I would like to know an example showing that second-order mathematical induction is stronger than first-order mathematical induction.
For a condition $P(x)$ that can be written in the language of Peano arithmetic, we can construct the set $\{x\in\mathbb{N};P(x)\}$.
Which subset of $\mathbb{N}$ cannot be constructed this way?
The examples in the question suggested as a duplicate are not what I expected.
I would like to know an example that is actually used in number theory, not ones that are only used as counterexamples.
Here, "actually used in number theory" is defined as being written in an article published in the Journal of Number Theory.