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jimrandomh comments on But Somebody Would Have Noticed - Less Wrong
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I think you are going to run into serious problems. Consider the predicate that always returns true. Then if I'm following Russell's original formulation of the paradox involving the powerset of the set of all sets will still lead to a contradiction.
I'm not deeply familiar with set theory, but cousin_it's formulation looks valid to me. Isn't the powerset of the set of all sets just the set of all sets of sets? (Or equivalently, the predicate X=>Y=>Z=>true.) How would you use that to reconstruct the paradox in a way that couldn't be resolved in the same way?
The powerset of the set of all sets may or may not be the set of all sets (it depends on whether or not you accept atoms in your version of set theory). However, Cantor's theorem shows that for any set B, the power set of B has cardinality strictly larger than B. So if B=P(B) you've got a problem.