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Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.
The induction, bounding and least number principles are commonly used in reverse mathematics and second-order arithmetic. For example, I Σ 1 {\displaystyle {\mathsf {I}}\Sigma _{1}} is part of the definition of the subsystem R C A 0 {\displaystyle {\mathsf {RCA}}_{0}} of second-order arithmetic.
In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. [15]
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proof by induction A method of mathematical proof used to establish the truth of an infinite number of cases, based on a base case and an inductive step. proof theory The branch of mathematical logic that studies the structure and properties of mathematical proofs, aiming to understand and formalize the process of mathematical reasoning.
Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
Induction (play), an opening scene; Induction (teachers), support of novice teachers; Inductive reasoning aptitude; Collective Induction, in psychology; Hypnotic induction, causing hypnosis "Induction", a song by Broken Spindles from Fulfilled/complete "Induction" (short story), a short story by Greg Egan
Coinduction is the mathematical dual to structural induction. [ citation needed ] Coinductively defined data types are known as codata and are typically infinite data structures , such as streams .