# What's the strength of weakening induction in PA by relativizing it to naturals?

If we weaken induction in PA to the following scheme:

$$\forall m [\phi(0) \land \forall n < m (\phi(n) \to \phi(n+1)) \to \forall n \leq m (\phi(n))]$$

Where $$\phi$$ is any formula in the language of PA.

Then how much that would weaken PA?

## 1 Answer

This is equivalent to the usual induction scheme. Indeed, suppose a formula $$\phi$$ satisfies $$\phi(0)$$ and $$\forall n(\phi(n)\to\phi(n+1))$$. Given any $$m$$, then, $$\phi(0)$$ and $$\forall n are true, and therefore by your scheme, $$\forall n\leq m(\phi(n))$$. In particular, $$\phi(m)$$ is true. Since $$m$$ was arbitrary, we conclude $$\forall m(\phi(m))$$.