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Gödel's Incompleteness Theorem and Gentzen's Proof

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Gödel's Incompleteness Theorem and Gentzen's Proof

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Questions and Answers

What is the significance of Gentzen's theory in relation to first-order Peano arithmetic (PA)?

Gentzen's theory is stronger than PA.

Gentzen's theory is a subtheory of PA.

Gentzen's theory is a weaker theory than PA.

Gentzen's theory is incomparable to PA. (correct)

What can be proved in Gentzen's theory that cannot be proved in PA?

The consistency of PA. (correct)

Ordinary mathematical induction for all formulae.

The completeness of PA.

The inconsistency of PA.

What is the significance of the notion of interpretability?

It is used to prove the inconsistency of a theory.

It is used to show the completeness of a theory.

It is used to compare the strength of two theories. (correct)

It is used to prove the consistency of a theory.

What can be conclude about theory T if it is interpretable in another theory B?

T is consistent if B is consistent. Signup and view all the answers

What is the consequence of the second incompleteness theorem?

A theory cannot prove its own consistency. Signup and view all the answers

What is the relationship between the consistency of theory T and theory B if T is interpretable in B?

T is consistent if B is consistent. Signup and view all the answers

What is the idea that motivates the use of interpretability to compare theories?

That, if B interprets T, then B is at least as strong as T Signup and view all the answers

What does Pudlák's result state about a consistent theory T that contains Robinson arithmetic, Q?

T cannot interpret Q plus Con(T) Signup and view all the answers

What is true about Q+Con(T) and T, according to the arithmetized completeness theorem?

Q+Con(T) is stronger than T Signup and view all the answers

Why can't PA interpret Gentzen's theory, according to Pudlák's result?

Because Gentzen's theory interprets Q+Con(PA) and interpretability is transitive Signup and view all the answers

What is true about Gentzen's theory and PA, in terms of consistency strength?

Gentzen's theory is stronger than PA Signup and view all the answers

What is required for a theory to be stronger than another theory, in terms of consistency strength?

That the stronger theory interprets the weaker theory Signup and view all the answers