Re: Which theories hold a true but unprovable statement?

On Feb 7, 6:34 pm, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:
Before you can make sense of the idea of
a sentence being true, you
have to specify a model for the theory.

"The theory" implies that THEORIES are what the missing
quantifer in this sentence is quantifying over. The fact that
no qualifications or characteristics are HEREin ATTRIBUTED
to the theory means that unless this generalization holds
FOR ALL [of the relevant class of theories, in this case
recursively axiomatizable first-order] theories, IT IS FALSE.
Wherefore I rebutted,
Self-refuting as usual. I continue to be amazed by who
graduated when I didn't.

Rupert continues:
What I said is quite correct.

No, it isn't, and the whole point is, THE VERY THEORY YOU
OFFER here IS A COUNTEREXAMPLE to your generalization.
That is why I called you self-refuting.

In the study of first-order languages,
to define the notion of truth for a sentence in a given first-order
language, you have to specify in which model.

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,
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.

This is not self-refuting at all.

Consider the theory of real-closed fields, for example.

THAT is self-refuting. THAT, as a counter-example to your claim that
models mattered IN GENERAL for determining truth-values of statements
in the theory, is a refutation of it.

.

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