[FOM] First Order Logic

[Stephen G Simpson]

Possibly relevant to this discussion is our recent paper

     Stephen G. Simpson and Keita Yokoyama, Reverse mathematics and
     Peano categoricity, Annals of Pure and Applied Logic, 164, 2013,
     pp. 284-293.  http://dx.doi.org/10.1016/j.apal.2012.10.014.
 
Carl Hewitt writes:
 > Date: Fri, 30 Aug 2013 17:28:29 -0400
 > 
 >    I am having trouble understanding why the proponents of first-order
 >    logic think that second-order systems are unusable.
 > 
 > 
 >    [Dedekind 1888] and [Peano 1889] thought they had achieved success
 >    because they had presented axioms for natural numbers and real numbers
 >    such that models of these axioms are unique up to isomorphism with a
 >    unique isomorphism.  And later generations of mathematicians were happy
 >    to use these axioms.
 > 
 >    [....]