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?
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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)
Brian M. Scott
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No fair I am writing from an iPhone. Also we read the question in two very different ways. – Asaf Karagila Dec 06 '12 at 01:12
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@Asaf: Your interpretation may be right; I really can’t tell. You might want to add a link for Scott’s trick. – Brian M. Scott Dec 06 '12 at 01:16
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Well. I hope to find out in six hours. Now I have an appointment with the Sandman. Also could you please edit the link in? Doing that from the iPhone is a true pain in one lower back. – Asaf Karagila Dec 06 '12 at 01:17
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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:
Asaf Karagila
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I should add the remark that you have to understand class well in ZFC (to the point where you won't ask this, and much beyond it) to work with classes without feeling that you are cheating. – Asaf Karagila Dec 06 '12 at 01:13