Axiom |
In epistemology, an axiom is a self-evidence truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist.
In .
=Etymology=
The word axiom comes from the Greek language word αξιωμα ( axioma ), which means that which is deemed worthy or fit or that which is considered self-evidence. The word comes from αξιοειν ( axioein ), meaning to deem worthy, which in turn comes from αξιος ( axios ), meaning worthy. Among the ancient Greece philosophers an axiom was a claim which could be seen to be true without any need for proof.
=Mathematics=
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms.
==Logical axioms==
These are certain in the language.
===Examples===
In the propositional calculus it is common to take as logical axioms all formulas of the following forms, where phi, psi, and chi can be any formulas of the language:
#phi o (psi o phi) #(phi o (psi o chi)) o #(lnot phi o lnot psi) o (psi o 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 o (B o A) and (A o lnot B) o (C o (A o 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 .
These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed.
Example. Let mathfrak{L}, be a first-order language. For each variable x,, the formula
x = x
is universally valid.
This means that for any does indeed do that.
Another, more interesting example, is that which provides us with what is known as universal instantiation:
Example. Given a formula phi, in a first-order language mathfrak{L},, a variable x, and a for x, in phi,, the formula
forall x. phi o phi^x_t
is universally valid.
Informally speaking, this example allows us to state that if we know that a certain property P, holds for every x, and that if 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 o 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. Given a formula phi, in a first-order language mathfrak{L},, a variable x, and a term t, that is substitutable for x, in phi,, the formula
phi^x_t o 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 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 group (algebra)). Thus non-logical axioms, unlike logical axioms, are not tautologies . Another name for a non-logical axiom is postulate .
Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas. This turned out to be impossible and proved to be quite a story ( #role ).
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 group (algebra), 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.
===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 (sometimes referred to as the theory of functions of one real variable ), linear algebra, and complex analysis (a.k.a. complex variables ), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms.
Geometries such as Euclidean geometry, projective geometry, symplectic geometry. Interestingly one of the results of the fifth Euclidean axiom being a non-logical axiom is that the three angles of a triangle do not by definition add to 180°. Only under the umbrella of Euclidean geometry is this always true.
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 arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous Gödel s second incompleteness theorem.
We have a language mathfrak{L}_{NT} = {0, S}, 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 o x = y) # ) o forall x.phi(x) for any mathfrak{L}_{NT}, formula phi, with one free variable.
The standard structure is mathfrak{N} = langleN, 0, S angle, 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. This set of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (David Hilbert used 23).
The axioms are referred to as 4 + 1 because for nearly two millennia the 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 geometry, Euclidean geometry, and hyperbolic geometry 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 complete ordered field . 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.
==Role in mathematical logic==
===Deductive systems and completeness===
A deductive system consists of a set Lambda, of logical axioms, a set Sigma, of non-logical axioms, and a set {(Gamma, phi)}, 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 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 lnotphi, can be proved from the given set of axioms.
There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms . The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.
==Further discussion==
Early mathematicians regarded 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. Ã?variste Galois showed just before his untimely death that these efforts were largely wasted, but that the grand parallels between axiomatic systems could be put to good use, as he algebraically solved many classical geometrical problems. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and abstract 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.
=See also=
=External links=
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