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 .