I've been trying to understand in a more formal way what a set actually is, but I have some questions. According to the axiom of regularity for every non-empty set A there exists an element in the set that's disjoint from A. That would mean that such element is also a set, right?
I read here: Axiom of Regularity , that in axiomatic set theory everything is a set, I understand that natural numbers are constructed from the empty set, integers are constructed from the naturals, rationals from the integers, and reals from the rationals. I can see how every element in such sets is also a set. But, for example, in the set of all the letters of the alphabet, or the sample space of an experiment when the possible results are not numbers, or the set of my classmates; it's not clear to me how their elements are also sets. So, are they really sets? Is every element in a set also a set?
Thank you
Best Answer
In the context of axiomatic set theory, we (usually) only allow sets whose elements are sets. So in that context, there is no such thing as "the set of all the letters of the alphabet", or your other examples. However, that doesn't mean you can't talk about these concepts in axiomatic set theory--you just have to encode them in sets. For instance, instead of talking about the letters of the alphabet, you could talk about the set of natural numbers from $1$ to $26$, with the understanding that $1$ secretly stands for A and $2$ secretly stands for B and so on. This isn't really any different from the idea of "constructing" numbers as sets which you said you were familiar with: we have an intuitive idea of numbers, and we find a way to talk about them using only sets whose elements are all sets. Similarly, we can find such a way to talk about letters, or classmates, or whatever else you want to talk about that is not intuitively a set.