Re: A simple paradox in Godels incompleteness theorem that invalidat

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.


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