History of $0 \in \mathbb N$.

Question

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?

Answer 1

For modern times:

• Richard Dedekind, Was sind und was sollen die Zahlen? (1888), page 20:

§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 $N$.

• Giuseppe Peano, Arithmetices Principia Novo Methodo Exposita (1889), page 1:

Axiomata

1. $1 \in N$

• Gottlob Frege, Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl (1884):

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

• Bertrand Russell, The Principles of Mathematics (1903):

§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.

• Mario Pieri, Sopra gli assiomi aritmetici (1907):

$\gamma.$ 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.

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For Bourbaki, see:

• Nicolas Bourbaki, Théorie des ensembles (2nd ed 1970), Ch.III, §3:

1. Le cardinal d'un ensemble

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

Answer 2

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.