# Some questions on Slate syntax

David Hopwood david.nospam.hopwood at blueyonder.co.uk
Mon Mar 28 09:39:54 PST 2005

John Leuner wrote:
> http://www.sparknotes.com/math/prealgebra/integersandrationals/section2.rhtml

# The natural numbers, also called the counting numbers, are the numbers
# 1, 2, 3, 4, and so on. They are the positive numbers we use to count
# objects. Zero is not considered a "natural number."

Well, this seems to be a matter of definition.

Wikipedia <http://en.wikipedia.org/wiki/Natural_number>:

# Natural number can mean either a positive integer (1, 2, 3, 4, ...) or
# a non-negative integer (0, 1, 2, 3, 4, ...).
#
# [...]
# In the nineteenth century, a set-theoretical definition of natural numbers
# was developed. With this definition, it was more convenient to include
# zero (corresponding to the empty set) as a natural number. Wikipedia
# follows this convention, as do set theorists, logicians, and computer
# scientists. Other mathematicians, primarily number theorists, often prefer
# to follow the older tradition and exclude zero from the natural numbers.

MathWorld <http://mathworld.wolfram.com/NaturalNumber.html>:

# 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 \in P."

PlanetMath <http://planetmath.org/encyclopedia/NaturalNumber.html>:

# Given the Zermelo-Fraenkel axioms of set theory, one can prove that there
# exists an inductive set $X$ such that $\emptyset \in X$. The natural
# numbers $\mathbb{N}$ are then defined to be the intersection of all
# subsets of $X$ which are inductive sets and contain the empty set as an
# element.
#
# The first few natural numbers are:
#
#   * $0 := \emptyset$
[...]
# In some contexts (most notably, in number theory), it is more convenient
# to exclude 0 from the set of natural numbers, so that $\mathbb{N} = # \{1,2,3,\dots\}$. When it is not explicitly specified, one must determine
# from context whether 0 is being considered a natural number or not.

FWIW, I find the definition that includes 0 much more useful.

--
David Hopwood <david.nospam.hopwood at blueyonder.co.uk>