On Mon, Aug 12, 2019 at 4:36 PM Jason Resch <[email protected]> wrote:
> In "The Universal Numbers. From Biology to Physics" Bruno writes > > "The universal dovetailing can be seen as the proofs of all true Sigma_1 > propositions there exists x,y,z such that P_x(y) = z, with some sequences > of such propositions mimicking the infinite failing or proving some false > Sigma_1 propositions." > > This is something I was thinking about recently in the context of > universal Diophantine equations. It seems more correct to me to say these > equations don't themselves represent the execution traces of the programs, > but rather represent proofs of the outputs of programs. > > This can be seen from the fact that the work of verifying a Diophantine > equation requires only a finite and constant number of arithmetical > operations, while the computation itself could involve much more work, in > terms of arithmetical steps. > > So is it right to say that the proof of the result of some computable > function is different from the computable function itself? In other words, > a fixed Diophantine equation, regardless of the values of its variables, > does not itself yield conscious mind states, though it points to the > existence of another object in math (the universal machine) whose operation > would yield the conscious mind states? > > I am just trying to develop a more clear picture in my mind of the > relation between arithmetic, proofs, computational traces, and mind states. > > Jason > Another comment/question: "with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions." Assuming the undecidability of the Mandelbrot set, does this undedicability have the same implications as the above "mimicking the infinite failing or proving some false Sigma_1 propositions"? In a poetic manner of speaking, does the shape of the Mandelbrot set's edge somehow mirror the shape of non-computable functions in the UD? Can we infer anything about the existence of all computations from the mathematical existence of the complete Mandelbrot set alone? Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUgxAv1sF5T2dS8%2Bi2y8GLi%2BmMQKxZZUzukrxuKjZRJNyw%40mail.gmail.com.
