Definition:Set/Historical Note

Historical Note on Set

The concept of a set first appears in Bernhard Bolzano's posthumous ($1851$) work Paradoxien des Unendlichen (The Paradoxes of the Infinite).

The first investigation into the concept in any depth was made by Georg Cantor in his two papers called Beiträge zur Begründung der transfiniten Mengenlehre ($1895$ and $1897$).

It was Georg Cantor who, in $1874$, defined a set thus:

By a set $M$ we understand any collection into a whole of definite and separate objects $m$ of our intuition or our thought.

Hence the definition of a set as:

a Many that allows itself to be thought of as a One.

-- Georg Cantor, A. Fraenkel and E. Zermelo, Gesammelte Abhandlungen (Berlin: Springer-Verlag, $1932$)

This definition was directly inspired by a problem raised by Bernhard Riemann in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe of $1854$, on the subject of Fourier series.

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
• 1972: Patrick Suppes: Axiomatic Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.1$ Set Theory and the Foundations of Mathematics
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Introduction