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.
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.
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.
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.
Deductive systems and completeness
A deductive system consists, of a set of logical axioms, a set of non-logical axioms, and a set rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for any statement that is a logical consequence of the set of axioms of that system, there actually exists a deduction of the statement from that set of axioms. 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 that statement nor its negation 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.
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 nineteenth 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.
All links retrieved May 4, 2016.
- Metamath axioms page