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Given a theory (say $T$) which satisfies the hypotheses of the Godel- Rosser theorem. How to show whether a Rosser sentence $R$ for $T$ is true for the standard interpretation ($\Bbb N$) or not?.

(I know it is true but how we can prove this claim!)

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Isn't Rosser's sentence: "If this sentence is provable then its negation is provable with a proof of smaller Gödel number than this one's"? If so, if arithmetic is consistent then if $R$ was provable then its negation would be provable. This is impossible. Thus it's not provable therefore it's true. –  Apostolos Mar 6 '11 at 14:44
    
Rosser's sentence: "If this sentence is provable then its negation is provable with a proof of smaller Gödel number than this one's"! yes it is. –  user7863 Mar 6 '11 at 15:07

1 Answer 1

The way that you prove that something is true in the standard interpretation is, essentially, by just considering what the sentence means in the standard interpretation. In the standard model, when the Rosser sentence uses the "Pvbl" predicate, this actually refers to the "is provable" predicate of the metatheory.

Now you reason in the metatheory as Apostolos did in a comment: if the sentence was provable in $T$, then its negation would also be provable in $T$, which is impossible because $T$ was assumed to be consistent. So the sentence is not provable. The sentence itself is an implication whose hypothesis is "if this sentence is provable". Since it isn't provable, the hypothesis is false, which makes the entire implication vacuously true. That's how you prove the Rosser sentence is true in the standard interpretation.

One thing that can be confusing about this is that when we prove the sentence is true in the standard interpretation, we don't just work in $T$. We know the sentence is not provable in $T$, so we can't do that anyway: we have to add extra metatheoretic assumptions about a particular model if we want to show the sentence is true in that model.

The way we make those assumptions here is by working with a particular model of $T$ in which the formal provability predicate "Pvbl" corresponds to actual provability. That is the only special property of the standard model that we need in our proof. So the proof that the Rosser sentence is true in the standard model can be viewed equally well as a proof that the Rosser sentence is true in any model whose formal provability predicate agrees with the standard one for standard sentences.

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