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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. Not all epistemologists agree that any axioms, understood in that sense, exist.
In mathematics, an axiom is not a self-evident truth but rather, a formal logical sentence that constitutes a fundamental brick in the development of a theory. 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 ancient Greek philosophers 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 logically valid, that is, 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.
In order to offer something as a logical axiom, we must be certain that it is logically valid. That is, it is necessary to guarantee its satisfiability in every model. This might challenge the classical notion of axiom and is, at least, one of the reasons why axioms are not regarded as obviously true or self-evident statements.
Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in any 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 what.
An example, used in virtually every deductive system, is the:
Axiom of equality.
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 (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.
Another, more interesting example, is that of:
Axiom of universal instantiation. Given a formula in a first order language , a variable and a term that is substitutable for in , the formula
This axiom simply states that if we know for some property , and is particular term in the language (i.e., it stands for a particular object in our structure), then we should be able to claim .
Likewise, we have the:
Axiom of existential generalization. Given a formula in a first order language , a variable and a term that is substitutable for in , the formula
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).
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 logically valid sentences, nor as if they were assumptions claimed to be true. For example, in some groups, the law of composition is commutative, and this is achieved with the introduction of an additional axiom; but without this axiom we can do quite well developing (the more general) group theory.
Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.
What we mean by example in this section, is a mathematical theory that is developed entirely from the ground up from only a list of non-logical axioms (axioms, henceforth). Any rigorous exposition to any such subject always starts by specifying this list, such as in textbooks, research papers or lectures.
Basic theories, such as Arithmetic, real analysis (sometimes pompously referred to as the theory of functions of one real variable in basic levels), linear algebra, and complex analysis (a.k.a. complex vaiables), are often introduced vaguely in mostly technical studies by prescinding of its formal development, but any well form course in these subjects always starts by presenting its list of axioms.
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.
The magnificent list simply goes on and on (as an aside, demonstrating the power of axiomatic reasoning), and could be offered in any subject classification of mathematics, but just to mention a few more standard theories, we have axiomatic set theory, measure theory, ergodic theory, probability, representation theory, differential geometry, ..., etc.
All these theories, study objects described by a list of axioms, all of which are always offered at the beginning of any treatise.
In all its formalism, the Peano axioms constitute the most widely used axiomatization of arithmetic; these are a set of axioms strong enough to prove several relevant facts of number theory and they allowed Gödel to establish his famous second incompleteness theorem.
We have a language where is a constant symbol and is a unary function and the following axioms:
- for any formula with one free variable.
The standard structure is where is the set of natural numbers, is the successor function and is naturally interpreted as the number 0.
Probably the oldest, and most famous list of axioms are the 4 + 1 Euclid's postulates of plane geometry. This collection of axioms turns out to be incomplete, and many more postulates are necessary to rigorously 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 object of study are the real numbers, their properties and what we can do with them. By properties we really mean the axioms of the set or real numbers, and this can be given by the those of a complete ordered field, in the sense that this define them uniquely up to isomorphism. 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. (This and every paragraph above, would awesomely deserve a whole section by itself; let's do something...)
Role in mathematical logic
Deductive systems and completeness
The formal issue arises in the need to derive what logicians call a deductive system, which consists of a set of logical axioms, a set of non-logical axioms and a set of rules of inference. Gödel's completeness theorem establishes that every deductive system with a consistent set of non-logical axioms is complete,
that is, for any statement that is a logical consequence of Σ there actually exists a deduction of the statement from . 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 recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist a true arithmetic statement such that neither nor can be proved (the later is not the same as being disproved - it simply means what it says, that there cannot be a deduction from to ) 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.
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.
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.
- Axiomatic system
- Peano axioms
- Axiom of choice
- Axiom of countability
- Axiomatic set theory
- Parallel postulate
- Continuum hypothesis
- List of axioms
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