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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 induction principle then "automates" log 2 n applications of this inference in getting from P(0) to P(n). In fact, it is called "prefix induction" because each step proves something about a number from something about the "prefix" of that number — as formed by truncating the low bit of its binary representation. It can also be viewed as ...
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 ...
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.
Because 0 is the least element of N, it must be that 0 ∉ X. For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X. Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N. Thus X has a least element.
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Now in , the least number principle is just one of many statements equivalent to the induction principle. The proof below shows how L N P {\displaystyle {\mathrm {LNP} }} implies P E M {\displaystyle {\mathrm {PEM} }} , and therefore why this principle also cannot be generally valid in H A {\displaystyle {\mathsf {HA}}} .
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