Tõestussüsteemides

Tõestussüsteemid are formal systems used in logic and mathematics to demonstrate the validity of statements, known as theorems. A proof system consists of a set of axioms and a set of inference rules. Axioms are fundamental statements that are accepted as true without proof. Inference rules are logical operations that allow new statements to be derived from existing ones. The goal of a proof system is to construct a sequence of statements, starting from axioms and applying inference rules, that logically leads to a desired theorem. This sequence of statements constitutes the proof.

Different types of proof systems exist, each with its own characteristics and applications. Propositional logic proof systems deal with simple propositions and logical connectives like AND, OR, and NOT. Predicate logic proof systems extend this by incorporating quantifiers (like "for all" and "there exists") and variables, allowing for more complex statements about objects and their properties. Hilbert-style proof systems are known for their large number of axioms and simple inference rules. Natural deduction proof systems, on the other hand, aim to mimic human reasoning with more intuitive inference rules. Sequent calculus is another important proof system that focuses on deriving sequents, which are statements of the form "if P then Q."

The properties of a proof system are crucial for its usefulness. A proof system is considered sound if every provable statement is indeed true. It is considered complete if every true statement within its logical framework can be proven. Decidability is another important property, meaning there exists an algorithm that can determine whether any given statement is provable. The study of proof systems is fundamental to understanding the foundations of mathematics and computer science, particularly in areas like automated theorem proving and formal verification.