what even is a set? where has it had any application? it seems to be just an abstract concept for placing numbers that have a similar property, such as a set of reals and complex numbers, like you're just making a bunch of rules then saying "all of the numbers that follow them are all contained in this thing". but then you have axioms which limit the extent this applies, so it's a rule on rules. is this wrong?
>>7789405 Set theory has no "numbers" in the sense you're thinking. Instead it has certain sets with certain operations defined on them that make them act in the way that we want numbers to act (this is how we formally define numbers in set theory). From then we get the set of natural numbers (each of which is also a set) and then by augmenting and remixing this set we can get integers and rationals. Then by defining sets of rationals and defining more operations on these sets we can get a set of real numbers (in set theory, a real number is just a set of rationals).
If you want to see how this is done in excruciating detail (with only high school math background required) then look up Landau's Foundations of Analysis.
The main takeaway is that when one approaches mathematics through set theory, everything is a set.
I like category theory and I'm in the process of learning type theory, however I don't see how we can construct any of the categories we wish to talk about without already having set theory. It's almost like category is a meta-theory that is particularly useful for studying the objects and relationships built in set theory amongst other theories.
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