[FOM] Re: The Myth of Hypercomputation

[Toby Ord]

On 10 Feb 2004, at 19:35, JoeShipman at aol.com wrote:

> 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.

Two technical points here:

1. You may have been pointing to this yourself, but there is an 
important issue regarding what we mean by 'definable' here. For 
example, it is possible (if apparently unlikely) that such a constant 
may code some set that is not definable within, say, ZFC but is easily 
definable from our context (such as truth within ZFC). This is the same 
for all fixed notions of definability as we can always diagonalize out 
in an 'obvious' manner. It seems that the notion of definability we 
would like here should be closed under such obvious diagonalizations, 
but no such notion is available.

2. We may have a number that is indefinable in whatever sense is 
appropriate and yet still have a case of useful non-recursive 
computation. For example, the odd digits in its expansion could code 
the halting problem while the even ones are some 'undefinable' decimal 
number. This would still be very useful for us.

Toby Ord.