FOM: sterility?

[Kanovei]

> Date: Mon, 22 Mar 1999 16:45:50 -0700
> From: Randall Holmes <holmes at catseye.idbsu.edu>

>Second-order logic allows <...> a categorical axiomatization 
>of N and other familiar structures; 
>first order logic, even formalization in a first-order
>theory as strong as ZFC, does not.

a) 2nd order logic axiomatizes N caregorically 
through appeal to subsets of N. 

b) 1st order logic does it with equal  
elegancy: simply require, in addition to 
the 1st order Peano axioms, that every  
proper nonempty initial segment has a 
maximal element. 

Is there any clear, well defined reason why 
2nd logicians consider a) as so remarkably 
superior to b) as to make it the sole 
cornerstone of a competitive system of f.o.m. ? 
Apparently, the only mathematical point here is that 
some properties of mathematical structures S, 
not expressible in the *corresponding* 1st-order 
language, 
become expressible if quantification 
over P(S) is allowed. 

V.Kanovei