1. We've had primitive notions of what it means to have a collection of stuff, but the formalism I'm interested in started with Cantor. Some kinks in Cantor's business had to get smoothed out, and then we got the Zermelo-Fraenkel business we have today. So by that, rigorous treatments of sets started at least before 1880. But the foundations didn't get a rigorous treatment until about 1900-1920.
2. Cantor sent out the party invitations. Russell, Zermelo & Fraenkel kept the party alive after Cantor vomited and passed out.
3. Nobody invents some sort of mathematics without any notion of what it could be about. Intuition precedes the praxis, the praxis precedes the formalism.
4. Numbers started out as the formalization we used to describe our concepts of quantity. So the collection of our numbers is a list of names There is a strict ordering on these names, and there is a binary operation on these names (addition) that plays well with the ordering.
But there's a large family of groups, semigroups, rings, fields, and other things that extend that basic intuition in different direction. Nearly all of these things are of some interest. But some of them (e.g. p-adic integers) may not jive with the original intuition. To each their own.
5. The multiplicative identity if we've defined multiplication. The generator of our semigroup if we're in that world. The positive generator if we're in a group. It's the same 'flavor' of thing in almost all situations, but it still depends on what sort of math-world we're working in.
6. 'Set' is one of those primitive concepts. If I want to talk about some things, I have to find some way of pointing to the things I'm talking about. I have to... sort of wrap them up in brackets, hold them up and say, "these things, right here." So I can't describe what a set is without appealing to my audience's notions of what a set is. The axioms don't provide a definition of a set. They're a framework for constructing and manipulating sets.
2. Cantor sent out the party invitations. Russell, Zermelo & Fraenkel kept the party alive after Cantor vomited and passed out.
3. Nobody invents some sort of mathematics without any notion of what it could be about. Intuition precedes the praxis, the praxis precedes the formalism.
4. Numbers started out as the formalization we used to describe our concepts of quantity. So the collection of our numbers is a list of names There is a strict ordering on these names, and there is a binary operation on these names (addition) that plays well with the ordering.
But there's a large family of groups, semigroups, rings, fields, and other things that extend that basic intuition in different direction. Nearly all of these things are of some interest. But some of them (e.g. p-adic integers) may not jive with the original intuition. To each their own.
5. The multiplicative identity if we've defined multiplication. The generator of our semigroup if we're in that world. The positive generator if we're in a group. It's the same 'flavor' of thing in almost all situations, but it still depends on what sort of math-world we're working in.
6. 'Set' is one of those primitive concepts. If I want to talk about some things, I have to find some way of pointing to the things I'm talking about. I have to... sort of wrap them up in brackets, hold them up and say, "these things, right here." So I can't describe what a set is without appealing to my audience's notions of what a set is. The axioms don't provide a definition of a set. They're a framework for constructing and manipulating sets.
