Re: Looking for Undecidable Propositions in Systems without a certain amount of arthimetic.

On Aug 13, 1:27 pm, Scott <ToaTe...@xxxxxxxxx> wrote:
On Aug 12, 2:43 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

The pure predicate calculus in any given language is one. The pure
predicate calculus in any given language is the least theory in that
langauge. Then any contingent sentence in that language is undecidable
in the theory.

What about FOL using only logical axioms?

That's exactly what I'm talking about. The set of sentences that are
provable from only logical axioms is the theory I'm calling "the pure
predicate calculus" (a bit of a misnomer, since I'm actually referring
to the theory itself (the set of sentences closed under entailment)
rather than the system of deduction, but I've made clear what I mean).
The set of sentences provable from only the logical axioms is
incomplete, just as we WANT it to be, in the sense that the contingent
sentences (those that are not logically true) are not in the set of
sentences provable only from logical axioms.

MoeBlee
.

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