Single Idea 9904

[catalogued under 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique]

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 Reference

'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.