Sets -[construct-into]-> Relations -[construct-into]-> Functions
The 'destructive' definitions are a bit less common
> the definitions of sets from functions and from relations are "destructive".
so a destructive definition of a set:
- start with a relation: define unary relations... I can see how this could be 'destructive'; it has a feel like an 'overkill' (or a 'waste') to use relations and then use only half of what they provide to regard them as sets
- start with a function: to define a relation use a singleton co-domain (this is really very similar to characteristic functions. This has been quite tricky for me to wrap my head around: how in logic there's a tacit assumption of 'truth' in the sense that a predicate defines a set and if this set is empty then the predicate is 'false'. I'm studying this from the lens of modal logic and formal semantics)
to define a relation use a power set.
I think all this is still relying on sets, because a co-domain is a set, as well as a power set.
> Since I can't think of any "constructive" definition of sets from functions or relations, while I can think of "constructive" definitions in the opposite direction, I would say sets are more fundamental than functions and relations
As I understand this, the sets really are more fundamental because in the end they bring to the system (for the lack of a better word) the capability to identify things, and when dealing with functions, which come with variables, the concept of a set allows for identifying the same variables in the function's call (parametric denotation?) and the functions actual body (its computable definition)
What I'm trying to say, is why it may well be the case that sets really are more fundamental than functions.