TheInfoList

An axiom, postulate or assumption is a statement that is taken to be
true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherw ... , to serve as a
premise A premise or premiss is a statement that an argument claims will induce or justify a Logical consequence, conclusion. It is an assumption that something is true. Explanation In logic, an argument requires a Set (mathematics), set of (at least) ...
or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which commends itself as evident.' The term has subtle differences in definition when used in the context of different fields of study. As defined in
classic philosophy This page lists some links to ancient philosophy, namely philosophical thought extending as far as early post-classical history (c. 600 CE). Overview Genuine philosophical thought, depending upon original individual insights, arose in many cult ...
, an axiom is a statement that is so
evident Evidence, broadly construed, is anything presented in support of an assertion, because evident things are undoubted. There are two kind of evidence: intellectual evidence (the obvious, the evident) and empirical evidence Empirical evidence i ...
or well-established, that it is accepted without controversy or question. As used in modern
logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit ... , an axiom is a premise or starting point for reasoning. As used in
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, �έχνη ''tiké échne', 'art' or 'cra ...
). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g.,
parallel postulate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ... in
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a smal ...
). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there may be multiple ways to axiomatize a given mathematical domain. Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the
philosophy of mathematics Philosophy (from , ) is the study of general and fundamental questions, such as those about reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
.

Etymology

The word ''axiom'' comes from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of 2018; Athens is ...
word (''axíōma''), a
verbal noun A verbal noun or gerundial noun is a verb form that functions as a noun. An example of a verbal noun in English grammar, English is 'sacking' as in the sentence "The sacking of the city was an epochal event" (''sacking'' is a noun formed from the ...
from the verb (''axioein''), meaning "to deem worthy", but also "to require", which in turn comes from (''áxios''), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages () ...
philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek thinker Pythagoras ( ... s an axiom was a claim which could be seen to be self-evidently true without any need for proof. The root meaning of the word ''postulate'' is to "demand"; for instance,
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης ; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often referred to as the "founder of ge ... demands that one agree that some things can be done (e.g., any two points can be joined by a straight line). Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books,
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Greek may refer to: Greece Anything of, from, or related to Greece ...
remarks that "
Geminus Geminus of Rhodes Rhodes (; el, Ρόδος, translit=Ródos ) is the largest of the Dodecanese islands of Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ... held that this
th Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property."
Boethius Anicius Manlius Severinus Boëthius, commonly called Boethius (; also Boetius ; 477 – 524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', and philosopher of the early 6th century. He was born about a ye ...
translated 'postulate' as ''petitio'' and called the axioms ''notiones communes'' but in later manuscripts this usage was not always strictly kept.

Historical development

Early Greeks

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (
syllogisms A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a Logical consequence, conclusion based on two or more propositions that are asse ...
, rules of inference) was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (
theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms ''axiom'' and ''postulate'' hold a slightly different meaning for the present day mathematician, than they did for
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit ... and
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης ; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often referred to as the "founder of ge ... . The ancient Greeks considered
geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ... as just one of several
science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and predictions about the u ... s, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's
posterior analytics The ''Posterior Analytics'' ( grc-gre, Ἀναλυτικὰ Ὕστερα; la, Analytica Posteriora) is a text from Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher an ...
is a definitive exposition of the classical view. An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that
''When an equal amount is taken from equals, an equal amount results.''
At the foundation of the various sciences lay certain additional hypotheses that were accepted without proof. Such a hypothesis was termed a ''postulate''. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates. The classical approach is well-illustrated by
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖον ''Stoikheîon'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), ...
, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions). :;Postulates :# It is possible to draw a
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ... from any point to any other point. :# It is possible to extend a line segment continuously in both directions. :# It is possible to describe a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a ...
with any center and any radius. :# It is true that all
right angle In geometry and trigonometry, a right angle is an angle of exactly 90Degree (angle), ° (degrees), corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the adjacent an ... s are equal to one another. :# ("
Parallel postulate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ... ") It is true that, if a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two rays lie in the plane (ge ...
s less than the two right angles. :;Common notions: :# Things which are equal to the same thing are also equal to one another. :# If equals are added to equals, the wholes are equal. :# If equals are subtracted from equals, the remainders are equal. :# Things which coincide with one another are equal to one another. :# The whole is greater than the part.

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,
propositions In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include ...
, theorems) and definitions. One must concede the need for primitive notions, 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 Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian language, Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new ... ,
Mario Pieri Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (nu ...
, and
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ... were pioneers in this movement. Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory,
group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...
,
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
). 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
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 mathematics itself can be regarded as a branch of
logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit ... .
Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meanin ...
, Russell, Poincaré,
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
, and 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
collection Collection or Collections may refer to: * Cash collection, the function of an accounts receivable department * Collection agency, agency to collect cash * Collections management (museum) ** Collection (artwork), objects in a particular field fo ...
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 In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...
; it should be impossible to derive a contradiction from the axiom. 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 of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a smal ...
, and the related demonstration of the consistency of those axioms. In a wider context, there was an attempt to base all of mathematics on Cantor's
set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...
. Here, the emergence of
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ... and similar antinomies of
naïve set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, formal ...
raised the possibility that any such system could turn out to be inconsistent. The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms ( Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a
corollary In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, Gödel proved that the consistency of a theory like
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
is an unprovable assertion within the scope of that theory. It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s, an
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...
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 (
Cohen ) one can show that the
continuum hypothesis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

Other sciences

Axioms play a key role not only in mathematics but also in other sciences, notably in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental phy ...
. In particular, the monumental work of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his time as a "natural philosophy, natural philosopher") ... is essentially based on
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης ; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often referred to as the "founder of ge ... 's axioms, augmented by a postulate on the non-relation of
spacetime In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
and the physics taking place in it at any moment. In 1905, Newton's axioms were replaced by those of
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...
's special relativity, and later on by those of
general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
. Another paper of Albert Einstein and coworkers (see
EPR paradox The Einstein–Podolsky–Rosen paradox (EPR paradox) is a thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomeno ...
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of ... , concerned the interpretation of
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quan ...
. This was in 1935. According to Bohr, this new theory should be
probabilistic Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
, whereas according to Einstein it should be
deterministic Determinism is the philosophical view that all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and consider ...
. The underlying quantum mechanical theory, i.e. the set of "theorems" derived by it, seemed to be identical. Einstein even assumed that it would be sufficient to add to quantum mechanics "hidden variables" to enforce determinism. However, thirty years later, in 1964, John Bell found a theorem, involving complicated optical correlations (see Bell inequalities), which yielded measurably different results using Einstein's axioms compared to using Bohr's axioms. And it took roughly another twenty years until an experiment of
Alain Aspect Alain Aspect (; born 15 June 1947) is a French physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classical ...
got results in favor of Bohr's axioms, not Einstein's. (Bohr's axioms are simply: The theory should be probabilistic in the sense of the
Copenhagen interpretation The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scal ...
.) As a consequence, it is not necessary to explicitly cite Einstein's axioms, the more so since they concern subtle points on the "reality" and "locality" of experiments. Regardless, the role of axioms in mathematics and in the above-mentioned sciences is different. In mathematics one neither "proves" nor "disproves" an axiom for a set of theorems; the point is simply that in the conceptual realm identified by the axioms, the theorems logically follow. In contrast, in physics, a comparison with experiments always makes sense, since a
falsified In the philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methodology, methods, and implications of science. The central questions of this study concern Demarcation problem, what qualifies a ...
physical theory needs modification.

Mathematical logic

In the field of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
, a clear distinction is made between two notions of axioms: ''logical'' and ''non-logical'' (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).

Logical axioms

These are certain
formulas In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term ''formula'' in science refers to the Commensurability (philosophy of science), general ...
in a
formal language In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms ''at least'' some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantificat ...
more logical axioms than that are required, in order to prove
logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...
s that are not tautologies in the strict sense.

=Propositional logic

= In
propositional logic Propositional calculus is a branch of logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos' ...
it is common to take as logical axioms all formulae of the following forms, where $\phi$, $\chi$, and $\psi$ can be any formulae of the language and where the included primitive connectives are only "$\neg$" for
negation In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argume ... of the immediately following proposition and "$\to$" for implication from antecedent to consequent propositions: #$\phi \to \left(\psi \to \phi\right)$ #$\left(\phi \to \left(\psi \to \chi\right)\right) \to \left(\left(\phi \to \psi\right) \to \left(\phi \to \chi\right)\right)$ #$\left(\lnot \phi \to \lnot \psi\right) \to \left(\psi \to \phi\right).$ Each of these patterns is an ''
axiom schemaIn mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
'', a rule for generating an infinite number of axioms. For example, if $A$, $B$, and $C$ are
propositional variable In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
s, then $A \to \left(B \to A\right)$ and $\left(A \to \lnot B\right) \to \left(C \to \left(A \to \lnot B\right)\right)$ are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and ''
modus ponens In propositional logic Propositional calculus is a branch of logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ ... '', one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with ''modus ponens''. Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed. These axiom schemata are also used in the
predicate calculus Predicate or predication may refer to: Computer science * Syntactic predicate (in parser technology) guidelines the parser process Linguistics *Predicate (grammar) In traditional grammar and syntax, the predicate is the portion of a sentenc ...
, but additional logical axioms are needed to include a quantifier in the calculus.

=First-order logic

=
Axiom of Equality. Let $\mathfrak$ be a first-order language. For each variable $x$, the formula
$x = x$
is universally valid.
This means that, for any variable symbol $x\,,$ the formula $x = x$ can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by $x = x$ (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol $=$ has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that. Another, more interesting example
axiom schemeIn mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
, is that which provides us with what is known as Universal Instantiation:
Axiom scheme for Universal Instantiation. Given a formula $\phi$ in a first-order language $\mathfrak$, a variable $x$ and a
term $t$ that is substitutable for $x$ in $\phi$, the formula
$\forall x \, \phi \to \phi^x_t$
is universally valid.
Where the symbol $\phi^x_t$ stands for the formula $\phi$ with the term $t$ substituted for $x$. (See
Substitution of variables Substitution may refer to: Arts and media *Chord substitution, in music, swapping one chord for a related one within a chord progression *Substitution (poetry), a variation in poetic scansion *Substitution (song), "Substitution" (song), a 2009 so ...
.) In informal terms, this example allows us to state that, if we know that a certain property $P$ holds for every $x$ and that $t$ stands for a particular object in our structure, then we should be able to claim $P\left(t\right)$. Again, ''we are claiming that the formula'' $\forall x \phi \to \phi^x_t$ ''is valid'', that is, we must be able to give a "proof" of this fact, or more properly speaking, a ''metaproof''. These examples are ''metatheorems'' of our theory of mathematical logic since we are dealing with the very concept of ''proof'' itself. Aside from this, we can also have Existential Generalization:
Axiom scheme for Existential Generalization. Given a formula $\phi$ in a first-order language $\mathfrak$, a variable $x$ and a term $t$ that is substitutable for $x$ in $\phi$, the formula
$\phi^x_t \to \exists x \, \phi$
is universally valid.

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 File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s and the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
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
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
). Thus non-logical axioms, unlike logical axioms, are not ''Tautology (logic), tautologies''. Another name for a non-logical axiom is ''postulate''. 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. 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 example, 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. Thus, an ''axiom'' is an elementary basis for a Formal system#Logical system, formal logic system that together with the rules of inference define a deductive system.

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 Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, �έχνη ''tiké échne', 'art' or 'cra ...
, 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. 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.Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2 We have a language $\mathfrak_ = \$ where $0$ is a constant symbol and $S$ is a unary function and the following axioms: # $\forall x. \lnot \left(Sx = 0\right)$ # $\forall x. \forall y. \left(Sx = Sy \to x = y\right)$ # $\left(\phi\left(0\right) \land \forall x.\,\left(\phi\left(x\right) \to \phi\left(Sx\right)\right)\right) \to \forall x.\phi\left(x\right)$ for any $\mathfrak_$ formula $\phi$ with one free variable. The standard structure is $\mathfrak = \langle\N, 0, S\rangle$ where $\N$ is the set of natural numbers, $S$ is the successor function and $0$ 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
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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 theorems 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.

Role in mathematical logic

Deductive systems and completeness

A deductive system consists of a set $\Lambda$ of logical axioms, a set $\Sigma$ of non-logical axioms, and a set $\$ of ''rules of inference''. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas $\phi$,
$\text\Sigma \models \phi\text\Sigma \vdash \phi$
that is, for any statement that is a ''logical consequence'' of $\Sigma$ there actually exists a ''deduction'' of the statement from $\Sigma$. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system. Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no ''recursive'', ''consistent'' set of non-logical axioms $\Sigma$ of the Theory of Arithmetic is ''complete'', in the sense that there will always exist an arithmetic statement $\phi$ such that neither $\phi$ nor $\lnot\phi$ can be proved from the given set of axioms. There is thus, on the one hand, the notion of ''completeness of a deductive system'' and on the other hand that of ''completeness of a set of non-logical axioms''. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Further discussion

Early mathematicians regarded Foundations of geometry, axiomatic geometry as a model of physical space, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra (logic), Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Évariste Galois, Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and abstract algebra, modern algebra was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.

* Axiomatic system * Dogma * First principle, axiom in science and philosophy * List of axioms * Model theory * Regulæ Juris * Theorem * Presupposition * Physical law * Principle