[FOM] Re: The Myth of Hypercomputation

[JoeShipman@aol.com]

I like the distinction drawn here between functions which are violations of some form of CT because somehow an infinite amount of computational work is done, and those which violate CT and involve only a finite physical process.

An example of the latter possibility would be if some yet-to-be-discovered physical theory implied that a physically measurable dimensionless real number (such as the fine-structure constant, or the ratio of two particle masses or two half-lives, or the ratio of occurence of two different modes of decay of a particle, etc.) had a non-recursive decimal expansion.  

For such a result to be established, the number in question would have to be DEFINABLE but nonrecursive; then ZFC would only determine the value of finitely many places in the decimal expansion of the number, and more precise measurement would allow us to derive new mathematical truths from experiment, that could not be proven in ZFC.  

(Of course we would have to believe that the physical theory was correct and the experimental setup was appropriate, but that is not so different from the current situation where we accept computer-assisted proofs because we believe the theory behind the construction of the computers and believe that the algorithms have been appropriately implemented.)

-- Joe Shipman