Editing Gödel's incompleteness theorems (section)

=== Conflicting goals ===
In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers {{harv|Smith|2007|p=2}}. In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called the [[principle of explosion]]), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a [[maximal set]] of non-[[Contradiction|contradictory]] theorems.{{Citation needed|date=May 2023}}

The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the continuum hypothesis, which is unresolvable<ref>in technical terms: [[Independence (mathematical logic)|independent]]; see [[Continuum hypothesis#Independence from ZFC]]</ref> in ZFC + "there exists an inaccessible cardinal".

The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.