Editing Metalogic (section)

=== Formal systems ===
{{Main|Formal system}}

A ''formal system'' (also called a ''logical calculus'', or a ''logical system'') consists of a formal language together with a [[Deductive system|deductive apparatus]] (also called a ''deductive system''). The deductive apparatus may consist of a set of [[Rule of inference|transformation rule]]s (also called ''inference rules'') or a set of [[axiom]]s, or have both. A formal system is used to [[Proof theory|derive]] one expression from one or more other expressions.

A ''formal system'' can be formally defined as an ordered triple <α,<math>\mathcal{I}</math>,<math>\mathcal{D}</math>d>, where <math>\mathcal{D}</math>d is the relation of direct derivability. This relation is understood in a comprehensive [[Sense and reference|sense]] such that the primitive sentences of the formal system are taken as directly [[formal proof|derivable]] from the [[empty set]] of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are so chosen that every first place member of <math>\mathcal{D}</math>d is a member of <math>\mathcal{I}</math> and every second place member is a finite subset of <math>\mathcal{I}</math>.

A ''formal system'' can also be defined with only the relation <math>\mathcal{D}</math>d.   Thereby can be omitted <math>\mathcal{I}</math>  and α in the definitions of ''interpreted formal language'', and ''interpreted formal system''. However, this method can be more difficult to understand and use.<ref name = "itslaia"/>