Bertrand Russell’s paradox wasn't taken as some little pleasurable game or puzzle. It "forced logicians to recast set theory and logic in a different way". Why was that? Because it kept ‘creeping in’ to set theory and logic generally. Despite that, logicians wanted to "still retain the bulk of what was useful and descriptive in the original systems" (326).
It was largely as a result of Russell’s Paradox, and the desire to create a solid foundation for mathematics, that Russell himself and Whitehead wrote the Principia Mathematica. At the heart of this book we can say is
"his first full-scale attempt to describe all of mathematics as a formal axiomatic system – an organisation of mathematical ideas based on a small number of statements assumed to be true".
At the core of an axiomatic system is a short list of simple statements called axioms. What do they do? They're combined in specifically defined ways to derive a much larger set of statements called theorems...
No. No, what are axioms? Not what do they do.
Anyway, forget the axioms themselves, what about the axiom system or systems to which they belong? What are they? More to the point, what did Russell and Whitehead want from their axiomatic system or systems? Take these examples:
i) A system powerful enough to derive sophisticated statements about mathematics as theorems.
ii) A system that avoided all inconsistencies, such as Russell’s Paradox.
iii) A system which could show that all possible mathematical truths could be derived as theorems.
Number iii) is quite amazing. A single system from which all possible mathematical truths could be derived as theorems. Would that have really been just a single axiomatic system rather than a collection of systems?
It can now be said that "their system also eliminated paradoxes of self-reference, such as Russell’s Paradox" (326). Perhaps this isn't surprising since it was his own paradox he was trying to counteract. However, the writer goes onto say
"whether the Principia Mathematica could avoid all inconsistencies and provide a method to derive all of mathematics remained to be seen". (326)
In retrospect we can now say that even
"though the axiom set of the Principia did solve the problem of Russell’s Paradox, in practice it was awkward, so it didn’t catch on with mathematicians". (327)
This was something that the young Quine realised in those early days. However, a different set of axioms solved the same problem. This well-known and important set of axioms is called the Zermelo-Frankel axioms (ZF axioms). It solved the problem of Russell’s Paradox and other problems of self-reference by "distinguishing sets from more loosely defined objects known as classes" (327).
How did these axioms do that?
I think that primarily it was a case of Frankel’s Foundation Axiom which showed that sets can never have other sets as members. Perhaps Russell’s Theory of Types and Tarski’s ideas about meta-languages and object-languages are also connected to the solution of Russell’s Paradox and other self-referential paradoxes. In any case, today
"the words set theory usually refers to one of several versions of set theory based in the ZF axioms". (327)