Proof theory

1. Proof theory is a branch of mathematical logic that focuses on the structure of mathematical proofs. It deals with the study and analysis of formal mathematical deduction systems and the rules used to establish the validity of mathematical statements. Proof theory is concerned with the fundamental question of how mathematical arguments can be formulated in a rigorous and systematic way, with the ultimate goal of providing a clear and precise understanding of the reasoning behind mathematical theorems and their proofs.

What is a Herbrand interpretation in proof theory?

A) An interpretation of a first-order logic formula by assigning concrete values to variables.
B) An interpretation based on mathematical induction.
C) An interpretation that relies on axiomatic systems.
D) An interpretation used in software engineering.

2. What is the goal of normalisation in proof theory?

A) To transform a proof into a canonical form for easier analysis.
B) To standardize the notation used in mathematical proofs.
C) To eliminate the need for formal proofs.
D) To add complexity to a proof in order to make it more convincing.

3. What is a proof complexity in proof theory?

A) The study of the resources required to prove mathematical theorems.
B) Counting the number of logical connectives in a formula.
C) Measuring the length of a mathematical proof.
D) Determining the truth value of a proposition.

4. What is the principle of cut elimination in proof theory?

A) The principle that cuts cannot be used in formal logic.
B) The rule that cuts are necessary for valid proofs.
C) Every proof containing a cut can be transformed into a cut-free proof.
D) The property that all proofs must eliminate cuts.

5. What is the Curry-Howard correspondence in proof theory?

A) A historical event in proof theory.
B) A rule for constructing mathematical proofs.
C) A correspondence between proofs and computer programs in intuitionistic logic.
D) A type of logical inference.

6. What are the logical connectives in propositional logic?

A) FOR, WHILE, DO.
B) IF, THEN, ELSE.
C) ADD, SUBTRACT, MULTIPLY.
D) AND, OR, NOT.

7. Who introduced the concept of sequent calculus in proof theory?

A) Gerhard Gentzen.
B) Henri Poincaré.
C) Alonzo Church.
D) Alfred Tarski.

8. What is the connection between Gödel's incompleteness theorems and proof theory?

A) The theorems establish standard axiomatic systems.
B) The theorems show the limitations of formal proof systems.
C) The theorems provide new techniques for proof construction.
D) The theorems eliminate the need for proof complexity.

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