# Thread: ZF Axiom of regularity

1. ## ZF Axiom of regularity

I am a little confused over the axiom. Does it mean we cannot have set such that: A = {B, C} and B = {C, D}

According to Venn diagram, this seems trivial but is against the foundation axiom ∀A≠∅: ∃B∈A: A∩B≠∅

I mean how can a member of set be disjoint from the set ??

2. Hi
The axiom of regularity just states that a set has to contain an element which has no common element with the set.

Does it mean we cannot have set such that: A = {B, C} and B = {C, D}
No, sets of this form exist: if $\displaystyle C$ does not contain (as elements) $\displaystyle B$ nor $\displaystyle C,$ then $\displaystyle C\cap A=\emptyset$ and there is no problem.

A consequence of this axiom is that for any set $\displaystyle x,$ $\displaystyle x\notin x,$ that $\displaystyle x_1\in x_2\in ...\in x_n\in x_1$ is impossible for any $\displaystyle n$ sets $\displaystyle x_i$ and more generally, there can't be any infinite downward $\displaystyle \in$-sequence between elements of a set.

But this axiom does not belong to the theory Z, and I've heard that some researchers are exploring set theories with its negation.

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