Is Zero a Natural Number?

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 my page on Construction of Sets and Peano's Axioms, I was faithful to this original formulation of the set.

Website http://xrefer.com/entry.jsp?xrefid=643157 defines a natural number as: "One of the whole numbers 1, 2, 3, ... Some authors also count zero as a natural number."

Another respected website, http://mathworld.wolfram.com/NaturalNumber.html, elaborates,

"A positive integer 1, 2, 3, ... (Sloane's A000027). The set of natural numbers is denoted N. Unfortunately, 0 is sometimes also included in the list of "natural" numbers (Bourbaki 1968, Halmos 1974), and there seems to be no general agreement about whether to include it. In fact, Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 is an element of P."

Since it is clear from these references that zero is only begrudgingly included in the set of natural numbers, I have included this information on this page.  See The Peano Postulates (reformulated for the modern set of natural numbers) for remarks that specifically refer to this issue.  The author of that website actually reformulated Peano's axioms to conform to the modern notion that zero is not a natural number.  A more traditional formulation of Peano's Axioms can be found at http://mathworld.wolfram.com/PeanosAxioms.html , where it is stated that zero is a natural number.

The moral of this story is this: read every text carefully to see the author's definition of "natural number", and be alert to clues such as references to "positive natural numbers" (which indicate that this author includes zero) or statements such as "n is a natural number, so it must be greater than zero" (which indicate that this author does not include zero).

Related pages in this website

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.)

Construction -- Construction of sets of numbers, starting with the original Peano Axioms, formulated so that zero is included in the set of natural numbers.

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.

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.

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