LECTURE NOTES ON MATHEMATICAL INDUCTION
PETE L. CLARK
2. The (Pedagogically) First Induction Proof
3. The (Historically) First(?) Induction Proof
4. Closed Form Identities
5. More on Power Sums
7. Extending binary properties to n-ary properties
9. The Principle of Strong/Complete Induction
10. Solving Homogeneous Linear Recurrences
11. The Well-Ordering Principle
12. Upward-Downward Induction
13. The Fundamental Theorem of Arithmetic
13.1. Euclid’s Lemma and the Fundamental Theorem of Arithmetic 13.2. Rogers’ Inductive Proof of Euclid’s Lemma
13.3. The Lindemann-Zermelo Inductive Proof of FTA
Principle of Mathematical Induction for sets
Let S be a subset of the positive integers. Suppose that:
(i) 1 ∈ S, and
(ii) ∀ n ∈ Z+ , n ∈ S =⇒ n + 1 ∈ S.
Then S = Z+ .
The intuitive justiﬁcation is as follows: by (i), we know that 1 ∈ S. Now apply (ii) with n = 1: since 1 ∈ S, we deduce 1 + 1 = 2 ∈ S. Now apply (ii) with n = 2: since 2 ∈ S, we deduce 2 + 1 = 3 ∈ S. Now apply (ii) with n = 3: since 3 ∈ S, we deduce 3 + 1 = 4 ∈ S. And so forth.
This is not a proof. (No good proof uses “and so forth” to gloss over a key point!) But the idea is as follows: we can keep iterating the above argument as many times as we want, deducing at each stage that since S contains the natural number which is one greater than the last natural number we showed that it contained. Now it is a fundamental part of the structure of the positive integers that every positive 1
PETE L. CLARK
integer can be reached in this way, i.e., starting from 1 and adding 1 suﬃciently many times. In other words, any rigorous deﬁnition of the natural numbers (for instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical induction. Alternately, the principle of mathematical induction is a key ingredient in any axiomatic characterization of the natural numbers. It is not a key point, but it is somewhat interesting, so let us be a bit more speciﬁc. In Euclidean geometry one studies points, lines, planes and so forth, but one does not start by saying what sort of object the Euclidean plane “really is”. (At least this is how Euclidean geometry has been approached for more than a hundred years. Euclid himself gave such “deﬁnitions” as: “A point is that which has position but not dimensions.” “A line is breadth without depth.” In the 19th century it was recognized that these are descriptions rather than deﬁnitions, in the same way that many dictionary deﬁnitions are actually descriptions: “cat: A small carnivorous mammal domesticated since early times as a catcher of rats and mice and as a pet and existing in several distinctive breeds and varieties.” This helps you if you are already familiar with the animal but not the word, but if you have never seen a cat before this deﬁnition would certainly not allow you to determine with certainty whether any particular animal you encountered was a cat, and still less would it allow you to reason abstractly about the cat concept or “prove theorems about cats.”) Rather “point”, “line”, “plane” and so forth are taken as undeﬁned terms. They are related by certain axioms, or abstract properties that they must satisfy.
In 1889, the Italian mathematician and proto-logician Gisueppe Peano came up with a similar (and, in fact, much simpler) system of axioms for the natural numbers. In slightly modernized form, this goes as follows: The undeﬁned terms are zero, number and successor.
There are ﬁve axioms that they must satisfy, the Peano axioms. The ﬁrst four are: (P1)
Zero is a number.
Every number has a successor, which is also a number.
No two distinct numbers have the same successor.
Zero is not the successor of any number.
References: [Ac00] F. Acerbi, Plato: Parmenides 149a7-c3. A Proof by Complete Induction? Archive for
History of the Exact Sciences 55 (2000), 57–76.
[Li33] F.A Lindemann, The Unique Factorization of a Positive Integer. Quart. J. Math. 4, 319–
[Mu63] A.A. Mullin, Recursive function theory (A modern look at a Euclidean idea). Bulletin of
the American Mathematical Society 69 (1963), 737.
[Ro63] K. Rogers, Classroom Notes: Unique Factorization. Amer. Math. Monthly 70 (1963), no.
[Ze34] E. Zermelo, Elementare Betrachtungen zur Theorie der Primzahlen. Nachr. Gesellsch. Wissensch. G¨
ottingen 1, 43–46, 1934.
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