FOM: Re:mathematical induction

Harvey Friedman friedman at
Thu Feb 25 16:12:23 EST 1999

Colin McLarty 6:15PM 2/25/99 writes:

>        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.

This is not correct.

THEOREM. ZFC proves the consistency of each of its finitely axiomatized

Furthermore, this Theorem is provable in Peano Arithmetic, and even in weak
fragements thereof such as EFA = exponential function arithmetic.

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

It often does include all cases of induction that are expressible in the
language of that system.

I think that you need to reformulate your point more carefully.

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