I have found two different statements on axiom of foundation of Zermelo–Fraenkel set theory in two different books as:

1) every nonempty set contains an element that is not an element of any other element in the set.

2) Every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets.

These two are not equivalent statements. Am I correct ?

  • 1
    $\begingroup$ Which books did you find these formulations in? When written formally with only $\in$ it is easy to misread the axiom. It is also possible that whoever wrote the book made a mistake (which may or may not have been corrected since then). $\endgroup$ – Asaf Karagila Apr 20 '13 at 12:58
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    $\begingroup$ I’ve a suspicion that a $\in$ got turned around in (1). $\endgroup$ – Brian M. Scott Apr 20 '13 at 13:06

They are not equivalent. In fact, (1) is actually refutable under a weak subtheory of ZF. Consider any inductive set $X$. Then for each $u \in X$ we have that $u \in u \cup \{ u \} \in X$. Thus $X$ has no element as described in (1).


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