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Full Idea
In no consistent theory is there a class of all classes with seventeen members. The existence of the paradoxes is a good reason to deny to 'seventeen' this univocal role of designating the class of all classes with seventeen members.
Gist of Idea
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
Source
Paul Benacerraf (What Numbers Could Not Be [1965], II)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.284
A Reaction
This was Frege's disaster, and seems to block any attempt to achieve logicism by translating numbers into sets. It now seems unclear whether set theory is logic, or mathematics, or sui generis.