Re: poly: Infinite calculations in an open universe?

"Perry E. Metzger" <perry@piermont.com> writes:

> You are making an implicit assumption, which is that the number of
> problems is countably infinite. It is not. Cantorian set theory shows
> that many sets are NOT mappable into the integers.

Yes, but Turing machines can only deal with countably infinite
problems; if you allow devices which use true continuum calculations
you can deal with uncountable problems (and do things that are
Turing-uncomputable, I seem to recall?), but due to physics they are
not very likely. All problems we can generate are countable, since we
express them as symbol strings; if we start from a finite countable
set of problems we can only get a countable set of new problems.

> However, it is trivial to state classes of problems that map into
> the reals

But are they interesting? Most problems are just trivial
generalizations of other problems ("If x has property P, does
x+epsilon have property P?"). Can you suggest a class of problems that
map into the reals where every problem is of interest?

-- 
-----------------------------------------------------------------------
Anders Sandberg                                      Towards Ascension!
asa@nada.kth.se                            http://www.nada.kth.se/~asa/
GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y