First use of curly braces to denote a set?

Question

I was wondering who was the first person to

• Use curly braces to represent a finite set. Exempli gratia, $\{1,2,3\}$.
• Use set builder notation. Such as $\{2n:n \in \mathbb{Z}\}$ to represent the even numbers. I am not sure if the colon or vertical pipe is the original form for set builder notation, I have seen both used, but the colon comes out better in $\TeX$.

Answer 1

Georg Cantor was the first; see at least:

• "Beiträge zur Begründung der transfiniten Mengenlehre (1)" (1895), Mathematische Annalen, 46, Engl.transl.P.E.B.Jourdain (1915, also Dover reprint):

By an "aggregate" (Menge) we are to understand any collection into a whole $M$ of definite and separate objects $m$ of our intuition or our thought. These objects are called the "elements " of $M$.

In signs we express this thus :

$$M = \{ m \}.$$

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The formal set builder notation:

$\{ x \mid \varphi(x) \}$

is an evolution of W&R's Principia Mathematica notation for "abstraction": $x \in \hat z(\psi z) \equiv \psi x$.

An early example is in:

• Paul Bernays, Axiomatic set theory (1958, Dover reprint), page 46:

The class operator $\{ x \mid \mathfrak A(x) \}$ "the class of the $x$ such that $\mathfrak A(x)$" [footnote : This symbol (used in some newer papers) is here taken, for the sake of easier print, instead of Russell's class symbol $\hat x \mathfrak A(x)$, whose adoption was first intended.]

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The 2nd French edition of:

• Nicolas Bourbaki, Théorie des ensembles (2éme ed., 1970), page E.II.4 adopts it:

On introduira [...] un symbole fonctionnel; dans ce qui suit, nous utiliserons le symbole $\{ x \mid R \}$; le terme correspondant ne contient pas $x$. C'est de ce terme qu'il s'agira quand on parlera de «l'ensemble des $x$ tels que $R$». [...] par suite la relation $R$ est équivalente à $x \in \{ x \mid R \}$.

The notation was not present in the English translation of 1968.