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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?

Blue
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xoux
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  • Btw, the statement you're trying to "prove" is wrong. $n+1$ not $n-1$. – Deepak May 11 '22 at 16:24
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    When there are multiple free variables, it helps to say which variable you are doing induction on. And I think that when someone says "induction" without explicitly stating which well-ordered set they are using, everyone assumes it's N, so it's fine leaving that implicit. – JonathanZ May 11 '22 at 16:27
  • Sure it is fine leaving it implicit but my question is just to confirm that the actual correct way is to say "on the set of natural numbers", right? – xoux May 11 '22 at 16:32
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    No, that's not actually necessary at all. In common mathematical usage, the bare phrase "induction" already refers to the statement about the natural numbers that you know. There are other types of induction, but they are always referred to with qualifying words attached, for instance transfinite induction. – Lee Mosher May 11 '22 at 16:41
  • evianpring - Yes, that is the right English Mathematical language to use, and it correctly describes the situation. And, as @LeeMosher points out, it is usually unnecessary, and borders on being redundant. – JonathanZ May 11 '22 at 16:52
  • @JonathanZsupportsMonicaC How can it be redundant if we are able to use the technique of proof by induction on other inductively defined sets? – xoux May 11 '22 at 18:01
  • It's a question of language. It's like saying "a four wheeled car". If you say "car", everyone is going to assume it has four wheels. If it's a specialty car, with just three wheels, then you should definitely say "three wheeled car". But if you say "four wheeled", lots of people are going to be left wondering why you bothered to specify that – JonathanZ May 11 '22 at 20:23
  • @JonathanZsupportsMonicaC Consider the set $S={3, 16, 29, 42, ... }$. We can define it inductively. Base clause: $3 \in S$. Inductive clause: if $x \in S$ then $x+13 \in S$. Final (closure) clause: S is the smallest set satisfying base and inductive clauses. Say we now want to prove $\forall x (x \in S \implies x>0)$. We can use induction. What is the phrase that we use here? "Let's use induction on _"? Fill in the blank. – xoux May 12 '22 at 08:15
  • You seem to be arguing from the point of view of a person who uses mathematical induction a lot and usually on the set of natural numbers. This person would just say "induction on n", and the meaning would be implicit that the induction is being done on the set of natural numbers. This is fine. But for people trying to understand how and why induction works, it is important to be able to know what the explicitly correct way to say it is. By actually saying "on the set of natural numbers" one is being explicit about the underlying concept used: that of an inductively defined set. – xoux May 12 '22 at 08:22
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    If stating this explicitly clarifies things for you, excellent! I'm just telling you what the common usage is. If you were learning about motors (let's say 20 years ago), and started saying "internal combustion engine car" all the time, you'd be right, and have a better understanding of what's happening when you pushed the accelerator, but if you dogmatically insisted on that phrasing every time, well, it'd be unusual. But you're certainly free to do so. – JonathanZ May 12 '22 at 14:50

2 Answers2

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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:

  1. decides on the proper variable (i.e., the mathematical object) for induction according to the problem.
  2. may decide on more than one variable (see, for instance, double induction).
  3. finds out the inductive relation.
  4. 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.
Tankut Beygu
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  • Do we not presuppose an inductively defined set? We inspect the base elements of such a set, and then we inspect the elements presupposed in the inductive clauses of the inductive definition and finally we inspect the newly generated elements. "n" simply denotes a variable. It could be "x" or "y" or "zebra". The objects denoted by the variable don't have to be numbers, they could be letters for example. Thus when we say "we will use induction on zebra", very little information is being given on what we will be doing. – xoux May 12 '22 at 14:13
  • For example, consider a set called A which we define inductively as follows: $ac \in A$, $x \in A \implies axc \in A$. This is the set ${ ac, aacc, aaaccc, ... }$. Now suppose we want to prove $\forall zebra (zebra \in A \implies \text { zebra has even number of letters })$. If we say "let's use induction on zebra", something is now implicit. In a math class it would be implicit that the inductive set is $\mathbb{N}$, but here the implicit thing is that the inductive set is $A$. In the normal course of things, I think both ways are fine, but I'm not questioning the communication aspect. – xoux May 12 '22 at 14:18
  • Indeed I am asking: if the goal is to be very precise and explicit about what is happening, then we must specify the inductive set, otherwise either i) we are mechanically doing the induction procedure without knowing what's happening under the hood or 2) we must do extra computations in our brain to figure out what is in fact implicit. – xoux May 12 '22 at 14:20
  • Right. I've done additions to make the answer more comprehensive. I hope it responds to your concerns now. – Tankut Beygu May 12 '22 at 17:41
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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.

Lee Mosher
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  • I understand the pros and cons of the communication aspect of saying it one way or another. However, as far as I can tell, your formal statement of the principle of induction (which is how I first saw it as well in mathematics books and classes) is not the same as the formal statement that I saw in a course on logic. It's not that they are incompatible, it's just that what I saw in logic seems more general. My question, in spirit, is asking: when we strip away the assumption that we are dealing with mathematicians who use induction all day, and we focus solely on induction as a concept – xoux May 12 '22 at 13:52
  • in isolation, and we make everything explicit, then fundamentally we are applying a technique to an inductively defined set ${x | P(x) }$. The technique involves proving that the inductively defined set is a subset of another set ${ x | Q(x) }$, and thus that $\forall x (x \in {x | P(x) } \implies x \in { x | Q{x} })$, or $\forall x (P(x) \implies Q(x))$. – xoux May 12 '22 at 13:57
  • Perhaps then you might wish to rewrite your post to explain the alternate formulation of induction to which you are referring. For example, if you saw a formal statement in a course on logic, then you should put that statement into your post. Otherwise, we are left to guess at what you might actually be asking about. – Lee Mosher May 12 '22 at 14:12
  • No, but see that's the thing. It's not that there are two types of formulations. Instead, it is the case that the one introduced in math classes is a very specific (but very commonly used) case of the general "framework". Until I read about the general working of induction I just used it for many years without having ever really thought about why exactly it is justified. – xoux May 12 '22 at 14:21
  • Then please tell us what the general framework is that you want a name for. – Lee Mosher May 12 '22 at 14:44
  • See Chapter 16 "Mathematical Induction" of the book "Language, Proof, and Logic". I said "framework", but what I meant was, something like an exposition of what induction is in a more general sense that just induction on the set of natural numbers. – xoux May 12 '22 at 15:43
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    I did mention a generalized induction in my post, namely transfinite induction which is induction with respect the general ordinal numbers. But if there's something specific from that chapter that you can transcribe into your post, that would improve your post greatly. Otherwise, you're likely to just get the same answer that you've already gotten in the comments and the posted answers. – Lee Mosher May 12 '22 at 16:08