When working in set theory, we often want to work with large collections of sets, so large that they themselves do not form a set. This requires us to have a notion of classes. But then we might want to form a collection of classes, and so on… (For examples, look at the category of all categories, or the collection of gaps of surreal numbers.) This requires stronger and stronger theories (indeed, stronger *languages), if we want to be rigorous.
Instead though, could we just use a model $ (M_1, \in_1)$ of set theory? Instead of sets, we would have elements of $ M$ , and instead of classes, we would have regular sets. If we want to go deeper, we just need a model $ (M_2,\in_2) \in M$ , and so on…
In particular, if we choose an $ (M_2, \in_2)$ such that $ (M_1, \in_1) \cong (M_2, \in_2)$ , (for example, see Lemma 7 here), then we can form an infinite sequence of models such that $ M_1 \ni M_2 \ni_1 M_3 \ni_2 M_4 \ni_3 \dots$ and $ (M_1, \in_1) \cong (M_2, \in_2) \cong (M_3, \in_3) \cong (M_4, \in_4) \cong \dots$ . We get all this without ever changing our formal system!
So for example, the category of all categories would be a set in $ M_1$ containing elements in $ c \in M_2$ (corresponding to categories) which would consist of elements of $ M_3$ (corresponding to the objects and morphisms of $ c$ ).
Due to isomorphism, we would also have a category of all categories in $ M_2$ , which would be isomorphic to one in $ M_1$ . It would also be an element of the category of all categories in $ M_1$ , since the category of all categories is itself a category.
My question is, does this actually work out? Or would issues crop up to stop this construction?
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