Is induction a metalogical concept?

Question

Concepts like induction... are these metalogical concepts? Axioms? Within-logic?

For example proving the deduction theorem often resorts to principles of induction but I don't see where this "fits" within a system of logic. Like the whole idea of case $P(0)$ being true and case $P(k) \to P(k+1)$ being true implies $P(n)$ is true for all $n$... says who? I'm not saying I am disagreeing with induction, I'm just curious how we "know" it's true and where it actually fits in with our rigorously-defined systems of logic.

Is induction something you can "derive" from the rules of logic or is it something we assume is true on a metalogical level? Where does it fit exactly?

The Deduction Theorem is a metalogical result: it is not a formula of the propositional calculus but a result about propositional calculus. — Mauro ALLEGRANZA, Sep 19 '18 at 13:48
Thus, the proof of DT is "produced" in the meta-theory and the resources used in the proof, like induction on the lenght of derivations, are metalogical resources. — Mauro ALLEGRANZA, Sep 19 '18 at 13:49
Induction is formalized in e.g. first-order theory of arithmetic. — Mauro ALLEGRANZA, Sep 19 '18 at 13:50
@MauroALLEGRANZA Would the set theory used in the proof of the deduction theorem also be considered metalogical resources? — user525966, Sep 19 '18 at 14:09
@MauroALLEGRANZA So the deduction theorem for propositional logic relies on a principle defined in first-order logic? — user525966, Sep 19 '18 at 14:10
Metalogic is "usual" informal mathematics: it uses numbers, sets, trees and so on. If we want to formalize it we have to use a (quite weak) set theory. If so, set theory is able to "reproduce" natural numbers (as sets) and prove Peano's axioms, induction included (they are theorems of set theory). — Mauro ALLEGRANZA, Sep 19 '18 at 14:13
"the deduction theorem for propositional logic relies on a principle defined in first-order logic?" Not exactly: formalized in first order arithmetic. Induction has been used in mathematics at least from Fermat. — Mauro ALLEGRANZA, Sep 19 '18 at 14:15
@MauroALLEGRANZA So any reasoning, logic, set theory, principles like induction, etc, used on a metalogical level aren't "circular" in that they're meant to be interpreted informally? — user525966, Sep 19 '18 at 14:16
@MauroALLEGRANZA not sure I understand. What's the difference between formalize and define? — user525966, Sep 19 '18 at 14:59

Is induction a metalogical concept?

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