Axiom

There is algebra software named Axiom.

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 do not stand for self-evident truths, but they are dealt with in two different scenarios: as logical axioms and as non-logical axioms. Axiomatic reasoning is today most widely used in mathematics.

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 philosophers of the ancient Greeks 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

Shortly speaking, logical axioms are those formulas that are universally valid, i.e., formulas that are satisfied by every structure (a.k.a. model) 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. An example of such a formula is the Axiom of Equality, used in virtually every deductive system:

\forall x(x=x)

Of course, in this particular example, for this not to fall into subsequent vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by "=" (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 - and Mathematical Logic does indeed that, properly delegating the "meaning" of "=" to Axiomatic Set Theory.

Logical axioms, as the mere formulas that they are, are void of any meaning; but when they become interpreted in any given universe, they will always hold no matter what values are assigned to the variables. Thus, despite the notice in the paragraph above, this notion of axiom is perhaps the closest to the intended meaning of the word.

Non-logical axioms

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.

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 proved to be quite a story.

This is the role of non-logical axioms, and in the practical discourse they are simply regarded as axioms, but not in the sense that they are true or as if they were assumptions claimed to be true. They simply serve to set up the table.

Deductive systems

The formal issue arises in the need to derive what logicians call a deductive system, which consists of a set Λ of logical axioms, a set Σ of non-logical axioms and a set {(Γ, φ)} of rules of inference. Gödel's completeness theorem establishes that every deductive system with a consistent set of non-logical axioms is complete,

if

\Sigma \models \phi

then

\Sigma \vdash \phi

i.e., for any statement that is a logical consequence of Σ there actually exists a deduction of the statement from Σ. Again, more simply, anything that is true from a given set of assumptions can be proved from those assumptions (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 of the Theory of Arithmetic is complete, in the sense that there will always exist true arithmetic statements that cannot be proved from the given set of non-logical 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.

Modern usage in mathematics (non-logical axioms)

As the word axiom is understood in modern mathematics, an axiom is not a proposition that is self-evident. Rather, it simply means a starting point in a logical system. For example, in some rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate. An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.

The moral is, any fact that we can derive from the axioms 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.

Mathematical examples

Arithmetic

In all of its 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 Incompletenss Theorem

$\exists 0(0\in \mathbb {N} )$
$\exists s:\mathbb {N} \longrightarrow \mathbb {N}$
$\lnot \exists x(s(x)=0)$
$\forall x\forall y(s(x)=s(y)\to x=y)$
$\forall S(S\subseteq \mathbb {N} \to ((0\in S\land \forall x(x\in S\to s(x)\in S))\to S=\mathbb {N} ))$

Geometry

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 since the fifth postulate (through a point outside a line there is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia. 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.

Consequences of separating axioms from "reality"

Early mathematicians regarded axiomatic geometry as a model of reality, 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.

In the twentieth century, Gödel's incompleteness theorem showed that no explicit (i.e. recursive) set of axioms sufficiently large to define traditional arithmetic could be both (1) complete (i.e. every statement can be either proved or disproved) and (2) consistent (i.e. no statement can be both proved and disproved).

External links

Metamath axioms page