I am reading Analysis from Terence Tao. The Axiom of regularity is defined as follows:
If A is a non-empty set, then there is at least one element x of A which is either not a set, or is disjoint from A
question: if I have a Set A such as
A:= {A,1}
since 1 is not a set, so axiom of regularity should be hold. But we all know actually it doesn't.

  • 3
    $\begingroup$ Consider $A\setminus\{1\}$. This exists if $A$ does, but if it exists it would be a counterexample to regularity. The point is that regularity can constrain us "indirectly" - $X$ might be "bad" because it lets us construct $Y$ which violates regularity. $\endgroup$ Jul 9, 2022 at 14:54
  • $\begingroup$ This question should contain more context. For example, In some contexts, a set can contain itself, whereas in other’s (ZFC for example) it cannot. In other words, whether or not your definition of $A$ is acceptable depends on the context. $\endgroup$ Jul 9, 2022 at 15:03
  • $\begingroup$ @NoahSchweber Why is a counterexample of regularity? {{A}} and {A} are still disjoint right? sorry i might be dumb. $\endgroup$
    – ICFSZ
    Jul 9, 2022 at 17:16
  • 1
    $\begingroup$ Let $B = A \setminus \{1\} = \{A\}$. Then $B$ contains at least one element, contains no elements that are not a set, and all its elements intersect it, as $A \cap B = \{A, 1\} \cap \{A\} = \{A\}$. $\endgroup$
    – mihaild
    Jul 9, 2022 at 18:58
  • $\begingroup$ @mihaild thanks! now i'm clear. $\endgroup$
    – ICFSZ
    Jul 9, 2022 at 19:44


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