# Who first discovered the concept corresponding to the symbol of class comprehension?

Who first discovered the concept corresponding to the symbol of class comprehension {x/𝛗}used today in set theory ?

• Was... that symbol introduced even before it was associated to a clear concept? I don't find the title or the question very clearly phrased – Qfwfq Apr 10 at 19:11
• Isn’t “the concept corresponding to the symbol of class comprehension” just “class comprehension”? – Emil Jeřábek Apr 10 at 19:28
• For what it's worth, the history of set-builder notation itself might be interesting - I can't find much about it after a quick google search. – Noah Schweber Apr 10 at 21:18
• However you make this precise, van Heijenoort’s Source Book in Mathematical Logic is a good place to look. – Matt F. Apr 10 at 22:32
• I completely agree with Emil Jerabek and Noah Schweber, that my question is really about the history of the notion of class comprehension and of the set-builder notation. Gérard Lang – Gérard Lang Apr 11 at 8:40

The concept is found with different notation in Peano (1894, p. 20) (translated):

Let $$p_x$$ be a proposition containing a variable letter $$x$$, that is, a condition on $$x$$. By the notation $$\overline{x\ \varepsilon}\,p_x$$ we shall indicate the class of those $$x$$ that satisfy the condition $$p_x$$.

Von Neumann (1923, Einleitung) uses $$M$$ for Menge:

Let $$E(x)$$ be a property, $$f(x)$$ a function defined for all $$x$$ possessing the property $$E(x)$$. Then $$M(f(x); E(x))$$ shall be the set of all $$f(x)$$ as $$x$$ runs over all $$x$$ possessing the property $$E(x)$$.

Our current $$\{\ ..\mid\,...\}$$ is in e.g. Lefschetz (1942, p. 2; 1949, p. 26; ...), Segal (1946; 1947; ...), Kelley (1947; 1953; 1955), Mostow (1949; 1950; ...), Nomizu (1950; 1954; ...), Godement (1952; ...), Loomis (1953; ...), etc. Kelley (1947, p. 683) seems to attribute the notation to Lefschetz, and Bernays (1958, § I.1) writes:

Our logical symbols are (...)

3) The class operator $$\{\mathfrak r\mid\mathfrak{A(r)}\}$$1)  « the class of the $$\mathfrak r$$ such that $$\mathfrak{A(r)}$$ »

(...)

The rules concerning the class terms are: the formula schema (“Church schema”) $$c\in\{\mathfrak r\mid\mathfrak{A(r)}\}\ \leftrightarrow\ \mathfrak A(c),$$ expressing a conversion law, in the sense of A. Church (1932), and (...)

1) This symbol (used in some newer papers) is here taken, for the sake of easier print, instead of Russell’s class symbol $$\hat{\mathfrak r}\mathfrak{A(r)}$$, whose adoption was first intended.

• Surely the concept of comprehension was already known to Cantor? – Emil Jeřábek Apr 14 at 8:00
• @EmilJeřábek ... or to Euler, who drew a “space” to depict what any notion “comprises”. I guess it depends how much explicitness we require. According to Ferreirós (2007, pp. 250-252), “Dedekind relied on the idea that every concept determines a set, but (...) did not formulate the principle of comprehension. His approach to sets remained as vague as those we find in Cantor (1895/97) or Schröder (1890/95, vol. 1).” – Francois Ziegler Apr 14 at 14:42
• (cont’d:) “It is sometimes said that Frege was the first to formulate in a precise form the principle of comprehension, with the basic law V of the Grundgesetze. However, as we shall see (...) that law was nothing but a principle of extensionality, while comprehension was implicit in the very notation employed by Frege. Thus, it seems that nobody pinned down the crucial principle of comprehension before the emergence of the paradoxes.” – Francois Ziegler Apr 14 at 14:42

Following G.Cantor ( see "Beiträge zur Begründung der transfiniten Mengenlehre" [Contributions to the founding of the theory of transfinite numbers], Mathematische Annalen (1895) ),the use of braces for sets was already present in E.Zermelo, “Untersuchungen über die Grundlagen der Mengenlehre,” Mathematische Annalen (1908) :

The set that contains only the elements $$a, b, c, \ldots, r$$ will often be denoted briefly by $$\{ a, b, c, \ldots, r\}$$.

The modern symbol for sets defined "via comprehension" evolved from Whitehead & Russell's Principia Mathematica (910-1913) :

$$\hat x (\phi x)$$,

meaning : the class of individuals satisfying the propositional function $$\phi x$$.

As referred in the answer above, Paul Bernays, in Axiomatic Set Theory (1958) says (page 46) :

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

The symbol were not used by Bernays in his previous work published in the Journ. of Symb. Log. (1937-1954).

Note : W&R's $$\hat x$$-notation for class-abstraction was also the source for Church's $$\lambda$$-notation.

See Alonzo Church,A Set of Postulates for the Foundation of Logic (1932) :

If $$\text M$$ is any formula containing the variable $$x$$, then $$\lambda x [ \text M ]$$ is the symbol for the function whose values are those given by the formula.

See Felice Cardone & J.Roger Hindley, History of lambda-calculus and combinatory logic (2006) :

By the way, why did Church choose the notation “$$\lambda$$”? In Church (1964) [unpublished letter] he stated clearly that it came from the notation “$$\hat x$$” used for class-abstraction by Whitehead and Russell, by first modifying “$$\hat x$$” to “$$\land x$$” to distinguish function abstraction from class-abstraction, and then changing “$$\land$$” to “$$\lambda$$” for ease of printing.

See also : J.B. Rosser, Highlights of the history of the lambda calculus, Annals of the History of Computing (1984), page 338.

IMO, the symbol slowly emerged during the 1950s.

Lefschez's Algebraic Topology (1942) has it (page 2) :

If $$P$$ is a property, the totality of all the elements $$x$$ which satisfy $$P$$ is denoted by $$\{ x \mid x \text { has the property } P \}$$.

Well-know set-theorists like A.Fraenkel do not used it; see Abraham Fraenkel, Abstract Set Theory (1953) and Abraham Fraenkel & Yehoshua Bar-Hillel, Foundations of Set Theory (1958).

But P.Suppes in Axiomatic set theory (1960) says (page 33) :

In many branches of modern mathematics it is customary to use the notation:

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

to designate the set of all objects having the property $$\varphi$$.

Bourbaki's Theory of sets (1968) has

$$\mathcal E_x(R)$$ [for the term denoting] "the set of all $$x$$ such that $$R$$".

The 1970 second French edition has instead :

le symbole $$\{x \mid R \}$$ [par] «l'ensemble des $$x$$ tels que $$R$$».

• Why do you believe that the source of Church's $\lambda x. \phi(x)$ was W&R's $\hat{x}.\phi(x)$? Does Church say so somewhere? – Michael Bächtold Apr 17 at 13:19
• @MichaelBächtold There are some references in en.wikipedia.org/wiki/… . – Emil Jeřábek Apr 17 at 13:32
• Interesting. Is there any evidence that W&R got the inspiration for their $\hat{x}.\phi(x)$ from Frege's $\grave x \phi(x)$, found in Funktion und Begriff, 1891? – Michael Bächtold Apr 17 at 13:39
• @MichaelBächtold - yes; added ref. – Mauro ALLEGRANZA Apr 17 at 13:44
• And yes; the origin is with Frege's symbol $`\epsilon ϕ(\epsilon)$ for Wertverlauf. – Mauro ALLEGRANZA Apr 17 at 13:45