[FOM] n-th order ZFC

[lubarsky.robert at comcast.net]

W Taylor wrote:

> I have read long ago that in some sense 3rd-order (or higher) is unnecessary, 
in that 3rd-order can be somehow "mirrored" in second-order, with no loss 
of fidelity. Or some such comment. Can anyone elaborate on that? And give us a brief idea of how such a coding is effected? 


Use a multi-sorted language. Have two sorts O for objects and S for sets. Axiomatize that S is the power set of O. Then in 2nd order logic, any model would have among its objects S as the power set of O, and so the power set of S too, obviating the need for 3rd order logic. 


Bob Lubarsky