Axioms

An axiom is any sentence, proposition, statement or rule that forms the basis of a formal system. Unlike theorems, axioms are neither derived by principles of deduction, nor are they demonstrable by formal proofs. Instead, an axiom is taken for granted as valid, and serves as a necessary starting point for deducing and inferencing logically consistent propositions. In many usages, "axiom," "postulate," and "assumption" are used interchangeably.

In certain epistemological theories, an axiom is a self-evident truth upon which other knowledge must rest, and from which other knowledge is built up. An axiom in this sense can be known before one knows any of these other propositions. Not all epistemologists agree that any axioms, understood in that sense, exist.

In logic and mathematics, an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results. To axiomatize a system of knowledge is to show that all of its claims can be derived from a small, well-understood set of sentences. This does not imply that they could have been known independently; and there are typically multiple ways to axiomatize a given system of knowledge (such as arithmetic). Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.

Etymology
The word "axiom" comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

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. 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 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 axioms (very basic, self-evident assertions) and postulates (common-sensical geometric facts drawn from our experience), are given.


 * Axiom 1: Things which are equal to the same thing are also equal to one another.
 * Axiom 2: If equals be added to equals, the wholes are equal.
 * Axiom 3: If equals be subtracted from equals, the remainders are equal.
 * Axiom 4: Things which coincide with one another are equal to one another.
 * Axiom 5: The whole is greater than the part.


 * Postulate 1: It is possible to draw a straight line from any point to any other point.
 * Postulate 2: It is possible to produce a finite straight line continuously in a straight line.
 * Postulate 3: It is possible to describe a circle with any center and distance.
 * Postulate 4: It is true that all right angles are equal to one another.
 * Postulate 5: 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.

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 the 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.

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)

Logical axioms
These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function. In colloquial terms, these are statements that are true in any possible universe, under any possible 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.

Propositional logic
In propositional logic it is common to take as logical axioms all formulae of the following forms, where $$\phi$$, $$\psi$$, and $$\chi$$ can be any formulae of the language:


 * 1) $$\phi \to (\psi \to \phi)$$
 * 2) $$(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))$$
 * 3) $$(\lnot \phi \to \lnot \psi) \to (\psi \to \phi)$$

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.

These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed.

Mathematical logic
Axiom of Equality. Let $$\mathfrak{L}\,$$ be a first-order language. For each variable $$x\,$$, the formula

$$x = x\,$$

is universally valid.

This means that, for any variable symbol $$x\,$$, the formula $$x = x\,$$ can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by $$x = x\,$$ (or, for all what matters, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol $$=\,$$ has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.

Another, more interesting example axiom scheme, is that which provides us with what is known as Universal Instantiation:

Axiom scheme for Universal Instantiation. Given a formula $$\phi\,$$ in a first-order language $$\mathfrak{L}\,$$, 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 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 $$\forall x \phi \to \phi^x_t$$ 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 first-order language $$\mathfrak{L}\,$$, a variable $$x\,$$ and a term $$t\,$$ that is substitutable for $$x\,$$ in $$\phi\,$$, the formula

$$\phi^x_t \to \exists x \phi$$

is universally valid.

Non-logical axioms
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural 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.

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.

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.

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 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.

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:


 * 1) $$\forall x. \lnot (Sx = 0) $$
 * 2) $$\forall x. \forall y. (Sx = Sy \to x = y) $$
 * 3) $$((\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x)$$ for any $$\mathfrak{L}_{NT}\,$$ formula $$\phi\,$$ with one free variable.

The standard structure is $$\mathfrak{N} = \langle\N, 0, S\rangle\,$$ 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. This set of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (Hilbert used 23).

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 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.

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 $$\lnot\phi\,$$ can be proved from the given set of axioms.

There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Further discussion
Early 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.