Sets of things are somewhat different from transfinitely iterated sets of sets of sets of sets, though… (I'll call these ZF-sets).
Discrete sets of atomic objects are in fact (special) weak omega-groupoids and aren't ZF-sets. (In a ZF-set, every element of a set is itself a set, which is not true of, say, {red, green, blue}, unless we impose some completely artificial and obfuscating coding). You could just as well say that when we were counting the elements of sets thousands of years ago, we were doing the first rung of building up weak omega-groupoids, rather than the first rung of building up ZF-sets.
The name "weak omega-groupoids" makes them sound more intimidating as a concept than they actually are. They're just certain kinds of shapes. A bunch of dots (like a discrete set) is a weak omega-groupoid. A circle is a weak omega-groupoid. Spheres and donuts are weak omega-groupoids.
That said, I don't assert that weak omega-groupoids are any more intrinsically a foundational concept than ZF-sets; rather, I just note that ZF-sets aren't particularly elementary, either.
Discrete sets of atomic objects are in fact (special) weak omega-groupoids and aren't ZF-sets. (In a ZF-set, every element of a set is itself a set, which is not true of, say, {red, green, blue}, unless we impose some completely artificial and obfuscating coding). You could just as well say that when we were counting the elements of sets thousands of years ago, we were doing the first rung of building up weak omega-groupoids, rather than the first rung of building up ZF-sets.
The name "weak omega-groupoids" makes them sound more intimidating as a concept than they actually are. They're just certain kinds of shapes. A bunch of dots (like a discrete set) is a weak omega-groupoid. A circle is a weak omega-groupoid. Spheres and donuts are weak omega-groupoids.
That said, I don't assert that weak omega-groupoids are any more intrinsically a foundational concept than ZF-sets; rather, I just note that ZF-sets aren't particularly elementary, either.