Editing Gödel's incompleteness theorems (section)

=== Systems which contain arithmetic ===

The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of [[Robinson arithmetic]] {{mvar|Q}}. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems.

The theory of [[algebraically closed field]]s of a given [[characteristic (algebra)|characteristic]] is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers.  A similar example is the theory of [[real closed field]]s, which is essentially equivalent to [[Tarski's axioms]] for [[Euclidean geometry]]. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.

The system of [[Presburger arithmetic]] consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication.

{{harvard citations |txt=yes |first=Dan |last=Willard |author1-link=Dan Willard |year=2001}} has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; that is to say, these systems are consistent and capable of proving their own consistency (see [[self-verifying theories]]).