Sorry for starting a lot of topics in set theory; I think this will be my last. I just want to know what are the basics I need to know about set theory to mathematical analysis.
Is what I have written below enough for set theory in relation to real analysis, and will it ensure that I only work with sets, and not so-called proper classes? And also, is what I write correct or wrong? I know almost nothing about ZF axiomatic theory and have no time to learn it right now, or even learn logic properly.
If we only use the basic set operations, like union, intersection, complement, difference, Cartesian-product, and also the axiom of choice, and only use these operations to create new sets on collections we already know are sets, we will not get any paradoxes or contradictions? (I assume in order to prove all of this you have to study a lot of logic and deep set theory, so I do not need a proof.)
In real analysis we sometimes meet very big sets, like the set of all functions on a given space, etc. The reason these sets are allowed to exist, is because if we look at the definition of a function as a relation, and the definition of a relation as a set, then all these big sets we meet in mathematical analysis can be created using known sets, and the basic operations in (1)? Hence they can exist without giving paradoxes like Russel's Paradox (but again, in order to prove this you probably need to study deep set theory). Is this correct?
The real numbers, and all the Euclidean vector spaces we work with are sets without paradoxes, because they are created by the basic operations described earlier of the natural numbers. And the collection of natural numbers is a set without paradoxes because it is easily defined using the basic set operations in (1)? (In this case you also need the axiom of infinity).
In summary, is the reason they don't encounter set-theory paradoxes in "ordinary mathematics" the fact that all their big sets (like the function spaces) can be shown to be created using the set operations on smaller sets like the natural numbers/real numbers, and it is proven in advanced set theory, that if you follow these rules, you do not get paradoxes?