Axiom
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This article is about a logical statement. For the vehicle, see Isuzu
Axiom. For other uses, see Axiom (disambiguation)
"Given" redirects here. For the usage in probability, see Conditional
probability.
In traditional logic, an axiom or postulate is a proposition that is
not proved or demonstrated but considered to be either self-evident, or
subject to necessary decision. 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 "non-logical 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 from which they
logically follow (otherwise they would be classified as theorems).
Logical axioms are usually statements that are taken to be universally
true (e.g., A and B implies A), while non-logical axioms (e.g., a + b =
b + a) 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 non-logical axiom is not a self-evident 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, well-understood 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.
Contents [hide]
1 Etymology
2 Historical development
2.1 Early Greeks
2.2 Modern development
3 Mathematical logic
3.1 Logical axioms
3.1.1 Examples
3.1.1.1 Propositional logic
3.1.1.2 Mathematical logic
3.2 Non-logical axioms
3.2.1 Examples
3.2.1.1 Arithmetic
3.2.1.2 Euclidean geometry
3.2.1.3 Real analysis
3.3 Role in mathematical logic
3.3.1 Deductive systems and completeness
3.4 Further discussion
4 References
5 Notes
6 See also
7 External links
[edit] Etymology
The word "axiom" comes from the Greek word ἀξίωμα (axioma), a verbal
noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also
"to require", which in turn comes from ἄξιος (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.
[edit] Historical development
[edit] 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), was developed by the ancient Greeks,
and has become the core principle of modern mathematics[citation
needed]. 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[citation needed], 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 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 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 real-world 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 (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 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.
[edit] 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[citation
needed]. 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 well-defined rules. In this view,
logic becomes just another formal system. A set of axioms should be
consistent; it should be impossible to derive a contradiction from the
axiom. A set of axioms should also be non-redundant; an assertion that
can be deduced from other axioms need not be regarded as an axiom.
It was the early hope of modern logicians that various branches of
mathematics, perhaps all of mathematics, could be derived from a
consistent collection of basic axioms. An early success of the formalist
program was Hilbert's formalization of Euclidean geometry, 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
Zermelo-Frankel 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 Zermelo-Frankel
axioms. Thus, even this very general set of axioms cannot be regarded as
the definitive foundation for mathematics.
[edit] Mathematical logic
In the field of mathematical logic, a clear distinction is made between
two notions of axioms: logical axioms and non-logical axioms (somewhat
similar to the ancient distinction between "axioms" and "postulates"
respectively)
[edit] Logical axioms
These are certain formulas in a formal language 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.
[edit] Examples
[edit] Propositional logic
In propositional logic it is common to take as logical axioms all
formulae of the following forms, where φ, χ, and ψ 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 A, B, and C 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, 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.[1]
These axiom schemata are also used in the predicate calculus, but
additional logical axioms are needed to include a quantifier in the
calculus.[2]
[edit] Mathematical 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 variable substitution.) 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. 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 in a
first-order language , a variable and a term that is substitutable for
in , the formula
is universally valid.
[edit] 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 integers, 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). Thus non-logical
axioms, unlike logical axioms, are not tautologies. Another name for a
non-logical axiom is postulate.[3]
Almost every modern mathematical theory starts from a given set of
non-logical axioms, and it was thought that in principle every theory
could be axiomatized in this way and formalized down to the bare
language of logical formulas. 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 neo-logicism.
Non-logical axioms are often simply referred to as axioms in
mathematical discourse. This does not mean that it is claimed that they
are true in some absolute sense. For example, in some groups, the group
operation is commutative, and this can be asserted with the introduction
of an additional axiom, but without this axiom we can do quite well
developing (the more general) group theory, and we can even take its
negation as an axiom for the study of non-commutative groups.
Thus, an axiom is an elementary basis for a formal logic system that
together with the rules of inference define a deductive system.
[edit] 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, 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, 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
Morse-Kelley 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 second-order arithmetic.
The study of topology in mathematics extends all over through point set
topology, algebraic topology, differential topology, and all the related
paraphernalia, such as homology theory, homotopy theory. The development
of abstract algebra brought with itself group theory, 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.
[edit] Arithmetic
The Peano axioms are the most widely used axiomatization of first-order
arithmetic. They are a set of axioms strong enough to prove many
important facts about number theory and they allowed Gödel to establish
his famous second incompleteness theorem.[4]
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.
[edit] 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.
[edit] 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 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.
[edit] Role in mathematical logic
[edit] 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 φ,
if then
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.
[edit] 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.
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