What is the purpose of the Axiom of regularity/foundation?

Solution 1:

True it adds nothing and can mostly be dispensed with. However it is equivalent to $$\forall x(x\in V)$$ and this is sometimes useful in set theory. In other words it says that every set has a rank.

Solution 2:

By transfinite recursion over $\Omega$ (the class of all ordinals) we can build the following sets in $ZFC$

1. $V_0:=\emptyset$
2. $V_{\alpha+1}=\mathcal{P}(V_{\alpha})$
3. $V_{\gamma}=\bigcup_{\alpha<\gamma}V_\alpha$ if $\gamma$ is a limit ordinal.

With this, you can build the class $WF:=\bigcup\{V_\alpha:\alpha\in\Omega\}$. This class is known as the class of well founded sets and this satisfies tha $x\in WF\Leftrightarrow (x$ is a well founded set).

You can prove that, under regularity axiom, $V=WF$ (even more, regularity is equivalen to $V=WF$). Maybe, in this context, is a bit clearest that says regularity axiom. Because, if regularity is not true, then $V\neq WF$, so in the universe of sets, there is a set that don't have a minumun element respect to $\in$.