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Full Idea
By analyzing the paradoxes to which Cantor's set theory had led, ..Russell brought to light the amazing fact that our logical intuitions (concerning such notions as truth, concept, being, class) are self-contradictory.
Gist of Idea
Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory
Source
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.452
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.452
A Reaction
The main intuition that failed was, I take it, that every concept has an extension, that is, there are always objects which will or could fall under the concept.