TheInfoList

OR:

An axiom, postulate, or assumption is a statement that is taken to be
true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...
, to serve as a
premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agr ...
or starting point for further reasoning and arguments. The word comes from the
Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...
word (), meaning '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, an axiom is a statement that is so
evident Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidenc ...
or well-established, that it is accepted without controversy or question. As used in modern
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, an axiom is a premise or starting point for reasoning. As used in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
). 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, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...
). 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 are typically many 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 The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
.

# Etymology

The word ''axiom'' comes from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
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 is 'sacking' as in the sentence "The sacking of the city was an epochal event" (''sacking'' is a noun formed from the verb ''sack''). ...
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 ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...
philosopher A philosopher is a person who practices or investigates 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 t ...
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, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
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 Neoplatonist philosopher, one of the last major classical philosopher ...
remarks that "
Geminus Geminus of Rhodes ( el, Γεμῖνος ὁ Ῥόδιος), was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the ''Introduction to the Phenomena'', still survives; it was intended as an in ...
held that this thPostulate 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 Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, '' magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
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 conclusion based on two propositions that are asserted or assumed to be true ...
,
rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
) 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, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...
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 and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
and
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
. The ancient Greeks considered
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
as just one of several
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidenc ...
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's '' Organon'' that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguish ...
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 A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
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, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...
, 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 In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segme ...
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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is co ...
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 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. ...
s are equal to one another. :# ("
Parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
") 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 rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
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 logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
, theorems) and definitions. One must concede the need for
primitive notion In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an ...
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 Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pi ...
, and
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The ...
were pioneers in this movement. Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory,
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closin ...
, 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, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
). 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 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 the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
.
Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philo ...
, Russell, Poincaré,
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, 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 (church), money donated by the congregation during a church service * Collection agency, agency to collect cash * Collectio ...
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, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is 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 of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...
, 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 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 a branch of mathematics, is mostly concern ...
. Here, the emergence of
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
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 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 and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
, 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 mathematician Giuseppe Peano. These axioms have been used nearly ...
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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
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 (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
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, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
(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

Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance, Newton's laws in classical mechanics,
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
in classical electromagnetism, Einstein's equation in general relativity, Mendel's laws of genetics, Darwin's
Natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Charl ...
law, etc. These founding assertions are usually called ''principles'' or ''postulates'' so as to distinguish from mathematical ''axioms''. As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (
falsified Falsifiability is a standard of evaluation of scientific theories and hypotheses that was introduced by the philosopher of science Karl Popper in his book ''The Logic of Scientific Discovery'' (1934). He proposed it as the cornerstone of a so ...
) the theory that the postulates install. A theory is considered valid as long as it has not been falsified. Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidian geometry or differential calculus that they imply. It became more apparent when
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
first introduced
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The la ...
where the invariant quantity is no more the Euclidian length $l$ (defined as $l^2 = x^2 + y^2 + z^2$) > but the Minkowski spacetime interval $s$ (defined as $s^2 = c^2 t^2 - x^2 - y^2 - z^2$), and then
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
where flat Minkowskian geometry is replaced with
pseudo-Riemannian In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
geometry on curved
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
. In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The ' Copenhagen school' (
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
,
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent seri ...
,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another ' hidden variables' approach was developed for some time by Albert Einstein,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
,
David Bohm David Joseph Bohm (; 20 December 1917 – 27 October 1992) was an American-Brazilian-British scientist who has been described as one of the most significant theoretical physicists of the 20th centuryPeat 1997, pp. 316-317 and who contributed ...
. It was created so as to try to give deterministic explanation to phenomena such as entanglement. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the EPR paradox in 1935). Taking this ideas seriously, John Bell derived in 1964 a prediction that would lead to different experimental results ( Bell's inequalities) in the Copenhagen and the Hidden variable case. The experiment was conducted first by
Alain Aspect Alain Aspect (; born 15 June 1947) is a French physicist noted for his experimental work on quantum entanglement. Aspect was awarded the 2022 Nobel Prize in Physics, jointly with John Clauser and Anton Zeilinger, "for experiments with entang ...
in the early 1980's, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc).

# 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 ...
, 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 a
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sy ...
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 quantifie ...
more logical axioms than that are required, in order to prove
logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whi ...
s that are not tautologies in the strict sense.

### =Propositional logic

= In
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
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, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fal ...
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 schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ...
'', a rule for generating an infinite number of axioms. For example, if $A$, $B$, and $C$ are
propositional variable In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of proposit ...
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, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It ...
'', 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: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
, 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 scheme, 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.) 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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s and the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 persons 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 ide ...
). Thus non-logical axioms, unlike logical axioms, are not '' tautologies''. Another name for a non-logical axiom is ''postulate''. Almost every modern
mathematical theory A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms. It can also simultaneously be a body of knowledge (e.g., based on known axioms and definitions), and so in this sense can refer to an area o ...
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 Discourse is a generalization of the notion of a conversation to any form of communication. Discourse is a major topic in social theory, with work spanning fields such as sociology, anthropology, continental philosophy, and discourse analysis ...
. 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, 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 logic system that together with the
rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
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 () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
,
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conve ...
and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
with choice, abbreviated ZFC, or some very similar system of
axiomatic set theory 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 a branch of mathematics, is mostly concern ...
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 In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precur ...
. The study of topology in mathematics extends all over through point set topology,
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classi ...
, differential topology, and all the related paraphernalia, such as
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topo ...
,
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
. The development of ''abstract algebra'' brought with itself
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
, rings, fields, and
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory t ...
. This list could be expanded to include most fields of mathematics, including
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
, ergodic theory,
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
,
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In ess ...
, and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
.

### =Arithmetic

= The
Peano axioms 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 mathematician Giuseppe Peano. These axioms have been used nearly ...
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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
and they allowed Gödel to establish his famous 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 A unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may or may not coincide with its range. Exam ...
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 In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
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
plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. The axioms are referred to as "4 + 1" because for nearly two millennia the 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 In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
add up to exactly 180 degrees or less, respectively, and are known as Euclidean and
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
geometries. If one also removes the second postulate ("a line can be extended indefinitely") then
elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
. The real numbers are uniquely picked out (up to
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
) 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 In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies ...
. The Löwenheim–Skolem theorems tell us that if we restrict ourselves to
first-order 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 quantifi ...
, 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 Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
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
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History ...
s regarded 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
made elaborate efforts to derive them from traditional arithmetic. 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 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 In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...
*
Dogma Dogma is a belief or set of beliefs that is accepted by the members of a group without being questioned or doubted. It may be in the form of an official system of principles or doctrines of a religion, such as Roman Catholicism, Judaism, Isla ...
*
First principle In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
, axiom in science and philosophy * List of axioms *
Model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
* Regulæ Juris *
Theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...
*
Presupposition In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions includ ...
*
Physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow ...
*
Principle A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, such as the la ...