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"Set theory" itself is a bit of an overloaded term; most practical uses of naïve set theory in math do not make use of properties like extensionality or the glo

"Set theory" itself is a bit of an overloaded term; most practical uses of naïve set theory in math do not make use of properties like extensionality or the global membership relation, that characterize axiomatic set theory over TT. Thus, one could also describe the sort of "more natural" systems you reference in your comment as 'structural set theory' https://ncatlab.org/nlab/show/structural+set+theory https://ncatlab.org/nlab/show/structural+set+theory It's true however that one can do material-set theory inside structural-set theory or type theory, by defining a type of material sets (not necessarily "iterative" however, that's a matter of whether you want to model the axiom of foundation!) and a membership relation on them.

gylterud 7y ago
Structural set theory, which as you say throws out the iterative structure, is more like type theory than set theory (this is pointed out in the nlab page you linked). Naïve set theory as practised in say a course in discrete mathematics certainly uses the iterative structure to construct say a pair. You will typically find the definition 〈x,y〉 ≔ {{a},{a,b}} instead of defining it through a universal property. All the axioms of material set theory take this iterative approach, not just foundation. In fact foundation is not very critical (The Axiom of Anti-foundation is just as workable), as you can always just use the class of well-founded sets and work with those.
dwheeler 7y ago
I agree that "set theory" itself is a bit of an overloaded term, as there are in fact many set theories. However, as you're probably well aware, naïve set theory can quickly lead to disaster because it leads to well-known paradoxes (like Russell's paradox). So if you're going to have foundations for mathematics (and it needs foundations!), it needs to avoid the paradoxes. ZFC does so, and in a way that many people have found satisfying enough.

gylterud 7y ago

Structural set theory, which as you say throws out the iterative structure, is more like type theory than set theory (this is pointed out in the nlab page you linked). Naïve set theory as practised in say a course in discrete mathematics certainly uses the iterative structure to construct say a pair. You will typically find the definition 〈x,y〉 ≔ {{a},{a,b}} instead of defining it through a universal property. All the axioms of material set theory take this iterative approach, not just foundation. In fact foundation is not very critical (The Axiom of Anti-foundation is just as workable), as you can always just use the class of well-founded sets and work with those.

dwheeler 7y ago

I agree that "set theory" itself is a bit of an overloaded term, as there are in fact many set theories. However, as you're probably well aware, naïve set theory can quickly lead to disaster because it leads to well-known paradoxes (like Russell's paradox). So if you're going to have foundations for mathematics (and it needs foundations!), it needs to avoid the paradoxes. ZFC does so, and in a way that many people have found satisfying enough.