# Historical Instances of Set Theory

Context: I've been reading a lot about Set Theory lately, and how it suddenly sprung onto the mathematical scene in the late 1800's, thanks largely to Cantor. But it seems strange to me that no one had ever done anything similar to it before, so I was wondering...

Are there any historical instances of someone inventing something similar to Set Theory? Or is it actually the case that no one had considered anything like it until the 1800's?

• What do you mean by "set theory"? I'm sure people had considered concepts like "the concept of grouping objects together", or intersections and unions. Commented Dec 21, 2014 at 1:34
• I mean anything resembling the concept of grouping objects together and then studying them. It seems that there truly was no one who thought of it before Cantor and his contemporaries, which is very counterintuitive Commented Dec 21, 2014 at 3:01
• See The Early Development of Set Theory and at least : José Ferreirós, Labyrinth of Thought : A history of set theory and its role in modern mathematics (1999). Commented Dec 21, 2014 at 8:47
• People probably used the idea of a set or collection casually and didn't realize that it was nontrivial to formulate a theory of such collections. The extremely hostile reaction to Cantor's work suggests that people weren't ready for concepts such as different sizes of infinities.
– user466
Commented Dec 21, 2014 at 19:18
• "I mean anything resembling the concept of grouping objects together and then studying them." Then you might be interested in this answer, which shows Euler studying at least set operations (intersections, unions) and "Venn" diagrams. Commented Jan 1, 2015 at 17:06

There were no historical instances. This is exactly why this discovery is considered so great. It is a feature of truly great discoveries that after they are made, people ask: why did nobody think like this before ?

Set theory is the best example of this I can think of.

Comparable perhaps only to Kepler's First law and General Relativity. I do not know any other examples of such radical changes in our thinking, made by one person.

Perhaps other members will give other examples.

• "Why did nobody think like this before?" Isn't the answer in the case of general relativity pretty clear? Before special relativity 1905 there wasn't the necessity, and before Minkowski there wasn't the idea to use the geometric math. Following these events, the first theory seems to be Nordströms. Commented Dec 25, 2014 at 1:36
• @Nikolaj-K: well, perhaps general relativity is not such a clean example, though it seems that a) Nordstrom's theory was not quite correct (not consistent with observations), and b) Einstein developed his version independently (as far as I know). Commented Mar 20 at 13:31
• Just so you know, I expect you to reply to this comment in 2034. Commented Mar 20 at 18:55
• @Nikolaj-K: As you see, I replied 10 years before your deadline. Commented Mar 21 at 10:13

Set theory and the term set (German: Menge) has been invented by Bernard Bolzano (1781-1848). In his posthumous book Paradoxien des Unendlichen, Reclam, Leipzig (1851), but also in his Wissenschaftslehre, Friedrich Frommann Verlag, Stuttgart (1985), Bolzano-Gesamtausgabe, Series I Vol. 11,1, he considers sets and the properties of their elements. It is just another set theory than that invented by Cantor but certainly not worse. Bolzano does not accept the one-to-one correspondence as a tool to measure sets. But he accepts different infinities: There are infinitely many tetrahedrons but there are four times as many corners of tetrahedrons.

The main idea of set theory however, grouping certain objects with common properties, is much older than the term set / Menge. Euclid for instance considered "every given number of prime numbers".

• I strongly disagree with your evaluation "not worse". This kind of "set theory" had no following. Every modern mathematician will agree that the set of tetrahedron is of the same cardinality as the set of vertices of tetrahedra. Commented Mar 20 at 13:26