From: scerir (scerir@libero.it)
Date: Tue Apr 03 2001 - 15:52:16 MDT
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.
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