Re: Which theories hold a true but unprovable statement?



george wrote:

Of course, but IN A DECIDABLE RECURSIVELY AXIOMATIZABLE
FIRST-ORDER THEORY, YOU DON'T need to consider any models
to discern the truth or falsity of any sentence. EVERY SENTENCE
in such theories is true (xor false) IN EVERY model of the theory,

Every sentence of every theory is true in every model of that theory.

This is a tautology, in the bad sense of tautology. Structures
in which the sentences of a theory are not all true are not models of
the theory.

I guess we know what you mean, though, although the OP might be
confused.


SO MODELS ARE JUST IRRELEVANT. You can decide all truths
SYNTACTICALLY FROM THE THEORY, WITHOUT EVER KNOWING
OR CARING about Semantics or Models. Truth is *originally*
defined in a model-DEpendent way, but IF BY CHANCE (or by design)
the truth-value HAPPENS to come up the SAME in all models, then THAT
is a proof that the truth-value of THOSE sentences IS INDEPENDENT of
choice of model. If this happens for ALL sentences over the language
of
the theory (i.e. when the theory is decidable),then models are almost
COMPLETELY IRRELEVANT, and if in addition to being decidable, the
theory should also be categorical, then models REALLY are completely
irrelevant. The theory already captures everything that any model
might've
hoped to add; the model is irrelevant because it is redundant.

I think you might want to phrase this more carefully in terms
of models, structures, sentences of a language, sentences of a
theory.

I agree that implication is structure-independent.

--
hz
.

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