Nothing of mathematics was disturbed or spoilt by Russell's paradox. Not even a simple lemma. The reason ist that set therory is not foundational to mathematics (1,2) because it even contradicts mathematics (3).
(1) "set theory is largely irrelevant to the practice of most mathematics. Most professional mathematicians never have occasion to use the Zermelo-Fraenkel axioms, while others do not even know them. [Saunders Mac Lane: "Mathematical models: A sketch for the philosophy of mathematics", American Mathematical Monthly, Vol. 88,7 (1981) p. 467f]
(2) "I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique." [T. Skolem: "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre", Akademiska Bokhandeln, Helsinki (1923) 217-232, reprinted as "Some remarks on axiomatized set theory" in J. van Heijenoort: "From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931", Harvard University Press, Cambridge, Mass. (1967) 290-301]
(3) Here is the gist of an argument made by Mückenheim: Scrooge McDuck every day receives 10 \$ and issues 1 \$. If he issues always the dollars received first and if he applies modern set theory, then he will go bankrupt because each of the dollars received will be spent. The set-limit is empty. According to mathematics we have to take the limit of the cardinality of the dollars held. This limit is infinite and stands in clear contradiction to set theory.
EDIT: Downvotes will not change the facts:
The actual infinite is not required for the mathematics of the physical world. [S. Feferman: "Infinity in mathematics: Is Cantor necessary?" in "In the light of logic", Oxford Univ. Press (1998) p. 30]
In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part if not all of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, "Is Cantor Necessary?", is answered with a resounding "no". [S. Feferman, "In the light of logic", Oxford Univ. Press (1998) description from the jacket flap]