[FOM] First Order Logic

[MartDowd at aol.com]

The second-order axiom of induction is
  $\forall S ( 0\in S \wedge \forall n ( n\in S\Rightarrow n+1\in S )  
\Rightarrow \forall n ( n\in S ))$
It is provable in ZFC that $N$ (the integers with 0,1,+,x) is the only  
structure satisfying Peano's axioms with the second order induction axiom.   
However, the statements in the language of number theory which can be proved 
in  ZFC to hold in this structure are recursively enumerable.  The true  
statements are not.  Any attempt to remedy the situation by means of  providing 
axioms for second order validity cannot succeed.
 
- Martin Dowd
 
 
 
In a message dated 8/30/2013 2:29:24 P.M. Pacific Daylight Time,  
hewitt at concurrency.biz writes:

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
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