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Okay, this may seem a crazy question. If it does, it's because I've been thinking too much about foundations, recently. I've read a bit on ZFC, on the need for universes for categories, on HoTT (though not far in that region), and on ETCS. I recently thought up a fairly different way to think about sets, which could perhaps act as an intuitive foundation for mathematics. I'm wondering if:

a) This idea has been tried before, and if it was found wanting.

b) If any immediate issues can be seen to appear via this foundation.

c) If I'm only imagining any advantages this may have over ZFC.

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Before I start, I'll just briefly explain the motivation behind this:

I recently discovered what I viewed as a flaw in my algebra textbook, which I only recently read after a long absence of set theory and categories, namely the innocuous statement $\mathbb{Q}\subseteq \mathbb{R}$.

This statement isn't true in ZFC. The objects of $\mathbb{Q}$ are equivalence classes of ordered pairs of equivalence classes of ordered pairs of natural numbers, as given by $n+1=n\cup\{n\}$ and $0=\varnothing$, recursively. The elements of $\mathbb{R}$, on the other hand, are equivalence classes of Cauchy sequences of rational numbers. There exists a subset of $\mathbb{R}$ which we commonly call "the rational numbers", but the simple fact is that $\mathbb{R}\cap\mathbb{Q}=\varnothing$. The way around this, I've been told, is to simply abuse notation and identify $\mathbb{Q}$ with this subset.

A bit dissatisfied, and recalling that I'd read about the idea of a sub$object$ from category theory, I did a bit of digging. The issue I'd had with the notion of sub$object$ which turned me off of Lawvere's book, "Sets for Mathematics", when I'd first tried tro read it is that it implied that $\{0,1\}\subseteq\{2,3,4\}$, which itself intuitively makes it seem like $0=2$, or something. Nonsense, I thought, until I came across this excellent answer by Martin Brandenburg. It made me think that perhaps these categorical notions aren't that bad, after all, but the issue still was that when sets are labelled it just seems weird to think of subsets this way - even though it's exactly what we are doing when we say that $\mathbb{Q}\subseteq \mathbb{R}$.

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My answer to this is to scrap labels entirely and merely let sets be defined by the structures we build out of them. We have some primitive notion of the counting numbers, and of cardinals, and merely say that for every cardinal $n$, there is exactly one set $Set_n = \{a_1,a_2,\dots, a_n\}$ (obviously looking a bit different for infinite cardinals) with a notion of equality within only each set to say that, for example, $a_1\neq a_2$. We also have the idea of an ordered $n$-tuple, which probably need be distinct from a set, which allows us to define relations and functions. In addition we may put in some primitive notion of a category.

In the usual manner we may now have binary operations $Set_n\times Set_n \to Set_n$, and so may now give labels to things if we wish by building up structures. Thus if we take the set with cardinality $\aleph_0$, and define an operation $+:Set_{\aleph_0}\times Set_{\aleph_0} \to Set_{\aleph_0}$ which corresponds to addition of the natural numbers, we end up with $(\mathbb{N}, +)$, a modification may then give us the group of integers under addition, for example. The transition then to the rational numbers is not so clear, I will admit. This is why I say that the idea of $n$-tuples may have to be more related to these sets.

This way of thinking allows for the categorical definitions of subset, and seemingly other things, too. We can say when two groups are not equal, which is precisely when they are not isomorphic as groups. At the same time it does seem to mean that two groups are in some way equal if they are isomorphic.

Perhaps the best way to express what I'm trying to get across is that perhaps it's best to define sets only by their structures and ignore labelling otherwise. We may not need to go as far as to literally strip sets down as I've indicated, but it seems more natural to me to talk of the natural numbers as merely the set of things which have the characteristic properties I pin to the natural numbers. Maybe some kind of idea of an isomorphism class is required, outside of set theory, to capture precisely what I mean. Perhaps a newer theory which we just attach to ZFC. I'm a bit tired at the moment, so I may not entirely make sense. It just seems as if a lot of things are getting a bit confused. Does something like this get established in UF/HoTT?

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I believe what you're looking for is structural set theory: see, for example:

(I'll add more sources as I remember them.)

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