Editing Theorem (section)

==Theoremhood and truth==
Until the end of the 19th century and the [[foundational crisis of mathematics]], all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every [[natural number]] has a successor, and that there is exactly one  [[line (mathematics)|line]] that passes through two given distinct points. These basic properties that were considered as absolutely evident were called [[postulate]]s or [[Axiom|axioms]]; for example [[Euclid's postulates]]. All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the [[interior angle]]s of a [[triangle]] equals 180°, and this was considered as an undoubtable fact.

One aspect of the foundational crisis of mathematics was the discovery of [[non-Euclidean geometries]] that do not lead to any contradiction, although, in such geometries, the sum of the angles of a triangle is different from 180°. So, the property ''"the sum of the angles of a triangle equals 180°"'' is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of [[set (mathematics)|sets]] leads to the contradiction of [[Russell's paradox]]. This has been resolved by elaborating the rules that are allowed for manipulating sets.

This crisis has been resolved by revisiting the foundations of mathematics to make them more [[mathematical rigor|rigorous]]. In these new foundations, a theorem is a [[well-formed formula]] of a [[mathematical theory]] that can be proved from the [[axiom]]s and [[inference rules]] of the theory. So, the above theorem on the sum of the angles of a triangle becomes: ''Under the axioms and inference rules of [[Euclidean geometry]], the sum of the interior angles of a triangle equals 180°''. Similarly, Russell's paradox disappears because, in an axiomatized set theory, the ''set of all sets'' cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is [[inconsistent]], and every well-formed assertion, as well as its negation, is a theorem.

In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even the significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas.

An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as [[mathematical object]]s, and to prove theorems about them. Examples are [[Gödel's incompleteness theorems]]. In particular, there are well-formed assertions than can be proved to not be a theorem of the ambient theory, although they can be proved in a wider theory. An example is [[Goodstein's theorem]], which can be stated in [[Peano arithmetic]], but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as [[Zermelo–Fraenkel set theory]].