Re: A simple paradox in Godels incompleteness theorem that invalidat

On 8 Oct, 02:48, Newberry <newberr...@xxxxxxxxx> wrote:
On Oct 6, 3:43 pm, Peter_Smith <ps...@xxxxxxxxx> wrote:

On 6 Oct, 23:30, Newberry <newberr...@xxxxxxxxx> wrote:
On Oct 5, 11:09 pm, Peter_Smith <ps...@xxxxxxxxx> wrote:
Sorry, perhaps I'm misunderstanding. What do you mean by "semantically
incomplete"?

That there wll be true but unprovable sentences.

By syntactically incomplete I mean that will be some formulas F such
that neither F nor ~F are provable.

But if T is syntactically incomplete, i.e. there is some sentence F
such that neither F nor ~F are provable, then whichever is the true
one out of F and ~F will be an example of a true but unprovable-in-T
sentence. Syntactic incompleteness trivially entails semantic
incompleteness (for classical theories, anyway).

We are not talking about classical theories, we are talking about ANY
theory that can encode sufficient arithmetic. For what we know we
could be talking about second order, 4-valued, modal, relevance logic.

For example if the theory is not bivalent then syntactic
incompleteness does not entail semantic incompleteness.

Oh I see what you have in mind! But don't get me started on dodgy
logics :-)) Dialethism anyone?

Just two points. Apologies, my previous post mis-spoke. The entailment
obviously depends on our meta-logic, not the logic in the object
theory.

Second, as to bivalence, we can say at least this. The Gödelian
argument for the negation incompleteness of a theory T satisfying the
usual conditions is constructively acceptable, so establishes the
truth of the unprovable-in-T Gödel sentence even for an intuitionist
who doesn't endorse excluded middle. So, for an intuitionist, the same
argument that demonstrates syntactic incompleteness reveals semantic
incompleteness.




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