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?
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.
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".