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The set of nonnegative integers used for counting and indexing.
The natural numbers are the set . (Some authors use instead; when needed, the chosen convention should be stated.)
Natural numbers are used to index a sequence and to formulate principles such as mathematical induction and the well-ordering principle. They embed naturally into the integers.
A set is an object for which statements of the form (read “ is an element of ”) are meaningful, and whose identity is determined by extensionality: for sets ,
A sequence in a set is a function , where is the natural numbers. One writes for , so the sequence is denoted .
Mathematical induction: Let be a statement about . If is true and for every the implication holds, then is true for all .
Well-ordering principle: Every nonempty subset has a least element (with respect to the usual order on ); that is, there exists such that for all .