Cartesian join between sets looks like "for every set A, and for every set B, there exists a set AxB such that for all x in A, and for all y in B, (x, y) is in AxB
I am not of course considered a luminary in the field of set theory. But I do know a bit.
Ordering of sets is an interesting topic.
If you have a set containing only natural numbers (positive counting numbers: 0, 1, 2...), then there's always a "least" element of that set, no matter how many numbers are in it (including if there are infinitely many). If 0 is in your set, then it has to be 0; if not, then the next number after 0, (i.e. 1), or the next after that, etc. This is the Well-Ordering principle.
If you have a set of other things, like all the rational numbers (fractions) greater than 0, and a function that uniquely maps every element of this set to a natural number, then you can pick out a "least" element of this set just by picking out the element that maps to the smallest natural number. This may seem counterintuitive. There isn't a "smallest" fraction greater than 0 - if you give me one, I can always come up with one smaller just by adding 1 to the denominator, e.g. you give me 1/1,000,000, and I go "yes, but 1/1,000,001 is even smaller" ad infinitum. But
there is a way to uniquely map the rational numbers to the natural numbers, and this means we can say there's an "
nth" rational number for every natural number
n, which means that for any set of rational numbers we can say which comes first in that ordering, even if it isn't the smallest number in the set.
Once we get into real numbers, things get gnarly. There's no unique mapping from the natural numbers to the real numbers (as Cantor proved), so we can't rely on that trick to pick a "least" real number. But we can well-order the reals using the axiom of choice, which says that for any set of sets, there's a set containing just one element from each set (example: I have a big bag containing three smaller bags - one of fruit, one of sex toys, and one of books. The axiom of choice says that there exists a "choice function" which will give me a bag containing one thing chosen from each of the smaller bags - an apple, a cock ring and a copy of
1984). How do we do it? We take the set containing only the set of real numbers, and use the choice function magically granted to us by the axiom of choice to give us a set containing just one element from that one set - picking out an arbitrary real. That's our "first" real. Then we take that number out of the set we started with, and do the "choice" thing again - that gets us our "second" real. And so on.
The axiom of choice is equivalent to the Well-Ordering Theorem, which says that
every set can be ordered in this way (but doesn't say how).
If we restrict our universe of sets to those which can be constructed by taking the empty set and repeatedly doing things to it to get larger sets, then every set in this universe has a unique "name" which is the smallest sequence of things you have to do to the empty set to arrive at it, and these "names" are countable, which means that both the axiom of choice and the Well-Ordering Theorem are automatically true in the "constructible universe". It's a bit like saying a) you can put computer programs in order, and b) every set we're prepared to consider is one that could be generated by a computer program, so c) the least element of any set is the element that is generated by the lowest-ordered computer program.