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The most common form of proof by mathematical induction requires proving in the induction step that (() (+)) whereupon the induction principle "automates" n applications of this step in getting from P(0) to P(n). This could be called "predecessor induction" because each step proves something about a number from something about that number's ...
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.
Principle of mathematical induction. Add languages. Add links. Article; ... Print/export Download as PDF; Printable version;
In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. [1] For example, of three gloves, at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put ...
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.
Most recursive definitions have two foundations: a base case (basis) and an inductive clause. The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some ...
Then, by the well-ordering principle, there is a least element ; cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n ...
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.