FOM: mathematical induction

[Colin McLarty]

Stephen G Simpson <simpson at math.psu.edu> writes

>Poincare and Detlefsen want to say that mathematical induction is
>illogical.  What are these people talking about?  What's going on
>here?

        I am quite sure of Poincare's point, and I believe it is Detlefsen's
also. They are not saying any single use of induction is inexpressible in
formal logic. They are saying that no formal expression can capture all of
your actual belief in induction.

        Suppose you really believed in the principle of induction only as it
is formalized in ZFC. Then you could not also believe that ZFC proves the
consistency of each of its finitely axiomatized fragments. That claim
involves inductions in the metalanguage that cannot be formalized in ZFC. 

        Of course the claim is provable in the extension of ZFC gotten by
taking the claim itself as a new axiom, and even in other more interesting
extensions. But each of those extensions admits exactly the same further
extension. 

        No consistent formal, first order, axiomatic theory includes every
case of induction that you yourself will want to accept.       

Colin