Re: clarification on recursively enumerable (R.E.)

From: shane miller (clearthink_at_attbi.com)
Date: 11/09/04

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    Date: 8 Nov 2004 20:14:53 -0800

    Will Twentyman writes:
    >For the purpose of 2, there is no such thing as an infinite amount of
    >time, much less whether or not 'A' will be accepted/rejected. There is
    >strictly "accept", "reject", "cannot know".
    basically this is what i said: in the case the claimed proof 'A' is
    not rejected in a finite amount of time, all that can remain is an
    infinite wait. but ...

    i thought i was told by a professor, however, that given an infinite
    amount of time (realizing this is IMPOSSIBLE AND BESIDE THE POINT FOR
    first order logic) the decision method would disposition 'A' as theorem
    or not theorem. But as this is beside the point for first order logic,
    i don't want to spend time on this issue!

    daryl@atc-nycorp.com (Daryl McCullough) wrote in message
    > That's not quite right. If A has a proof, then it has
    > a proof that can be discovered in a finite amount of time,
    > under the usual notion of what a proof is. The actual
    > situation for r.e. theories is this:
    >
    > <snip>
    >
    > 2. If A is *not* provable, then there may be no way to
    > demonstrate it in a finite number of steps.
    But why? I mean the decision method is probably looking for
    both the proof of 'A' and the proof of 'not A'. But what is
    the source of "then there may be no way to demonstrate it in
    a finite number of steps."

    It cannot be the case that an infinite number of premises
    are required because compactness eliminates this possibility!

    Regards,
    Shane

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