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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.
Almgren–Pitts min-max theory; Approximation theory; Arakelov theory; Asymptotic theory; Automata theory; Bass–Serre theory; Bifurcation theory; Braid theory
[1] [2] The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. [3] In modern logic, an axiom is a premise or starting point for reasoning. [4] In mathematics, an axiom may be a "logical axiom" or a "non ...
Dividing both sides by 2 yields b 2 = 2c 2. But then, by the same argument as before, 2 divides b 2 , so b must be even. However, if a and b are both even, they have 2 as a common factor.
In simple type theory objects are elements of various disjoint "types". Types are implicitly built up as follows. If τ 1,...,τ m are types then there is a type (τ 1,...,τ m) that can be thought of as the class of propositional functions of τ 1,...,τ m (which in set theory is essentially the set of subsets of τ 1 ×...×τ m). In ...
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. [5] The field was founded by Harvey Friedman . Its defining method can be described as "going backwards from the theorems to the axioms ", in contrast to the ordinary mathematical practice of deriving ...
Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences [1] (such as economics, psychology, sociology, political science). It ...