In proof by induction, is it more correct to say "we use induction on n" or "we use induction on the set of natural numbers"? When proof by induction is started, it is common to see a statement of the form "we use induction on _".
For example, say we were trying to prove
$$\forall n (n \in \mathbb{N} \implies \sum\limits_{i=1}^n i = \frac{n(n+1)}{2})$$
Would it be more correct to say "we use induction on n" or "we use induction on the set of natural numbers"?
After all, under the hood of proof by induction, we have an inductively defined set $\mathbb{N}$, and we are attempting to show that it is a subset of another set $S=\{x |  \sum\limits_{i=1}^x i = \frac{x(x-1)}{2}\}$.
What is the correct phrase?
I do think that "we use induction on n" is easier to use, especially if we are doing, for example, a double induction on $n$ and $k$ in a problem, both of which are natural numbers.
However, this usage seems to lack a bit of underlying meaning. Is this impression correct?
 A: Let us say which one might be preferable, rather than which one is correct or incorrect. A review of the origin of the term may help us clarify fine distinctions:
Induction is actually Latin translation of the method which Aristotle called epigoge (ἐπαγωγή). The root verb almost literally means lead on (but not by deception). We are led on to a general conclusion by inspecting individual cases. So, for example, we are aware of a property X and observing that the particular objects that are X are also Y, another property, we convince ourselves and others that X and Y are co-extensional.
Keeping with the analogy, we "inspect" each number $n$ and reach a judgement about the totality; we do not, as it were, presuppose a specific set. Seen this way, to say "induction on $n$" is more faithful a phrase to the original idea.
However, it should be remarked that mathematical induction is actually a deductive method, as opposed to scientific induction (and Aristotle's conception). We can see this clearly in the language of Peano Arithmetic. Written in the following form, induction stands, in ineffect, as a deductive rule of inference:
$$\dfrac{P(0), P(n)\to P(S(n))}{\forall nP(n)}$$
Thus in mathematics, it turns out that, when one says
"I prove this statement by induction on $n$",
one says essentially
"I've picked out the variable object (I could pick out another one or more than one) and denoted it by $n$, also found out a relation that allows me to show that the case for $n+1$ is true whenever the case for $n$ is true (i.e., $P(n)\to P(S(n))$)".
Hence, one does the following:

*

*decides on the proper variable (i.e., the mathematical object) for induction according to the problem.

*may decide on more than one variable (see, for instance, double induction).

*finds out the inductive relation.

*since the well-ordering property, the principle of mathematical induction and strong induction are provably equivalent (see Lars–Daniel Öhman's open access article Are Induction and Well-Ordering Equivalent?, thought-provoking), legitimately applies induction on any type of object to which the well-ordering property can be attributed.

A: When I teach mathematical induction, at some point I will get around to giving a formal statement of induction. The formal statement that I write might start with words like this:

The Principle of Induction: Given a statement $P(n)$ defined for all natural numbers $n$ ...

This sets the stage: if one wishes to prove that some statement $P(n)$, one that is defined for all natural numbers $n$, is actually true for all natural numbers $n$, here is a method for doing this. That method is called induction.
And, as you well know, the formal statement of the method continues something like this:

... if $P(1)$ is true, and if the implication $P(n) \implies P(n+1)$ is true for all $n \ge 1$, then $P(n)$ is true for all $n \ge 1$.

My point is this: as this common formulation of induction is stated, the fact that it is induction on "$\mathbb N$" is baked into the formulation, and so it is not necessary to keep inserting the prepositional phrase "on $\mathbb N$".
Now, having said that, as you learn more mathematics the language will change. As mathematical history as progressed, the language has already changed. There is almost never one explicitly correct way to say something in mathematics.
We are all human beings around here, and what's important is human communication of mathematics.
If you find that a mathematical phrase is too imprecise to clearly communicate the idea to others, then yes, you should feel free to recast the phrase into more precise language. You may have heard of transfinite induction, for example. In any context where transfinite induction is under discussion, then yes, it would probably be very wise to say "induction on $\mathbb N$" in order to distinguish it from induction on some more complicated ordinal number.
If on the other hand you find that a mathematical phrase is redundant, and that you can communicate just as clearly with an abbreviated phrase, then you should also feel free to adopt that abbreviation.
