Dynamic and Static Sets

Question

Are there any dynamic sets in Set Theory ? For example the total population of earth. It is always changing. We say that this is a finite set. I think such sets are finite only at a particular time or another parameter. Are there any axioms or specification for such sets in Set theory ? And is it fair to keep these sets as Finite when we are not able to count them as total, they are ever changing. I think there should be a classification as Static and Dynamic Sets and a way to express them. Also staticness should be a necessary property of Finite sets.

Answer 1

Set theory has nothing to say about time. It just defines what is, and what isn't, a set. Typically, the elements of a set are fixed, and are based on how the set is defined. The rules governing what is, and what isn't, a set are very strict. If the answer to "Is $1$ an element of this set?" isn't the same every time you ask the question, then you do not have a set.

If you want to talk about things that vary with time, then you're essentially talking about a function. And a function is also a kind of set, but it's a set of pairs of values - in the case of something like "world population", then it's a set of pairs of $(t, P)$ where $t$ is some indicator of time and $P$ is the population at that time.

Even then, it's not really something varying with time - nothing in mathematics cares about time passing in the real world - mathematics models things in the world, and it can model things that change with time, but not in the way you seem to be talking about.