Posted by: Alexandre Borovik | May 19, 2011

## Consistency of Peano Arithmetic

A dispute rages at the Foundation of Mathematics mailing list about consistency of Peano Arithmetic. I am an outsider in the foundations of mathematics domain, but I was struck by a quote from Edward Nelson’s Predicative Arithemtic:

The reason for mistrusting the induction principle is that
it involves an impredicative concept of number. It is not correct
to argue that induction only involves the numbers 0 to n; the
property of n being established may be a formula with bound
variables that are thought of as ranging over all numbers.
That is, the induction principle assumes the natural number
system as given. A number is conceived to be an object satisfying
every inductive formula; for a particular inductive formula,
therefore, the bound variables are conceived to range over
objects satisfying every inductive formula, including the one
in question.

This is something that has always made me uncomfortable with the principle of mathematical induction .