Deduction system in the context of Hilbert's program

Deduction system in the context of Hilbert's program

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⭐ Core Definition: Deduction system

A formal system (or deductive system) is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.

In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.However, in 1931 Kurt Gödel proved that any consistent formal system sufficiently powerful to express basic arithmetic cannot prove its own completeness. This effectively showed that Hilbert's program was impossible as stated.

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Deduction system in the context of Disjunction introduction

Disjunction introduction or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjunctions to logical proofs. It is the inference that if P is true, then P or Q must be true.

An example in English:

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← Disjunction introduction
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Abstract structure → Formalism (philosophy of mathematics) → Axiomatic system → Deductive reasoning → Rule of inference → Theorem → Axioms → David Hilbert → Mathematics → Kurt Gödel → Consistency → Completeness (logic) → Hilbert's program →