Editing Gödel's completeness theorem (section)

==Preliminaries==
There are numerous [[deductive system]]s for first-order logic, including systems of [[natural deduction]] and [[Hilbert-style deduction system|Hilbert-style systems]]. Common to all deductive systems is the notion of a ''formal deduction''. This is a sequence (or, in some cases, a finite [[tree structure|tree]]) of formulae with a specially designated ''conclusion''. The definition of a deduction is such that it is finite and that it is possible to verify [[algorithm]]ically (by a [[computer]], for example, or by hand) that a given sequence (or tree) of formulae is indeed a deduction.

A first-order formula is called ''[[Validity (logic)|logically valid]]'' if it is true in every [[structure (mathematical logic)|structure]] for the language of the formula (i.e. for any assignment of values to the variables of the formula). To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system. A deductive system is called ''complete'' if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense. Thus, in a sense, there is a different completeness theorem for each deductive system. A converse to completeness is ''[[soundness theorem|soundness]]'', the fact that only logically valid formulas are provable in the deductive system.

If some specific deductive system of first-order logic is sound and complete, then it is "perfect" (a formula is provable if and only if it is logically valid), thus equivalent to any other deductive system with the same quality (any proof in one system can be converted into the other).{{Citation needed|date=December 2024|reason=Provide reliable source defining 'perfect'.}}