Why does this inductive proof fail?

Question

I would like to spell out where the following inductive argument goes wrong, but I am not sure how to.

Goal: Show Ga₁₀

(1) Assume that Fa₁

(2) Assume that Fa_n $\to$ Ga_n+1

(3) By induction, Ga₁₀.

What makes this argument invalid?

Answer 1

The argument fails simply because it is not induction. An inductive argument is of the form

• Prove that $P(1)$ is true
• Prove that if $P(n)$ is true, then $P(n+1)$ is also true
• Conclude that $P(n)$ is true for all positive integers $n$

Your argument is not of this form.

(Here, $P(k)$ is a statement about $k$ which is unambiguously either true or false for any particular given positive integer $k$.)

Answer 2

Given a proposition $P(n)$, say you want to show that $P(n)$ is true for all $n\in\mathbb{N}$. Induction states that it suffices to show the following:

1. $P(1)$ is true
2. If $P(k)$ is true, then $P(k+1)$ is true

One failure of your argument is that you seem to be confused about what exactly is the proposition $P(n)$. Say you know that $F(1)\implies G(1)$, and that $F(k)\implies G(k)$. We cannot extrapolate to say that $G(k)\implies G(k+1)$. This seems to be the form of your argument.

In your example from the comments, say you assume ball $1$ to be red and ball $2$ to be orange. Also assume that if ball $k$ is red, then ball $k+1$ is orange. With this information, what color is ball $3$? Is it red? Orange? We don't know - we don't have enough information to make any statements about the color.