For a bunch of… initially strange but ultimately interesting reasons, mathematicians and philosophers in the late 18th and early 19th century would sometimes get really hung up on how exactly to define a Set, as a mathematical object.
For basically all parts of math besidesAxiomatic_Set_Theory , textbooks can just say “a Set is a collection of objects” and move on to bigger and better things like the actual topic of the textbook.
BUT, when Bertrand Russell and Alfred North Whitehead started to try and ground all of mathematics on axiomatic foundations… shit got real hectic, and mathematicians had to start thinking really hard about how to define this basic building block. To see why, the best I’m gonna be able to do here is to point you to Russell’s Paradox.
- On the one hand, some historians of math talk about this as if it led to some sort of “downfall of mathematical truth”, by linking it to Gödel’s_Incompleteness_Theorem.
- On the other (less melodramatic) hand, really it just meant that we had to work with Sets in a slightly different way from the standard “define-object-then-derive-theorems-about-it” approach. It’s summarized pretty well in @halmos_naive_1960:
One thing that [the book] will not include is a definition of sets. The situation is analogous to the familiar axiomatic approach to elementary geometry. That approach does not offer a definition of points and lines; instead it describes what it is that one can do with those objects.