formal system
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any well-defined system of abstract thought based on the model of mathematics
~11 min read
Encyclopedic overview
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.
Excerpted from Wikipedia’s “formal system” article, available under the CC BY-SA 4.0 licence.