Editing Axiom (section)

===Modern development===
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, [[propositional logic|propositions]], theorems) and definitions. One must concede the need for [[primitive notion]]s, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. [[Alessandro Padoa]], [[Mario Pieri]], and [[Giuseppe Peano]] were pioneers in this movement.

Structuralist mathematics goes further, and develops theories and axioms (e.g. [[Field theory (mathematics)|field theory]], [[group (mathematics)|group theory]], [[topological space|topology]], [[linear space|vector spaces]]) without ''any'' particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., [[hyperbolic geometry]]). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.

When mathematicians employ the [[Field (mathematics)|field]] axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and [[logicism|mathematics itself can be regarded as a branch of logic]]. [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]], [[Henri Poincaré|Poincaré]], [[David Hilbert|Hilbert]], and [[Kurt Gödel|Gödel]] are some of the key figures in this development.

Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.

In the modern understanding, a set of axioms is any [[Class (set theory)|collection]] of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be [[consistent]]; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization{{efn|Hilbert also made explicit the assumptions that Euclid used in his proofs but did not list in his common notions and postulates.}} of [[Euclidean geometry]],<ref>For more, see [[Hilbert's axioms]].</ref> and the related demonstration of the consistency of those axioms.

In a wider context, there was an attempt to base all of mathematics on [[Georg Cantor|Cantor's]] [[set theory]]. Here, the emergence of [[Russell's paradox]] and similar antinomies of [[naïve set theory]] raised the possibility that any such system could turn out to be inconsistent.

The formalist project suffered a setback a century ago, when [[Gödel's incompleteness theorems|Gödel showed]] that it is possible, for any sufficiently large set of axioms ([[peano arithmetic|Peano's axioms]], for example) to construct a statement whose truth is independent of that set of axioms. As a [[corollary]], Gödel proved that the consistency of a theory like [[Peano arithmetic]] is an unprovable assertion within the scope of that theory.<ref>{{Citation|last=Raatikainen|first=Panu|title=Gödel's Incompleteness Theorems|date=2018|url=https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2018|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-19}}</ref>

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of [[natural number]]s, an [[Infinite set|infinite]] but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern [[Zermelo–Fraenkel axioms]] for set theory. Furthermore, using techniques of [[forcing (mathematics)|forcing]] ([[Paul Cohen|Cohen]]) one can show that the [[continuum hypothesis]] (Cantor) is independent of the Zermelo–Fraenkel axioms.<ref>{{Citation|last=Koellner|first=Peter|title=The Continuum Hypothesis|date=2019|url=https://plato.stanford.edu/archives/spr2019/entries/continuum-hypothesis/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-19}}</ref> Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.