Elements of Algebra pp 18-37 | Cite as

# The Rational Numbers

Chapter

## Abstract

The natural numbers 0, 1, 2, ... are the numbers used for counting. They are generated from 0 by the

*successor*operation +1 (add one). In other words, the set ℕ = {0, 1, 2, ...} of natural numbers is the closure of the set {0} under successor, that is, the intersection of all sets*S*such that 0 ∈*S*and*n*+ 1 ∈*S*when*n*∈*S*. This definition yields several logically equivalent properties of ℕ known as*induction*. The most commonly used versions of induction are:- I.
If 0 ∈

*S*, and if*n*+1 ∈*S*when*n*∈*S*, then ℕ ⊆*S*. - II.
If 0 ∈

*S*, and if*n*+ l ∈*S*when 0, 1, ...,*n*∈*S*, then ℕ ⊆*S*. - III.
If

*T*⊆ ℕ is nonempty then*T*has a least member (that is, an*n*∈*T*such that the closure of {*n*} under successor includes all of*T*).

## Keywords

Natural Number Prime Divisor Rational Solution Algebraic Integer Congruence Class
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1994