The usual axioms ensure the existence of certain sets that serve as functions. For example (which is chosen arbitrarily) the function f which maps real values of x to x^2+2 can be represented by the set containing all the pairs of the form (x,x^2+2) where x is a real number. I wondered what the intuition behind the existence of such sets is, since it might feel weird that the second (or in general n-th) value depends on a preceding one. Are there some interpretations/reasons on this, for example when interpreting sets as collections? I guess that one could argue that it should be intuitive that the collection of exactly all objects of R matched with exactly those that are of the form x^2+2 should exist, but I wanted to see if there are better views on this. Thanks in advance.
Edit for clarification: This question came up when thinking about foundations from a platonistic viewpoint. Surely I can just accept this by a formalistic viewpoint, but I find that to be not satisfying.
I am aware that functions are intuitively viewed as processes, but I feel like this intuition is not well captured by the definition in set theory, given by a relation (maybe its just me but thats the way I feel about it). It is intuitive that one can have subsets where the objects satisfy certain properties. However, in this case one has a set of objects where the 2nd "coordinate" depends on the first, which made me ask why this dependency "should be allowed" or is "natural to allow". Surely the concept is useful and widely used in many areas, but I was looking for an intuitive explanation justified by the interpretation of sets and not the usefulness of the concept. This lead to the question: Why should collections of pairs be allowed where the second coordinate depends on the first?
