**Next message:**Technotranscendence: "ECON: The Fed's Y2K Recession of 2001"**Previous message:**J. R. Molloy: "Re: "analog computer" = useless hypothesis?"**In reply to:**hal@finney.org: "Re: MATH/COMP/PHIL: "Omega Man""**Next in thread:**Lee Corbin: "Re: MATH/COMP/PHIL: "Omega Man""**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

Hal Finney wrote:

*> I still think it is useful to think of Euclidean geometry minus the
*

*> parallel postulate to gain insight into these issues. No TM is ever
*

*> going to prove the PP starting with this trucated axiom base.
*

Actually the Parallel Postulate (or Ramsey's Theorem or the

Continuum Hypothesis, etc.) is an independent statement.

That is to say: it can not be proven within the formal axiomatic

system (and its negation can not be proven either).

A statement S is independent of the recursively axiomatizable

theory T iff a program P, which searches in T the proof (or the

refutation) of S, never halts.

Independent statements are a sub-set of the finitely refutable

statements set. A finitely refutable statement is equivalent to

the assertion that a program P, searching for some non-existent

object, never halts.

**Next message:**Technotranscendence: "ECON: The Fed's Y2K Recession of 2001"**Previous message:**J. R. Molloy: "Re: "analog computer" = useless hypothesis?"**In reply to:**hal@finney.org: "Re: MATH/COMP/PHIL: "Omega Man""**Next in thread:**Lee Corbin: "Re: MATH/COMP/PHIL: "Omega Man""**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

*
This archive was generated by hypermail 2b30
: Mon May 28 2001 - 09:59:44 MDT
*