| I think this is an interesting topic worth discussing, but also I don't think it's related to your post I replied to, which claimed that ZFC doesn't presuppose any more primitive kind of collection. > sets of beer mugs Set theory doesn't admit anything other than sets. Based on the way you're describing things, you'll need to encode each mug as a set, just as we encode numbers themselves as sets. But that's rather unnatural, and mathematicians generally go about their business without worrying about such encodings. They work with the axioms of their setting instead, comforted merely that they could find a model of their constructions in set theory if they really cared to carry out the encodings required. Confusingly, the "group" and "set" in "group theory" and "set theory" are referring to different parts of the theory. Group theory posits a universe with entities called elements, which when used in certain ways do nice things. Set theory posits a universe with entities called sets, which when used in certain ways do nice things. In one, a "group" is the universe; in the other, a "set" is a constituent of the universe. If you wanted your mugs to be a model of set theory, the same way we say that something is a group (is a model of group theory), you'd need to somehow prove that your collection of mugs satisfies the ZFC axioms. I don't think "is this mug an element of that mug" is the kind of question you're hoping to model, so your mugs are probably not a model of set theory. Your mugs might be a model of the theory of preorders, if you deign to order them by their fluid capacity. > Whereas for a category it's just impossible to get started - should my beer mugs be objects or arrows? It sounds like what you want is some kind of "mug theory", and the sibling comment has it right in asking what you want to model about mugs. The common thread with all of these theories is that you state first-order axioms, and then work with those axioms freely ignorant of whatever choice of model somebody might come along with that satisfies those axioms. This is really no different than what we do in something like Java. We define interfaces (theories) capturing the things we want to be able to do with objects, and worry later about what implementations (models) actually meet the requirements. If you want to work with mugs in Java, what are you going to do but define a new type with the operations you need? There's no reason to expect that some other interface is going to just happen to meet your needs. Not even set theory; not even category theory. |