[seanhunter]

It’s interesting to note that in regular mathematics, induction on the natural numbers is axiomatic.[1] That is, you can’t get there from within the framework you have built up to then so you need to build an extra axiom into the system to make it work. Both the mathematical technique of induction and the use of the word induction as per the article to mean deducing knowledge from experience are due to Descartes I believe.[2]

[1] I’m slightly simplifying. Some people teach “the principle of mathematical induction” as a “principle” (whatever that is, but it sounds like an axiom to me), and some people teach it as a theorem like any other theorem but derive it by adding an axiom to set theory called the “well-ordering principle” (that every non-empty set of natural numbers has a smallest element) that turns out to be equivalent to induction. If you don’t accept induction as an axiom and don’t take it to be axiomatic that you can choose the smallest element from a set of natural numbers then as far as I know you can’t do induction in a well-founded way.

[2] https://www.jstor.org/stable/43154967

[skybrian]

In first-order logic, it's an infinite axiom schema, which means you can take a finite subset of them, like assuming induction axioms hold for the formulas you're actually using and not in any other case.