History of $0 \in \mathbb N$.

There are different conventions around whether $0 \in \mathbb N$.

I know that $\mathbb N = \{0,1,2,3, \cdots\}$ is called the Bourbakian definition of $\mathbb N$. But did this convention really come form Bourbaki?

In the axioms of Peano, which were earlier, 0 was also taken as a natural number.

So where did each convention ($0 \in \mathbb N$ and $0 \not\in \mathbb N$) come form?

• I don't think Euclid ever used a symbol analogous to $\mathbb N$ in this context, but to Euclid, numbers were $2,3,4,5,\ldots$. The smallest number was $2$. $\qquad$ – Michael Hardy Feb 26 '16 at 18:24
• @MichaelHardy I know. There are even some conventions that gave three as the first number, because three is the smallest integer such that $n^2>2n$. – wythagoras Feb 26 '16 at 18:32
• Where does one find that convention? – Michael Hardy Feb 26 '16 at 18:57
• @MichaelHardy I don't remember. – wythagoras Feb 26 '16 at 19:00

For modern times:

§71. Definition. A system $N$ [Ein system $N$] is said to be simply infinite when there exists a similar transformation $\phi$ of $N$ in itself such that $N$ appears as chain of an element not contained in $\phi(N)$. We call this element, which we shall denote in what follows by the symbol $1$, the base-element of N and say the simply infinite system $N$ is set in order by this transformation $\phi$.

[...]

§73. Definition. If in the consideration of a simply infinite system $N$ set in order by a transformation $\phi$ [...] these elements [are] called natural numbers or ordinal numbers or simply numbers, and the base-element $1$ is called the base-number of the number-series $N$.

Axiomata

1. $1 \in N$

§74. Null [$0$] ist die Anzahl, welche dem Begriffe "sich selbst ungleich" zukommt.

§120. Peano’s three indefinables are $0$, finite integer and successor of. [...] Peano’s primitive propositions are then the following.

(1) $0$ is a number.

$\gamma.$ There is at least one number that is not the successor of any number. [...]

$f.$ The number $0$ that is not the successor of any number [...] is given the name zero.

For Bourbaki, see:

1. Le cardinal d'un ensemble

On note $0$ le cardinal $\text {Card} (\emptyset)$.

Originally zero was not considered a natural number. In fact it is one of the most unnatural numbers. Since the word "progression naturelle" was introduced by Chuquet, in 1484 and "natural number" by Emmerson in 1763 always the sequence starting with 1 has been addressed. "Natural number, defined as the numbers 1, 2, 3, 4, 5, etc., appears in the 1771 Encyclopaedia Britannica in the Logarithm article." http://jeff560.tripod.com/n.html Cantor for instance (although rarely using this wording but prefering "integers" (ganze Zahlen) mentiones the "natural sequence of numbers" (natürliche Zahlenreihe): $1, 2, 3, ...,\nu , ...$" in Über unendliche lineare Punktmannigfaltigkeiten Math. Annalen 15, (1879). Also Peano started with 1 in his first papers. Later he switched to 0 like many others. (See the answer by Mauro Allegranza)

Where did these conventions come from? What was the reason for this change? Cantor explained it in a letter to Dedekind of Juli 28,1899: The sequence of ordinal numbers $0, 1, 2, 3, ... ,\omega_0, \omega_0 + 1, ..., \gamma,..$ has the property that every number is the ordinal type of the preceding numbers. Without including zero this property would not be satisfied for the finite part.

I think that this is the reason.