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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that any axioms, understood in that sense, exist.
In mathematics, axioms are not self-evident truths. They are of two different kinds: logical axioms and non-logical axioms. Axiomatic reasoning is today most widely used in mathematics.
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 philosophers of the ancient Greeks an axiom was a claim which could be seen to be true without any need for proof.
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms.
These are formulas which are valid, i.e., formulas that are satisfied by every model (a.k.a. structure) under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values.
Now, in order to claim that something is a logical axiom, we must know that it is indeed valid. That is, it might be necessary to offer a proof of its validity (truth) in every model. This might challenge the very classical notion of axiom; this is at least one of the reasons why axioms are not regarded as obviously true or self-evident statements.
Logical axioms, as the mere formulas that they are, are void of any meaning; but the point is that when they become interpreted in any given universe, they will always hold no matter what values are assigned to the variables. Thus, this notion of axiom is perhaps the closest to the intended meaning of the word: that axioms are true, no matter when, where or why.
An example, used in virtually every mathematical logic does indeed that, properly delegating the meaning of "=" to axiomatic set theory.
Another, more interesting example, is that of:
Axiom of universal instantiation. Given a formula <math>\phi\,<math> in a first order language <math>\mathfrak{L}\,<math>, a variable <math>x\,<math> and a term <math>t\,<math> that is substitutable for <math>x\,<math> in <math>\phi\,<math>, the formula<math>\phi^x_t \to \exists x \phi<math>
is valid.
[OK. The later two are being presumed to actually be logical axioms, i.e. valid formulas. It would better be to say "valid formulas, as follows..." The proofs of these facte are definitely a technical issue, but interesting enough on their own.]
Non-logical axioms are formulas that play rather the role of assumptions up from which a theory is developed. These formulas need not be universally valid as above. Another name for a non-logical axiom is postulate.
It is the case that 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 fomalized down to the bare language of logical formulas. This turns out to be impossible and proved to be quite a story.
This is the role of non-logical axioms, they simply constitute a starting point in a logical system. Since they are so fundamental in the development of a theory, it is often the case that they are simply referred to as axioms in the mathematical discourse, but again, not in the sense that they are true propositions nor as if they were assumptions claimed to be true. For example, in some groups, the operation of multiplication is commutative; in others it is not.
Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.
Arithmetic, Euclidean geometry, linear algebra, real analysis, topology, group theory, set theory, projective geometry, symplectic geometry, von Neumann algebras, ergodic theory, probability, etc. All these theories are based on their respective set of non-logical axioms.
In all this formalism, the Peano axioms constitute the most widely used axiomatization of arithmetic; these are a set of non-logical axioms strong enough to prove several relevant facts of number theory and they allowed Gödel to establish his second incompleteness theorem
The language is <math>\mathfrak{L}_{NT} = \{0, S\}\,<math> where <math>0\,<math> is a constant symbol and <math>S\,<math> is a unary function. The postulates are:
There is a standard structure is <math>\mathfrak{N} = <\N, 0, S>\,<math> where <math>\N\,<math> is the set of natural numbers, <math>S\,<math> is the successor function and <math>0\,<math> is naturally interpreted as the number 0.
Probably the most famous very early set of axioms is the 4 + 1 postulates of Euclid. This turns out to be incomplete, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23).
"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. The general theory of relativity is essentially a claim that mass gives space hyperbolic geometry.
We are fortunate enough to have that the standard model of "real analysis", described by the axioms of a complete ordered commutative field, is unique up to isomorphism.
The formal issue arises in the need to derive what logicians call a deductive system, which consists of a set <math>\Lambda<math> of logical axioms, a set <math>\Sigma<math> of non-logical axioms and a set <math>\{(\Gamma, \phi)\}<math> of rules of inference. Gödel's completeness theorem establishes that every deductive system with a consistent set of non-logical axioms is complete,
if <math>\Sigma \models \phi<math> then <math>\Sigma \vdash \phi<math>
i.e., for any statement that is a logical consequence of <math>\Sigma<math> there actually exists a deduction of the statement from <math>\Sigma<math>. Again, more simply, anything that is true from a given set of axioms can be proved from those axioms (with reasonable rules of inference).
Note the subtle difference between this and the later and equally celebrated Gödel's first incompleteness theorem, which states that no set of recursive, consistent, set of non-logical axioms <math>\Sigma<math> of the Theory of Arithmetic is complete, in the sense that there will always exist a true arithmetic statement <math>\phi<math> such that neither <math>\phi<math> nor <math>\lnot\phi<math> can be proved (the later is not the same as <math>\phi<math> being disproved - it simply means what it says, that there cannot be a deduction from <math>\Sigma<math> to <math>\lnot\phi<math>) from the given set of axioms.
There is thus, in 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 moral is, any fact that we can derive from a set of axioms (logical or non-logical) is not needed as an axiom. Anything that we cannot derive from the axioms and for which we also cannot derive the negation might reasonably be added as an axiom.
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, 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 modern algebra was born.
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