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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.
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The argument above makes the implicit assumption that the set of + horses has the size at least 3, [3] so that the two proper subsets of horses to which the induction assumption is applied would necessarily share a common element. This is not true at the first step of induction, i.e., when + =.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms ...
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.
The ninth, final, axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with ...
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further generalized to arbitrary Noetherian induction .
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