# Is the symbol for set membership $\in$ derived from greek letter $\epsilon$?

Title self explains: Is the symbol for set membership $\in$ derived from greek letter $\epsilon$? What is their historical relationship? Obviously the letter must be older, since greek alphabet is extremely old. Paul Halmos Naive Set theory, page 2, second paragraph (equivalently page 2 lines 8-11)

This version of the Greek letter epsilon is so often used to denote belonging that its use to denote anything else is almost prohibited. Most authors relegate $\in$ to its set-theoretic use forever and use $\epsilon$ when they need the fifth letter of the Greek alphabet.

In the comments of this answer is found a discussion, but not in the form of answer. Also this site seems more appropriate than Math.SE.

Yes and no. Peano originally used $$\epsilon$$ in Arithmetices Prinicipia Nova Methodo Exposita (1889), and stated that the symbol was an abbreviation for Latin est (is), apparently using a Greek letter for a Latin word. However, as Mauro Allegranza pointed out, in Principi di Logica Matematica (1891) he changes the script, and explains the use of $$\varepsilon$$ by reference to the Greek ἐστι (is). In Formulaire de Mathematiques (1895) Peano goes back to $$\epsilon$$ (possibly the choice between the two depended on typographers).

In the first edition of Principia Mathematica (1903) Russell explicitly adopted Peano’s symbol $$\varepsilon$$, along with the Greek lineage. Extensional (sets) and intensional (classes) notions were not separated at the time, so Russell-Peano's $$a\,\varepsilon P$$ is more of intensional "$$a$$ is $$P$$", with $$P$$ as a class defining property, than modern extensional "$$a$$ is an element of the set $$P$$". Fully extensional interpretation of sets only appears in Hausdorff's Grundzuge der Mengenlehre (1914), see Kanamori's The empty set, the singleton, and the ordered pair. The stylized $$\in$$ and its crossed version $$\not\in$$ appear in Bourbaki's Theorie des Ensembles (1939), likely responsible for their widespread adoption, but they might have been used earlier.

See Jeff Miller's site Earliest Uses of Symbols of Set Theory and Logic and Cajori's History of Mathematical Notations for these and similar questions.

• @MauroALLEGRANZA Sorry, I do not follow. It is written in Latin, but he uses Greek letters. – Conifold Mar 22 '17 at 23:00
• He does not say that the Greek symbol $\varepsilon$ is an abbreviation for the Latin word "est" (is); he says that the Greek symbol $\varepsilon$ is an abbreviation for the Greek word "ἐστι" (is). At that time every learned man (mathematician included), especially in "latin" european countries, studied Latin and Ancient Greek and thus they were perfectly able to "do not mix Greek and Latin" :-) – Mauro ALLEGRANZA Mar 23 '17 at 7:00
• @MauroALLEGRANZA I see now, Jeff Miller's entry only accounts for Arithmetices Prinicipia version, I should have checked. I cleaned it up. – Conifold Mar 24 '17 at 18:31

Yes, it is.

In Giuseppe Peano's Arithmetices Principia (1889), the $$\epsilon$$ symbol is explained as follows (page x):

Signum $$\epsilon$$ significat est.

[The sign $$\epsilon$$ means is.]

In his Principi di Logica matematica (1891), Peano gives the full explanation (page 3) :

Per indicare la proposizione singolare « $$x$$ è un individuo della classe $$s$$ » scriveremo (8)
[To mean the proposition "$$x$$ is an individual of the class $$s$$" we will write] $$x \varepsilon s,$$

e il segno $$\varepsilon$$ si potrà leggere è, o è un, o fu, o sarà, a seconda delle regole grammaticali; però il suo significato è sempre quello spiegato.
[and the sign $$\varepsilon$$ will be read is, or is a, ...]

[...]

(8): Il segno $$\varepsilon$$ è l'iniziale di ἐστι.
[The sign $$\varepsilon$$ is the initial of ἐστι.]

The propositional function "$$x$$ is a member of the class $$\alpha$$" will be expressed, following Peano, by the notation
$$x \epsilon \alpha.$$ Here $$\epsilon$$ is chosen as the initial of the word ἐστι.
In 1888, Richard Dedekind, in Was Sind und was Sollen Die Zahlen? (art. 3), used a "reversed" $$\varepsilon$$ ($$A ∋ S$$) to mean "is part of", where this relation slid between membership and improper inclusion.