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I recently learned that $\emptyset$ or $\varnothing$ is a relatively new notation for the empty set created in 1939. I know $\{\}$ is also used along with $\{\cdot\}$ to denote empty sets. What was their origin? In general, what was the historical development of notation for the empty set?

Since set theory was given a rigorous treatment starting in the late 19th century, did mathematicians have different, non-standardized notations prior?

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    $\begingroup$ You may find the link below interesting. It has information about the earliest use of various mathematical notation including that of set theory. jeff560.tripod.com/mathsym.html $\endgroup$ – gjh Jul 15 '15 at 8:51
  • $\begingroup$ There was nothing to notate. $\endgroup$ – Mikhail Katz Jul 5 '16 at 7:39
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George Boole introduced the concept of empty set, or "nothing" as he called it, as the complement to the "universe" in his Mathematical Analysis of Logic (1847). His notations for them were somewhat boring, $0$ and $1$ respectively. Cantor wrote in 1880 "for the absence of points we choose the letter $O$". Frege, the founder of mathematical logic, interpreted "null class" as extension of the concept "not identical with itself" rather than a collection of objects, and used $\{\}$ for it. Peano used $\Lambda$ for both null class and "false" in his axiomatization of arithmetic in 1889, but Zermelo axiomatizing set theory in 1908 went back to Boole's $0$, and so did Hausdorff in his influential book Grundzuge der Mengenlehre (1914). Hausdorff was the first to use "empty set" in its modern purely extensional meaning, before him there was always an air of intensional "null class" to it, and even Zermelo stipulates extensionality as a separate assumption. See Kanamori's The empty set, the singleton, and the ordered pair.

The modern symbol $\emptyset$ or $\varnothing$ was indeed introduced in 1939 by Andre Weil, and gained wide acceptance after its use in the Bourbaki volumes.

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    $\begingroup$ Corresponding to Peano's $\Lambda$ for the empty set, there was also $V$ for the universal set (or class). $\endgroup$ – Gerald Edgar Jul 9 '15 at 1:18

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