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 .