[FOM] n-th order ZFC

[Roger Bishop Jones]

On Thursday 07 Jul 2011 04:21, 
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.
> 
> Can anyone elaborate on that?  And give us a brief idea
> of how such a coding is effected?

The principal reason for advocacy of second order logic is 
semantic.
It is that in second order logic one can give a categorical 
axiomatisation of the natural numbers and proceed from there 
by conservative extension through analysis, whereas in first 
order logic most interesting mathematical structures do not 
have a categorical axiomatisation.

In this respect adding more orders confers no additional 
advantages.
Instead of going to omega-order logic, you can go to second 
order set theory using axioms similar to those of ZFC in 
second order logic.  Second order set theory differs from 
first order set theory in being quasi-categorical, i.e. in 
having at most one model of any cardinality, and is much 
stronger than n-th order logic for any finite n.

However, this does not mean that 3rd order logic is 
conservative over second order, it is strictly stronger.
You have to add axioms to second order logic to make it as 
strong as nth-order logic for n>2.

Nor is it strictly necessary to go to second order logic to 
secure these benefits.
There is no reason why one could not do set theory in a first 
order language with an appropriate semantic, by nominating a 
class of standard models.
For many purposes (including the natural numbers and many 
other structures definable by induction) the constraint to 
well-founded models would suffice, though this does not suffice 
for cardinal arithmetic and hence problems like CH and GCH, 
for which a constraint equivalent to that in second order 
set theory, i.e. insisting on "full" power sets, is better 
and would achieve the same quasi-categoricity for a "first 
order" set theory as in second order set theory (though 
there would still be more models of the first order than of 
the second order system).

Shapiro's "Foundations without Foundationalism" is a 
thorough discussion of the merits of second order logic.

Roger Jones