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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 .
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'. [1][2]
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. [1] These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. [2]
First-order logic —also called predicate logic, predicate calculus, quantificational logic —is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables.
Peano axioms. In mathematical logic, the Peano axioms (/ piˈɑːnoʊ /, [1] [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical ...
Intuitionistic logic is a subsystem of classical logic. It is commonly formulated with as the set of (functionally complete) basic connectives. It is not syntactically complete since it lacks excluded middle A∨¬A or Peirce's law ( (A→B)→A)→A which can be added without making the logic inconsistent. It has modus ponens as inference rule ...
List or describe a set of sentences in the language L σ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in L σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An L σ ...
weakly and strongly Mahlo, α- Mahlo, and hyper Mahlo cardinals. λ-unfoldable, unfoldable cardinals, ν-indescribable cardinals and λ-shrewd, shrewd cardinals (not clear how these relate to each other). superstrong cardinals (=1-superstrong; for n -superstrong for n ≥2 see further down.) The following even stronger large cardinal properties ...