# [FOM] First Order Logic

MartDowd at aol.com MartDowd at aol.com
Sat Aug 31 14:12:25 EDT 2013

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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