User Contributed Dictionary
Noun
postulates Plural of postulate
Verb
postulates thirdperson singular of postulate
Extensive Definition
In traditional logic, an axiom or postulate is a
proposition that is not proved or demonstrated but considered to be
selfevident.
Therefore, its truth is taken for granted, and serves as a starting
point for deducing and inferring other (theory dependent) truths.
In mathematics, the
term axiom is used in two related but distinguishable senses:
"logical
axioms" and "nonlogical
axioms". In both senses, an axiom is any mathematical statement
that serves as a starting point from which other statements are
logically derived. Unlike theorems, axioms (unless
redundant) cannot be derived by principles of deduction, nor are
they demonstrable by mathematical
proofs, simply because they are starting points; there is
nothing else they logically follow from (otherwise they would be
classified as theorems).
Logical axioms are usually statements that are
taken to be universally true (e.g., ), while nonlogical axioms
(e.g, ) are actually defining properties for the domain of a
specific mathematical theory (such as arithmetic). When used in
that sense, "axiom," "postulate", and "assumption" may be used
interchangeably. In general, a nonlogical axiom is not a
selfevident truth, but rather a formal logical expression used in
deduction to build a mathematical theory. To axiomatize a system of
knowledge is to show that its claims can be derived from a small,
wellunderstood set of sentences (the axioms). There are typically
multiple ways to axiomatize a given mathematical domain.
Outside logic and mathematics, the term "axiom"
is used loosely for any established principle of some field.
Etymology
The word "axiom" comes from the Greek word (axioma), a verbal noun from the verb (axioein), meaning "to deem worthy", but also "to require", which in turn comes from (axios), 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 true without any need for proof.Historical development
Early Greeks
The logicodeductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, if we are talking about 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.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 logicodeductive method as a means of
avoiding error, and for structuring and communicating knowledge.
Aristotle's posterior
analytics is a definitive exposition of the classical
view.
An “axiom”, in classical terminology, referred to
a selfevident 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 which 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
realworld experience. Indeed, 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, where a list of postulates is given
(commonsensical geometric facts drawn from our experience),
followed by a list of "common notions" (very basic, selfevident
assertions).
 ;Postulates

 It is possible to draw a straight line from any point to any other point.
 It is possible to produce a finite straight line continuously in a straight line.
 It is possible to describe a circle 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 angles less than the two right angles.
 ;Common notions:

 Things which are equal to the same thing are also equal to one another.
 If equals be added to equals, the wholes are equal.
 If equals be 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, theorems) and definitions. This abstraction, one might even say formalization, makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts.Structuralist mathematics goes farther, and
develops theories and axioms (e.g. field
theory, group
theory, 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 we get theories
that have meaning in wider contexts, hyperbolic
geometry for example. We must simply be prepared to use labels
like “line” and “parallel” with greater flexibility. The
development of hyperbolic geometry taught mathematicians that
postulates should be regarded as purely formal statements, and not
as facts based on experience.
When mathematicians employ the axioms of a
field,
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 logic itself can be regarded as a
branch of mathematics. Frege,
Russell,
Poincaré,
Hilbert,
and Gödel are
some of the key figures in this development.
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 welldefined rules.
In this view, logic becomes just another formal system. A set of
axioms should be consistent; it should be
impossible to derive a contradiction from the axiom. A set of
axioms should also be nonredundant; 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 naive 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, 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 ZermeloFrankel
axioms for set theory. The axiom of
choice, a key hypothesis of this theory, remains a very
controversial assumption. Furthermore, using techniques of forcing
(Cohen)
one can show that the continuum
hypothesis (Cantor) is independent of the ZermeloFrankel
axioms. Thus, even this very general set of axioms cannot be
regarded as the definitive foundation for mathematics.
Mathematical logic
In the field of mathematical
logic, a clear distinction is made between two notions of
axioms: logical axioms and nonlogical axioms (somewhat similar to
the ancient distinction between "axioms" and "postulates"
respectively)
Logical axioms
These are certain
formulas in a formal
language that are universally valid, that is, formulas that
are satisfied by
under every interpretation
and with any
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 \phi, \chi, and \psi can be any formulae
of the language and where the included primitive
connectives are only "\neg" for negation of the immediately
following proposition and "\to" for implication from antecedent
to consequent propositions:
 \phi \to (\psi \to \phi)
 (\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))
 (\lnot \phi \to \lnot \psi) \to (\psi \to \phi).
Each of these patterns is an axiom
schema, a rule for generating an infinite number of axioms. For
example, if A, B, and C are propositional
variables, then A \to (B \to A) and (A \to \lnot B) \to (C \to
(A \to \lnot B)) 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, 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 schemas 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.
Mathematical logic
Axiom of Equality. Let \mathfrak\, be a firstorder
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 neverending 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 firstorder 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
variable substitution.) 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(t)\,. 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. Actually, 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 firstorder 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.
Nonlogical axioms
Nonlogical axioms are formulas that play the
role of theoryspecific assumptions. Reasoning about two different
structures, for example the natural
numbers and the integers, may involve the same
logical axioms; the nonlogical axioms aim to capture what is
special about a particular structure (or set of structures, such as
groups).
Thus nonlogical axioms, unlike logical axioms, are not
tautologies. Another name for a nonlogical axiom is
postulate.
Almost every modern mathematical
theory starts from a given set of nonlogical 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. This turned out to be impossible and proved to be quite a
story (see
below); however recently this approach has been resurrected in
the form of neologicism.
Nonlogical axioms are often simply referred to
as axioms in mathematical discourse. This does not mean that it is
claimed that they are true in some absolute sense. For example, in
some groups,
the group operation is commutative, and this can be
asserted with the introduction of an additional axiom, but without
this axiom we can do quite well developing (the more general) group
theory, and we can even take its negation as an axiom for the study
of noncommutative groups.
Thus, an axiom is an elementary basis for a
formal logic system that together with the rules of
inference define a deductive
system.
Examples
This section gives examples of mathematical
theories that are developed entirely from a set of nonlogical
axioms (axioms, henceforth). A rigorous treatment of any of these
topics begins with a specification of these axioms.
Basic theories, such as arithmetic, real
analysis and complex
analysis are often introduced nonaxiomatically, 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, most often
Von Neumann–Bernays–Gödel set theory, abbreviated NBG. This is
a conservative
extension of ZFC, with identical theorems about sets, and hence
very closely related. Sometimes slightly stronger theories such as
MorseKelley
set theory or set theory with a
strongly inaccessible cardinal allowing the use of a Grothendieck
universe are used, but in fact most mathematicians can actually
prove all they need in systems weaker than ZFC, such as secondorder
arithmetic.
The study of topology in mathematics extends all
over through point
set topology, algebraic
topology, differential
topology, and all the related paraphernalia, such as homology
theory, homotopy
theory. The development of abstract algebra brought with itself
group
theory, rings
and fields,
Galois
theory.
This list could be expanded to include most
fields of mathematics, including axiomatic
set theory, measure
theory, ergodic
theory, probability, representation
theory, and differential
geometry.
Arithmetic
The Peano axioms are the most widely used axiomatization of firstorder arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.We have a language \mathfrak_ = \\, where 0\, is
a constant symbol and S\, is a unary
function and the following axioms:
 \forall x. \lnot (Sx = 0)
 \forall x. \forall y. (Sx = Sy \to x = y)
 ((\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x) for any \mathfrak_\, formula \phi\, with one free variable.
The standard structure is \mathfrak = \langle\N,
0, S\rangle\, where \N\, is the set of natural numbers, S\, is the
successor
function and 0\, is naturally interpreted as the number
0.
Euclidean geometry
Probably the oldest, and most famous, list of
axioms are the 4 + 1 Euclid's
postulates of 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. Indeed, one can assume that no
parallels through a point outside a line exist, that exactly one
exists, or that infinitely many exist. These choices give us
alternative forms of geometry in which the interior angles of a triangle add up to less than,
exactly, or more than a straight line respectively and are known as
elliptic,
Euclidean,
and hyperbolic
geometries.
Real analysis
The object of study is the real
numbers. The real numbers are uniquely picked out (up to
isomorphism) by the
properties of a Dedekind complete ordered field, meaning that any
nonempty set of real numbers with an upper bound has a least upper
bound. However, expressing these properties as axioms requires use
of secondorder
logic. The
LöwenheimSkolem theorems tell us that if we restrict ourselves
to firstorder
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 nonstandard
analysis.
Role in mathematical logic
Deductive systems and completeness
A deductive system consists, of a set \Lambda\,
of logical axioms, a set \Sigma\, of nonlogical 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,
if \Sigma \models \phi then \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 commonlyused 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 nonlogical 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 nonlogical axioms. The completeness
theorem and the incompleteness theorem, despite their names, do not
contradict one another.
Further discussion
Early mathematicians 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 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 we may take as axioms any
set of formulas we like, as long as they are not known to be
inconsistent.
References
Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks. ISBN 0534066240Notes
See also
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