order to have a result, either the Turing machine has to halt after a finite, although large, number of steps, or to converge to a definite limit, although the latter may require infinite storage capacity.
> "such a mechanism may compute incomputable functions, for example, the characteristic function of the halting problem"
This may be true. The characteristic function of the halting problem requires infinite storage, and browsing through the output allows us to access only a finite number of them in finite time.
To read an infinite output in a finite time will require an infinite mind. Having this capacity already makes us equivalent to an accelerated Zeno-Turing machine.
- Do you think we could use such a machine to provide a finite length derivation of the axiom of choice, and of the continuum hypothesis, from Zermelo-Frankel's axioms?
I agree that if Quantum Mechanics is indeterministic, or even if it is deterministic, but subject to delayed initial conditions (I give some details in my essay, and I will be honored if you would be interested to reading it), we may obtain unbiased randomness. Unbiased randomness seems indeed to be impossible to obtain from deterministic Turing machines. This may be used in constructing non-deterministic Turing machines, but these are known to be equivalent with the deterministic ones.
Congratulations for your essay, and success with this contest,
Cristi Stoica