Formal 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.

The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation.