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Hello, I have just been reading about the Zermelo-Frankel (ZF) axioms for set theory and thinking about their consequences. I understand that the Axiom of Regularity is needed in order to prevent contradictions like Russell's Paradox arising. That axiom says that any non-empty set A must contain an element x that is disjoint from A.
I have seen the simple proof that this axiom prevents a set from being an element of itself.
Now I'm wondering whether it prevents other forms of infinite, self-referential recursion.
For example, are the following possible within ZF?
I couldn't see an obvious proof that these are impossible, but I haven't had much practice with set theory.
I have seen the simple proof that this axiom prevents a set from being an element of itself.
Now I'm wondering whether it prevents other forms of infinite, self-referential recursion.
For example, are the following possible within ZF?
- A is an element of B which is an element of A, hence giving infinite recursion with a two-step cycle: B = {A,x,y,...} = {{B,u,v,...},x,y,...} = {{{{B,u,v,...},x,y,...} ,u,v,...},x,y,...} etc
- {A} is an element of A, so that A = {{A},b,c,...} = {{{{A},b,c,...}},b,c,...} = {{{{{{A},b,c,...}},b,c,...}},b,c,...}, etc?
I couldn't see an obvious proof that these are impossible, but I haven't had much practice with set theory.
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