2.2 Resolution in Predicate Logic / 2.3 Completeness and Incompleteness

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What does Gödel's formula essentially state?

The formula with number n cannot be proved within the formal system. (correct)

The consistency of the formal system relies on the provability of the formula.

An unprovable formula is inconsistent within the formal system.

A provable formula has a specific number associated with it.

What does the statement 'This formula cannot be proved' indicate if it is true?

It implies that the formal system is complete.

It shows that the system is incomplete. (correct)

It reveals that Gödel's proposition is false.

It signifies that the formal system is consistent.

What conclusion can be drawn if Gödel's sentence is not true?

The proposition is unprovable.

The system is complete.

The system is inconsistent. (correct)

Gödel's formula lacks a specific number.

What does Gödel's formulation of the sentence within the system demonstrate?

The inconsistency and incompleteness of the formal system simultaneously. Signup and view all the answers

Why is determining whether a theorem is provable considered a central question of computability theory?

It reveals whether there exists an algorithm for proof determination. Signup and view all the answers

What is the purpose of unification in predicate logic?

To combine two clauses into a new clause Signup and view all the answers

Which of the following is a true statement about resolution in predicate logic?

Resolution can only be applied if the terms used in the predicates are unified Signup and view all the answers

What is the result of applying resolution to the formulas Px ⋁ Qy and ¬Pv ⋁ Rz with x = v?

Qy ⋁ Rz Signup and view all the answers

What is the definition of a unifier in predicate logic?

A unifier is a substitution that makes two literals equal Signup and view all the answers

How is unification achieved in predicate logic?

Through substitution of variables Signup and view all the answers

What is the result of substituting x with Socrates in the statement 'All humans are mortal'?

Socrates is mortal Signup and view all the answers

Who is known for his work on resolution and the unification algorithm?

John Alan Robinson Signup and view all the answers

What is a unifier from which all other unifiers can be obtained through substitution called?

Most general unifier Signup and view all the answers

What does the set of clauses being inconsistent mean?

It is possible to use resolution to derive a literal L as well as its negation ¬L Signup and view all the answers

What is Hilbert's program?

To base mathematics on a finite, complete set of axioms and provide a proof that these axioms are consistent Signup and view all the answers

Who is known for his incompleteness theorems?

Kurt Gödel Signup and view all the answers

What is the starting point of the antinomies?

Self-reference Signup and view all the answers

What is the significance of the soundness of a calculus?

It ensures that one cannot derive formulas that are not true for all interpretations Signup and view all the answers

What is the significance of the completeness of a calculus?

It ensures that a logical calculus for mathematics must be comprehensive enough Signup and view all the answers

What is the resolution rule used in?

All of the above Signup and view all the answers

What is the significance of the most general unifier not being unique?

It allows for multiple substitutions to be obtained from the most general unifier Signup and view all the answers

What did Kurt Gödel prove in his PhD thesis in 1929?

Predicate logic is complete Signup and view all the answers

What is the implication of Gödel's First Incompleteness Theorem?

A consistent formal system contains tautologies that cannot be proved within it Signup and view all the answers

What is the significance of Gödel's Second Incompleteness Theorem?

A consistent formal system cannot prove its own consistency Signup and view all the answers

What did mathematicians believe about Goldbach's conjecture before the publication of Gödel's incompleteness theorems?

Both a and c Signup and view all the answers

What is the idea for the proof of the first incompleteness theorem related to?

The theorem of diagonalization Signup and view all the answers

What is the fundamental limitation of Gödel's completeness theorem?

Both a and b Signup and view all the answers

What is the significance of the absence of a counterexample to Goldbach's conjecture?

It does not actually prove or disprove Goldbach's conjecture Signup and view all the answers

What is the main difference between Gödel's completeness theorem and his incompleteness theorems?

The completeness theorem states that predicate logic is complete, while the incompleteness theorems state that formal systems are either inconsistent or incomplete Signup and view all the answers

What is the main implication of Gödel's incompleteness theorems for mathematics?

There is no consistent and complete formal system that can serve as a basis for mathematics Signup and view all the answers

What is the significance of Gödel's proof technique known as diagonalization?

It proved that formal systems are either inconsistent or incomplete Signup and view all the answers