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An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
This is a list of axioms as that term is understood in mathematics. In epistemology , the word axiom is understood differently; see axiom and self-evidence . Individual axioms are almost always part of a larger axiomatic system .
The last Peano's axiom is the only one that induces logical difficulties, as it begin with either "if S is a set then" or "if is a predicate then". So, Peano's axioms induce a quantification on infinite sets, and this means that Peano arithmetic is what is presently called a Second-order logic.
The axioms of modules imply that (−1)x = −x, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
An axiomatic system is a set of axioms or assumptions from which other statements (theorems) are logically derived. [97] In propositional logic, axiomatic systems define a base set of propositions considered to be self-evidently true, and theorems are proved by applying deduction rules to these axioms. [98] See § Syntactic proof via axioms.
A deductive system, also called a deductive apparatus, [8] consists of the axioms (or axiom schemata) and rules of inference that can be used to derive theorems of the system. [1] Such deductive systems preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to ...
AC – Axiom of Choice, [1] or set of absolutely continuous functions. a.c. – absolutely continuous. acrd – inverse chord function. ad – adjoint representation (or adjoint action) of a Lie group. adj – adjugate of a matrix. a.e. – almost everywhere. AFSOC - Assume for the sake of contradiction; Ai – Airy function. AL – Action limit.
Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example axiomatic set theory and non-Euclidean geometry.