Natural numbers are a set of positive integers that are commonly used for counting and ordering. The set of natural numbers typically includes: - The positive integers: 1, 2, 3, 4, 5, ... Some definitions include zero in the set of natural numbers, making it: - 0, 1, 2, 3, 4, 5, ...
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The natural numbers are the nonnegative integersThey count the elements of finite sets: the empty set has cardinality zero, and a two-element set has cardinality two. Addition and multiplication preserve natural numbers, while subtraction need not: is an integer outside .
The convention here includes zero. The positive integers instead begin at one. The well-ordering principle for the natural numbers ensures every nonempty subset has a least element, supporting mathematical induction and proof by minimal counterexample.
