An axiom, postulate, or assumption is a
statement that is taken to be
true, 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 agre ...
or starting point for further reasoning and arguments. The word comes from the
Ancient Greek 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, an axiom is a premise or starting point for reasoning.
As used in
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 c ...
).
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
Euclidean geometry). 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 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 philosophers 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 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 remarks that "
Geminus 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 tra ...
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,
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 in ...
) 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 (
theorems, 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 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 ge ...
.
The ancient Greeks considered
geometry as just one of several
sciences, 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 distinguishe ...
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, 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, postul ...
, 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 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 con ...
with any center and any radius.
:# It is true that all
right angles are equal to one another.
:# ("
Parallel postulate") 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 a ...
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 ...
, 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 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 primitive noti ...
,
Mario Pieri, 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 stan ...
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 as ...
,
topology,
vector spaces) without ''any'' particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g.,
hyperbolic geometry). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.
When mathematicians employ the
field
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 grass ...
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.
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 p ...
,
Russell
Russell may refer to:
People
* Russell (given name)
* Russell (surname)
* Lady Russell (disambiguation)
* Lord Russell (disambiguation)
Places Australia
* Russell, Australian Capital Territory
* Russell Island, Queensland (disambiguation)
...
,
Poincaré,
Hilbert, 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 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 i ...
; 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, 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. Here, the emergence of
Russell's paradox 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 axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It de ...
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
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 nearl ...
, 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 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 numbers, an
infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern
Zermelo–Fraenkel axioms for set theory. Furthermore, using techniques of
forcing (
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 the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
in general relativity,
Mendel's laws
Mendelian inheritance (also known as Mendelism) is a type of biological inheritance following the principles originally proposed by Gregor Mendel in 1865 and 1866, re-discovered in 1900 by Hugo de Vries and Carl Correns, and later populariz ...
of genetics, Darwin's
Natural selection 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 sol ...
) 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 theory ...
first introduced
special relativity where the invariant quantity is no more the Euclidian length
(defined as
) > but the Minkowski spacetime interval
(defined as
), and then
general relativity 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 r ...
geometry on curved
manifolds.
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,
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 Über quantentheoretische Umdeutung kinematis ...
,
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 ...
) 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,
David Bohm. 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
EPR may refer to:
Science and technology
* EPR (nuclear reactor), European Pressurised-Water Reactor
* EPR paradox (Einstein–Podolsky–Rosen paradox), in physics
* Earth potential rise, in electrical engineering
* East Pacific Rise, a mid-oce ...
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 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 forma ...
, 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 sym ...
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 more logical axioms than that are required, in order to prove
logical truths that are not tautologies in the strict sense.
Examples
=Propositional logic
=
In
propositional logic it is common to take as logical axioms all formulae of the following forms, where
,
, and
can be any formulae of the language and where the included
primitive connectives are only "
" for
negation of the immediately following proposition and "
" for
implication from antecedent to consequent propositions:
#
#
#
Each of these patterns is an ''
axiom schema'', a rule for generating an infinite number of axioms. For example, if
,
, and
are
propositional variables, then
and
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 ...
'', 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, but additional logical axioms are needed to include a quantifier in the calculus.
=First-order logic
=
Axiom of Equality. Let
be a
first-order language. For each variable
, the formula
is universally valid.
This means that, for any
variable symbol the formula
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
(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
in a first-order language
, a variable
and a
term that is
substitutable for
in
, the formula
is universally valid.
Where the symbol
stands for the formula
with the term
substituted for
. (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), a 2009 song by Silversun Pic ...
.) In informal terms, this example allows us to state that, if we know that a certain property
holds for every
and that
stands for a particular object in our structure, then we should be able to claim
. Again, ''we are claiming that the formula''
''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
in a first-order language
, a variable
and a term
that is substitutable for
in
, the formula
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 numbers 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
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reason ...
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, 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 in ...
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 c ...
,
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
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
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
or set theory with a
strongly inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
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
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 classify ...
,
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 topolog ...
,
homotopy theory. 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 ...
,
rings,
fields, and
Galois theory.
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 simila ...
,
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, 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 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, "Ma ...
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
where
is a constant symbol and
is a
unary function and the following axioms:
#
#
#
for any
formula
with one free variable.
The standard structure is
where
is the set of natural numbers,
is the
successor function and
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. 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 a ...
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- colline ...
add up to exactly 180 degrees or less, respectively, and are known as Euclidean and
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
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. 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
of logical axioms, a set
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
,
that is, for any statement that is a ''logical consequence'' of
there actually exists a ''deduction'' of the statement from
. 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
of the Theory of Arithmetic is ''complete'', in the sense that there will always exist an arithmetic statement
such that neither
nor
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
On ...
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 i ...
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
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.
See also
*
Axiomatic system
*
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, Islam ...
*
First principle, axiom in science and philosophy
*
List of axioms
This is a list of axioms as that term is understood in mathematics. In epistemology, the word ''axiom'' is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.
ZF (the Zermelo� ...
*
Model theory
*
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 t ...
*
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 include ...
*
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 narro ...
*
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 l ...
Notes
References
Further reading
* Mendelson, Elliot (1987). ''Introduction to mathematical logic.'' Belmont, California: Wadsworth & Brooks.
*
External links
*
*
''Metamath'' axioms page
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