An axiom is a statement accepted without proof as a starting point within a mathematical or logical theory. Together with rules of inference, axioms provide the basis for deriving other statements through deductive reasoning. In modern mathematics, an axiom need not be self-evident or universally true: its role is to specify assumptions under which particular conclusions follow. A statement’s status as an axiom therefore depends on the theory and presentation in which it occurs. (ocw.mit.edu)
Historical development
The axiomatic approach has a foundational example in Euclid’s Elements. Book I distinguishes definitions, postulates, and common notions. Postulates concern geometrical constructions and relations; common notions express more general principles about equality and magnitude. For example, one common notion states that quantities equal to the same quantity are equal to one another. Subsequent propositions are established from these starting principles and earlier results. (maths.tcd.ie)
The parallel postulate became especially important because attempts to derive it from other geometrical assumptions eventually gave way to the development of non-Euclidean geometry. Different assumptions about parallels can support different geometrical theories rather than simply constitute errors within one supposedly mandatory framework. (maths.tcd.ie)
David Hilbert developed an influential modern treatment of axiomatic geometry. His approach specifies relations among objects called points, lines, and planes, making explicit which conclusions depend on which groups of assumptions. This helped separate the logical structure of geometry from any particular intuitive interpretation of its objects. (maths.tcd.ie)
Axioms, definitions, and theorems
An axiom differs from a theorem in its function within a presentation: an axiom is admitted as a starting assumption, whereas a theorem is established by mathematical proof. This distinction is relative rather than intrinsic. A proposition taken as an axiom in one presentation may be derived from different starting assumptions in another. Axioms themselves are available as premises in proofs; calling them “unproved” does not mean they are excluded from the theory’s provable statements. (ocw.mit.edu)
A definition normally introduces terminology or specifies the meaning of an expression. Axioms instead impose conditions on the objects under discussion. In the structural approach, they can also serve as implicit definitions: rather than explaining individually what “point” or “line” means, the theory specifies the relationships those objects must satisfy. Different collections of objects may fulfill the same conditions. (plato.stanford.edu)
“Postulate” is often used interchangeably with “axiom,” although historical treatments sometimes distinguish them, as Euclid did. Modern usage does not require axioms to possess a special degree of intuitive obviousness. (maths.tcd.ie)
Formal axiomatic systems
A formal system specifies a symbolic language, axioms, and rules governing permissible derivations. In first-order logic, axioms are typically sentences using logical symbols together with symbols for the theory’s relations, operations, or distinguished objects. A formal proof records a finite derivation according to the specified rules. (math.ucla.edu)
An axiomatization may contain finitely or infinitely many axioms. An axiom schema is a pattern specifying a family of axioms rather than just one sentence. For example, first-order Peano arithmetic uses an induction schema with an instance for each appropriate formula. An effectively axiomatized theory has axioms that can be generated by an algorithm, a condition central to results about the limits of formal proof. (math.ucla.edu)
Examples in mathematics
Axioms can describe an abstract structure or provide a foundation for a broader subject. Peano arithmetic describes natural numbers through principles concerning zero, succession, addition, multiplication, and induction. Its first-order formulation distinguishes the intended natural-number interpretation from other structures satisfying the same axioms. (math.ucla.edu)
In set theory, axiomatization specifies principles governing sets and their membership relations. Zermelo–Fraenkel set theory, abbreviated ZF, is a standard foundation; adding the axiom of choice produces ZFC. Choice asserts that a family of nonempty sets admits a function selecting one element from each set. It is independent of ZF, assuming ZF is consistent. (plato.stanford.edu)
In probability theory, the standard axioms require nonnegative probabilities, probability one for the entire sample space, and countable additivity for pairwise disjoint events. These constrain a probability assignment without specifying the probability of every individual event. A particular probabilistic model supplies those additional details. (probabilitycourse.com)
Consistency, independence, and completeness
Three properties distinguish important questions about an axiomatization:
- Consistency: no sentence and its negation can both be derived.
- Independence: an individual axiom cannot be derived from the remaining axioms.
- Completeness: for every sentence in the theory’s language, either that sentence or its negation is derivable. (ocw.mit.edu)
A model is a mathematical structure satisfying the axioms, a notion studied in model theory. To demonstrate an axiom’s independence, one can construct a model satisfying the other axioms but not that axiom. With sound inference rules, such a model shows that the disputed axiom cannot follow from the others. (maths.tcd.ie)
Limits of axiomatization
Gödel’s incompleteness theorems establish limitations for consistent, effectively axiomatized theories capable of expressing sufficient arithmetic. The first theorem shows that such a theory contains sentences it can neither prove nor refute. The second, under the relevant technical conditions, shows that the theory cannot prove its own consistency as expressed by its standard internal consistency statement. These results do not apply indiscriminately to every axiomatic system. (plato.stanford.edu)
This incompleteness differs from the completeness of first-order logic. Logical completeness means that every sentence true in all models of given axioms has a formal derivation from those axioms. An incomplete theory may nevertheless have models disagreeing about a particular sentence; the logical proof system can be complete while the theory does not decide that sentence. (math.ucla.edu)