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The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on the statement P ( n ) defined as " Q ( m ) is false for all natural numbers m less than or equal to n ", it follows that P ( n ) holds for all n , which means that Q ( n ) is false for every natural number n .
In first-order arithmetic, the induction principles, bounding principles, and least number principles are three related families of first-order principles, which may or may not hold in nonstandard models of arithmetic. These principles are often used in reverse mathematics to calibrate the axiomatic strength of theorems.
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 ...
By the principle of induction, it suffices to exhibit an F-closed set for which the property holds. Dually, suppose you wish to show that x ∈ ν F {\displaystyle x\in \nu F} . Then it suffices to exhibit an F-consistent set that x {\displaystyle x} is known to be a member of.
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.
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction.
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion.
The proof of this is derived from a game between the induction and the environment. Essentially, any computable induction can be tricked by a computable environment, by choosing the computable environment that negates the computable induction's prediction. This fact can be regarded as an instance of the no free lunch theorem.