There is a definition of ordered pairs such that $(a,b)$= $\{a,\{a,b\}\}$. Using the axiom of regularity, which states that every non-empty set has a member which is disjoint from the set, we can prove the ordered pair property for that construction. Is it really necessary? That is, is there a model of $ZFC$ - $Regularity$ where that construction does not have the ordered pair property?
2026-05-06 00:20:10.1778026810
Is the axiom of regularity really needed to prove this ordered pair property?
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If there are two sets $a\ne c$ such that $a=\{c\}$ and $c=\{a\}$, then with that funny definition of ordered pairs we would have $$(a,a)=\{a,\{a,a\}\}=\{a,\{a\}\}=\{a,c\}=\{c,a\}=\{c,\{c\}\}=\{c,\{c,c\}\}=(c,c).$$ I've been told that such a pair of sets can exist in a model of ZFC minus Regularity.