Re: compactness in angels/devil problem
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Fri, 01 Feb 2008 08:07:55 -0600
On Fri, 1 Feb 2008 03:11:45 -0800, William Elliot
<marsh@xxxxxxxxxxxxxxxxxx> wrote:
On Fri, 1 Feb 2008 pauldepstein@xxxxxxx wrote:
The literature on the angels-and-devil problem often refers to aThe compactness theorem for FOL logic is:
"compactness argument" for passing from conclusions about finite
boards to conclusions about the infinite case. What is this
"compactness argument" and which topology is the compactness concept
being applied to?
if P can be concluded from an infinite set S of statements,
then P can be concluded from a finite subset of S.
Formally. Within an FOL, if S is an infinite set of statements
and S |- P, then there's a finite F subset S with F |- P.
No, that's much too trivial to be called a theorem.
The Compactness Theorem is the corresponding statement
for |= in place of |-.
(The proof of the Compactness Theorem proceeds by using
the Soundness and Completeness Theorems, which say that
|- and |= are equivalent. But this equivalence is not trivial.)
David C. Ullrich
.
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