I am reading Naive set theory by P R Halmos. He says that "The axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging."(adsbygoogle = window.adsbygoogle || []).push({});

The example for that is

"Suppose we consider human beings instead of sets, and change our definition of belonging a little. If and are human beings, we write whenever is an ancestor of . Then our new (or analogous) axiom of extension would say if two human beings and are equal then they have the same ancestors (this is the “only if” part, and it is certainly true), and also that if and have the same ancestors, then they are equal (this is the “if” part, and it certainly is false"

How does this example substantiate the fact that axiom of extension is not a logically necessary property of sets but a non-trivial statement about belonging? I'd greatly appreciate it if someone can explain the above.

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# Axiom of extension

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