Re: Defining Consistency



Nam Nguyen wrote:


Are you saying that we don't know of a theory that has 2 non-
contradictory axioms? What about a theory whose axioms are
2 atomic formulae, or 2 elementary formulae? What about the
"geometry" theory, with or without the 5th postulate? [People do
have different mental (abstract) models in which the postulate could
be *interpreted* to be true or false!]. You can correct me if I'm wrong
but I think such a theory would be consistent; hence the kind of proof
I've presented would be _valid_. No?

I think I've come to a resolution for the issue of validity of such
"proof". Proofs (meta or otherwise) have to come from relevant
definition and the logical rules involved. In the case of inconsistent
theories, the definition (in whatever equivalent form) would render
a sound procedure (or proof) for the inconsistency: all it takes is
one instance of a theorem of the form F /\ ~F. For instance, given
any particular formula F, the theory { F /\ ~F} can be proven to be
inconsistent, based on the definition. On the other hand, if a general
theory (that would contain infinite number of axioms) is genuinely
consistent, the definition of consistency in whatever equivalent form
wouldn't help: it's impossible to have an effective procedure
(or non-vacuous proof) simply because we could not finitely exhaust
the list of infinite (non-logical) theorems, to find out that it has no
theorem of the form F /\ ~F.

Would this bring a closure to the issue?
--
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Why then do thoughts linger, long after everything
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Ryokan
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Relevant Pages

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