Talk:Proof of mathematical induction

message: I'm looking at this article, and I'm thinking that maybe this should be merged with Mathematical induction. -- Fogger 00:06, 13 September 2005 (UTC)
thread_title:  Merge? 

message: I believe I can state this proof in a way that would be easier for a reader less mathematically minded (like myself) to understand:
#Having proved that 0 has a certain property P: formally P(0)
#And, having proved that, if any natural number n has the property P then n+1 must also have this property: formally P(n)=>P(n+1)
#It is asserted that every natural number m must have this property: formally P(m)
Proof:
*Assume that there is/are some natural number(s) mˈ which does/do not have the P property: formally notP(mˈ)
*The smallest mˈ, call it mˈˈ, is not 0 (by #1)
*But then mˈˈ-1 must have the property P: formally P(mˈˈ-1)
*And mˈˈ = (mˈˈ-1)+1, so, by substitution in #2, P(mˈˈ-1)=>P((mˈˈ-1)+1), which is simply P(mˈˈ-1)=>P(mˈˈ)
*This says that since the largest natural number smaller than mˈˈ, having the property P, implies that the successor, mˈˈ must also have the property P (by #2)
*The assumption is thus contradicted, and #3 is proved.
I am sure that someone will correct any errors I have made.  Too Old 05:34, 22 April 2006 (UTC)
thread_title:  Clearer english statement 

message: A principled proof can be given by defining the natural numbers to be the carrier of the initial algebra in the class of algebras with signature
:zero : 1 → N
:succ : N → N
This defines a set with the appropriate "Peano structure" uniquely up to isomorphism. It can now be shown that for this set the proposition usually called the "axiom of induction" is actually a theorem. I've no idea if this is considered standard. --LambiamTalk 00:54, 24 July 2006 (UTC)
thread_title:  A principled proof 

message: This excerpt from Kolmogorov and Fomin may be of interest.
Theorem 4 (Mathematical induction). Given a proposition P(n) formulated for every positive integer n, suppose that
# P(1) is true;
# The validity of P(k) for all k ≤ n implies the validity of P(n+1).
Then P(n) is true for all n = 1, 2, …
Proof. Suppose P(n) fails to be true for all n = 1, 2, …, and let n1 be the smallest integer for which P(n) is false (the existence of n1 follows from the well-ordering of the positive integers). Clearly n1 > 1, so that n1−1 is a positive integer. Therefore P(n) is valid for all k ≤ n1−1 but not for n1. Contradiction! ∎
Replacing the set of all positive integers by an arbitrary well-ordered set, we get
Theorem 4′. (Transfinite induction). Given a well ordered set A, let P(a) be a proposition formulated for every element a ∈ A. Suppose that
# P(a) is true for the smallest element of A;
# The validity of P(a) for all a < a∗ implies the validity of P(a∗).
Then P(a) is true for all a ∈ A.
Proof. Suppose P(a) fails to be true for all a ∈ A. Then P(a) is false for all a in some nonempty subset A∗ ⊂ A. By the well-ordering, A∗ has a smallest element a∗. Therefore P(a) is valid for all a < a∗ but not for a∗. Contradiction! ∎
Remark. Since any set can be well-ordered, by the well-ordering theorem, transfinite induction can in principle be applied to any set M whatsoever. In practice, however, Zorn's lemma is a more useful tool, requiring only that M be partially ordered.
Both these proofs are trivial. Munkres, in Topology, mentions that the "Principle of Induction" follows from a definition of the positive integers (which he defines using "inductive sets"), but omits a proof, and only introduces transfinite induction in an exercise. Admittedly neither of these books is concentrating on foundations. (For example, K&F do not give a formal definition of the integers prior to these proofs, but are using the much broader well-ordering of sets Zermelo derived from the axiom of choice.)
Still, my feeling is that this article must do something more to justify its independent existence. For example, contrast the content here with that at Proofs of Fermat's little theorem or Proof that the sum of the reciprocals of the primes diverges. --KSmrqT 15:56, 26 July 2006 (UTC)
replies: text: In the article Well-ordering principle there is a sketch of this argument.--Pokipsy76 09:38, 27 July 2006 (UTC)
thread_title:  From ''Introductory Real Analysis''