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Full Idea
The paradoxes only seem to arise in connection with Frege's logical notion of extension or class, not Cantor's mathematical notion of set. Cantor never assumed that every condition determines a set.
Gist of Idea
The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set
Source
JP Burgess / G Rosen (A Subject with No Object [1997], III.C.1.b)
Book Ref
Burgess,J/Rosen,G: 'A Subject with No Object' [OUP 1997], p.224
A Reaction
This makes the whole issue a parochial episode in the history of philosophy, not a central question. Cantor favoured some sort of abstractionism (see Kit Fine on the subject).