[FOM] n-th order ZFC

[William Tait]

On Jul 6, 2011, at 10:21 PM, W.Taylor at math.canterbury.ac.nz 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.

In second-order logic with the epsilon relation, let T have the axioms
1. there is a (von Neumann) ordinal alpha such that all ordinals are <= alpha +n, 
2. for W=V_{alpha} and phi the conjunction of the axioms of second-erder  ZF(C), phi^W (the relativization of phi to W). 
[It is possible to do this because second-order ZF is finitely axiomatizable.] 
3. the definition of  V_{beta} is extended to beta greater or equal to alpha and V_{beta} is postulated to contain, for each class whose elements are of rank < beta, a set with the same elements (i.e. the second-order axiom of Separation is extended to all  the ordinals).

In this theory there is an obvious translation of  ZF of order n+2.

Best wishes,
Bill Tait