Equivalence relations on classes instead of sets
Can someone please explain to me how to deal with equivalence relations on classes instead of sets? Is there some sort of generalisation of relations?
Thank you
It’s not clear to me whether you’re asking what a class equivalence relation would look like formally, or how to work with one when you have it. I’ve given an answer based on the first interpretation, and Asaf has sketched an answer to the second interpretation.
Suppose that a class $\mathbf C$ is described by a formula $\varphi$: $x\in\mathbf{C}\leftrightarrow\varphi(x)$. A formula $\psi(x,y)$ describes a class equivalence relation on $\mathbf C$ if it satisfies the following conditions:
- $\forall x\Big(\varphi(x)\to\psi(x,x)\Big)$ (reflexivity)
- $\forall x,y\Big(\varphi(x)\land\varphi(y)\land\psi(x,y)\to\psi(y,x)\Big)$ (symmetry)
- $\forall x,y,z\Big(\varphi(x)\land\varphi(y)\land\varphi(z)\land\psi(x,y)\land\psi(y,z)\to\psi(x,z)\Big)$ (transitivity)
If you have an equivalence relation and each of the classes is a proper class, one can use Scott's trick and trim those classes into sets.
In concrete situations one may also be able to produce canonical representatives regardless to that fact. For examples cardinals in ZFC. There is a proper class of sets of each cardinality (except zero) but we have canonical sets to represent each class.
If you give more details it might be possible to give a more accurate answer. If this is just an idle curiosity then Scott's trick should satisfy it.
Also related:
- What can I do with proper classes?
- Are surreal numbers actually well-defined in ZFC?
- homeomorphism of topological spaces is an equivalence relation ?