Formal system

A formal system is the use of an <a href="/facts/Axiomatic_system/dwMz70d5">axiomatic system</a> utilized for <a href="/facts/Deductive_reasoning/HadA1zWS">deductive reasoning</a> (or alternatively an <a href="/facts/Mathematical_induction/KxAPqNrH">inductive</a> system) or an <a href="/facts/Abstract_structure/EkSkcwdQ">abstract structure</a> whose properties are specified. The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of <a href="/facts/Notation/IkqYLL0X">notation</a>, for example, <a href="/facts/Paul_Dirac/6bt5yRdk">Paul Dirac</a>'s <a href="/facts/Bra%25E2%2580%2593ket_notation/NDlKsaRh">bra–ket notation</a>.
In 1921, <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a> proposed to use formal systems as the foundation of knowledge in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>.
However, in 1931 <a href="/facts/Kurt_G%25C3%25B6del/P14DqKqK">Kurt Gödel</a> proved that any <a href="/facts/Consistency/IHkdnA9M">consistent</a> formal system sufficiently powerful to express basic arithmetic cannot prove its 
own <a href="/facts/Completeness_(logic)/GzK9zLnu">completeness</a>, i.e. some statements can never be proven, or disproven.
<a href="/facts/G%25C3%25B6del%2527s_incompleteness_theorem/du8PRu8g">Gödel's incompleteness theorem</a> showed that Hilbert's grand plan was impossible as stated.

Formal system open-in-new

Formal system