Full Idea
In the case of Russell's antinomy, the tacit and trusted pattern of reasoning that is found wanting is this: for any condition you can formulate, there is a class whose members are the things meeting the condition.
Gist of Idea
Russell's antinomy challenged the idea that any condition can produce a set
Source
Willard Quine (The Ways of Paradox [1961], p.11)
Book Reference
Quine,Willard: 'Ways of Paradox and other essays' [Harvard 1976], p.11
A Reaction
This is why Russell's Paradox is so important for set theory, which in turn makes it important for the foundations of mathematics.