A main reason why we use proof by induction is to show somet

A main reason why we use proof by induction is to show something is true about an infinite set.

(1) Why is the following “proof” incorrect?

Claim. The Earth is flat.

“Proof ”. When I stand in my office and look out the window, the Earth is flat. When I stand outside the library and look around, the Earth is flat. In fact, I can stand at n number of points, for any n, anywhere on the Earth and at each one of those points, the Earth is flat. Since the Earth is flat at every point, the Earth must be flat. (2) How do proofs by induction avoid the problem in part (1)?

Solution

Mathematical induction is used to prove the validity of a mathematical statement (of quantitative not qualitative nature) involving natural numbers, i.e., if there is a statement in the form of an equation or inequality, mathematical induction is used to prove that its an identity i.e., it holds for all natural numbers.

\"Earth is flat\" can only be expressed Mathematically by giving an equation for a plane relative to some frame of reference. In that case the consecutive statements given in problem (1) would indicate that at each of the n points you choose to observe the earth, the equation of the plane representing the earth remains same, provided the frame of referece is fixed and is situated outside the earth (offcourse). If that be the case, the statement would have been true for all n, i.e., the Earth would have been flat mathematically.

The basic idea behind Mathematical induction lies in the fundamental assumptions of Mathematics which says that the kind of relations defined for natural numbers would be considered valid, if they follow such an order as used in the proof by mathematical induction.