I am trying to self study Set Theory. While studying the ZFC axioms, I am introduced to Russell's Paradox and why the Universal Set does not exist. With the Axiom of Restricted Comprehension, Russell's Paradox ceases to exist and it can be shown that the Universal Set does not exist.
My confusion here is why include the Axiom of Regularity if possible cases of sets containing themselves have been eliminated by Restricted Comprehension. There are obviously many more implications of the axiom, but surely the main motivation was to eliminate such sets.
Are their sets that can be constructed using the axioms of ZFC excluding Regularity. Because this to me is the only potential reason why someone would include the Axiom of Regularity.
There are a few things here:
Russell's paradox predates the Axiom of Regularity. It comes to show how unrestricted Comprehension is inconsistent.
The Axiom of Regularity has nothing to do with the paradox. If $\sf ZF$ is consistent, then $\sf ZF-Reg+\lnot Reg$ is consistent. So the paradox shouldn't be affected from this. So it is also not true that the Axiom of Regularity was formulated to "avoid Russell's paradox" (an unfortunate mistake you can find all over the place).
In the usual proof of Russell's paradox, the Axiom of Regularity makes it one step shorter. $R=\{x\mid x\notin x\}$, then $R$ is the class of all sets in our case, and since $R\notin R$ by the Axiom of Regularity, then by definition $R\in R$. Oops... then it's not a set.