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

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

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

3. What is a proof complexity in proof theory?

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

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

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

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

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

6. What are the logical connectives in propositional logic?

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

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

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

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

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

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