Editing Formal system (section)

==== Proof system ====
{{Main|Proof system|Formal proof}}
Formal proofs are sequences of [[well-formed formula]]s (or WFF for short) that might either be an [[axiom]] or be the product of applying an inference rule on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a [[Theorem#Theorems in logic|theorem]].

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a [[decidability (logic)|decision procedure]] for deciding whether a given WFF is a theorem or not. 

The point of view that generating formal proofs is all there is to mathematics is often called ''[[Formalism (philosophy of mathematics)|formalism]]''. [[David Hilbert]] founded [[metamathematics]] as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a ''[[metalanguage]]''. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the ''object language'', that is, the object of the discussion in question. The notion of ''theorem'' just defined should not be confused with ''theorems about the formal system'', which, in order to avoid confusion, are usually called [[metatheorem]]s.