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 Skip Navigation LinksMath Help > Sets, Set theory, Number systems > Sets > Construction

Peano's Axioms and Induction

Important note: Most textbooks do not include zero among the set of natural numbers.  However, Peano, in his original formulation of these five postulates, did include zero in this set.  Therefore, in this page, I will be faithful to this original formulation.  See "Is Zero A Natural Number?" for more about this.

Peano's five axioms define the natural numbers starting with just 0 and s, the successor function. (The successor s(x) is the number immediately after x, ie x+1).

1. zero: 0 is in N

2. successor: s: N —> N 

3. nonzero successor: 0 ≠ s(x), any x in N

4. one-to-one: s(x) = s(y) ==> x = y 

5. induction:  Let F(x) be a statement about natural number x. Then F(0) and (for all x in N, F(x) ==> F(s(x))) ==> (F(x) for any x in N)

Other arithmetic operators can be defined, e.g. by recursion

0 + n = n base case for +

s(m) + n = s(m + n) recursive case for +

0 * n = 0 base case for *

s(m) * n = n + (m * n) recursive case for *

The ordering relations are defined

m ≥ n <=> m = n + p, some p in N

m > n <=> m = n + p, some p in N+

Here N+ = N − {0} is the positive integers.

From these axioms, and simple logical reasoning, all properties of the natural numbers can be deduced, including the fact that N has no maximum, and the well-ordering theorem:

S is a non-empty subset of N ==> S has a minimum element

which is crucial to proving the termination of recursive programs such as Euclid's gcd algorithm.

The set of Integers (Positive Naturals + 0 + Negatives) is usually denoted Z.

A ring is an algebraic system that it is closed under addition, subtraction, and multiplication (but not necessarily division). A ring is ordered if it has a relation of 'greater than', symbolized by ">", with 1>0, b>a if and only if b−a>0, and such that the positive elements (those greater than 0) are closed under addition and multiplication. An ordered ring is well-ordered if every nonempty subset of its positive elements has a (necessarily unique) smallest element. Then, the (positive and negative) integers Z form the unique (up to isomorphism) well-ordered ring.

The set of Rationals (a/b, where a, b are integers, b not zero) is usually denoted Q.

A field is a ring that is closed under the four basic arithmetic operations (addition, subtraction, multiplication, and division by nonzero elements). Then, the system Q of rational numbers is the unique (up to isomorphism) smallest ordered field.

The set of Reals (limits of converging series of Rationals) is usually denoted R.

An ordered field is complete if every nonempty set X of its elements that is bounded above (there is some element of the field that is greater than or equal to every element in X) has a smallest upper bound. Then, the system R of real numbers is the unique (up to isomorphism) complete, ordered field.

Internet references

The Foundations of Mathematics, an overview at the close of the second millennium, by William S. Hatcher

Math League: integers, whole numbers and their properties (Using Whole Numbers, Place value, Expanded form, Ordering, Rounding whole numbers, Divisibility tests; Operations and their properties: commutative, associative, distributive, zero property, identity, order of operations (PEMDAS))

Related pages in this website

Sets - how to construct sets of integers, reals, etc.

The Peano Postulates -- Proving the properties of natural numbers using the Peano Postulates, which have been formulated so that zero is not included in the set of natural numbers.  (There's quite a debate about this point.)

Is Zero a Natural Number? -- a discussion of the fact that some authors include zero, and others do not.

Introduction to Counting -- explains what mathematicians mean by "counting" -- that is, putting sets in one-to-one correspondence.

Set Theory -- an introduction to sets, including examples of some standard sets.

Integers -- a description of integers as a type of number.

Counting Ordered Pairs of Integers -- An explanation of the "square spiral" that puts the set of natural numbers in one-to-one correspondence with the set of rational numbers.

Abstract Algebra -- definitions of terms associated with Ring Theory and Group Theory

 


The webmaster and author of this Math Help site is Graeme McRae.