Editing Axiom (section)

===Non-logical axioms===
'''Non-logical axioms''' are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the [[natural number]]s and the [[integer]]s, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as [[group (algebra)|groups]]). Thus non-logical axioms, unlike logical axioms, are not ''[[Tautology (logic)|tautologies]]''. Another name for a non-logical axiom is ''postulate''.<ref name="properaxioms">Mendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2</ref>

Almost every modern [[mathematical theory]] starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.{{Citation needed|date=July 2011}}{{Explain|date=June 2019|reason=use of past tense without explanation of change}}<!-- This turned out to be impossible{{Citation needed|date=March 2010}} and proved to be quite a story (''[[#role|see below]]''); however recently this approach has been resurrected in the form of [[neo-logicism]].-->

Non-logical axioms are often simply referred to as ''axioms'' in mathematical [[discourse]]. This does not mean that it is claimed that they are true in some absolute sense. For instance, in some groups, the group operation is [[commutative]], and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

====Examples====
This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.

Basic theories, such as [[arithmetic]], [[real analysis]] and [[complex analysis]] are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of [[Zermelo–Fraenkel set theory]] with choice, abbreviated ZFC, or some very similar system of [[axiomatic set theory]] like [[Von Neumann–Bernays–Gödel set theory]], a [[conservative extension]] of ZFC. Sometimes slightly stronger theories such as [[Morse–Kelley set theory]] or set theory with a [[strongly inaccessible cardinal]] allowing the use of a [[Grothendieck universe]] is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as [[second-order arithmetic]].{{citation needed|reason=This claim should include a citation |date=April 2016}}

The study of topology in mathematics extends all over through [[point set topology]], [[algebraic topology]], [[differential topology]], and all the related paraphernalia, such as [[homology theory]], [[homotopy theory]]. The development of ''abstract algebra'' brought with itself [[group theory]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[Galois theory]].

This list could be expanded to include most fields of mathematics, including [[measure theory]], [[ergodic theory]], [[probability]], [[representation theory]], and [[differential geometry]].

=====Arithmetic=====
The [[Peano axioms]] are the most widely used ''axiomatization'' of [[first-order arithmetic]]. They are a set of axioms strong enough to prove many important facts about [[number theory]] and they allowed Gödel to establish his famous [[Gödel's second incompleteness theorem|second incompleteness theorem]].<ref>Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2</ref>

We have a language <math>\mathfrak{L}_{NT} = \{0, S\}</math> where <math>0</math> is a constant symbol and <math>S</math> is a [[unary function]] and the following axioms:

# <math>\forall x. \lnot (Sx = 0) </math>
# <math>\forall x. \forall y. (Sx = Sy \to x = y) </math>
# <math>(\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x)</math> for any <math>\mathfrak{L}_{NT}</math> formula <math>\phi</math> with one free variable.

The standard structure is <math>\mathfrak{N} = \langle\N, 0, S\rangle</math> where <math>\N</math> is the set of natural numbers, <math>S</math> is the [[successor function]] and <math>0</math> is naturally interpreted as the number 0.

=====Euclidean geometry=====
Probably the oldest, and most famous, list of axioms are the 4 + 1 [[Euclid's postulates]] of [[Euclidean geometry|plane geometry]]. The axioms are referred to as "4 + 1" because for nearly two millennia the [[parallel postulate|fifth (parallel) postulate]] ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior [[angle]]s of a [[triangle]] add up to exactly 180 degrees or less, respectively, and are known as Euclidean and [[hyperbolic geometry|hyperbolic]] geometries. If one also removes the second postulate ("a line can be extended indefinitely") then [[elliptic geometry]] arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

=====Real analysis=====
The objectives of the study are within the domain of [[real numbers]]. The real numbers are uniquely picked out (up to [[isomorphism]]) by the properties of a ''Dedekind complete ordered field'', meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use of [[second-order logic]]. The [[Löwenheim–Skolem theorem]]s tell us that if we restrict ourselves to [[first-order logic]], any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in [[non-standard analysis]].