Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I can't understand what is wrong with this paradox. How we should strictly mathematically explain it?

Mathematical induction:

1. The basis:

$n=1,n=2$. Through any two (one) points on a plane we can draw a straight line.

2. The inductive step:

$n=k$. Through any $k$ distinct points on a plane we can draw a straight line.

3. Fake-Paradox:

We have an arbitrary $k+1$ points on the plane: $P_1, P_2, ..., P_{k+1}$. From $2)$ (inductive step) we can draw a straight line $L_1$ through $k$ points $P_1, P_2, ..., P_{k}$ and line $L_2$ through $k$ points $P_2, ..., P_k, P_{k+1}$. Lines $L_1$ and $L_2$ have at least two common points $P_2$ and $P_k$. But any two distinct points of a straight line completely determine that line $\Rightarrow L_1=L_2$ and $P_{k+1} \in L_1$. And we prove that

Through any $n$ distinct points on a plane we can draw a straight line.

share|improve this question
    
Hint: Whenever someone gives you a set $\{a, b\}$ and claims it has two elements, you should frown and check whether $a\ne b$ is true. –  Hagen von Eitzen Sep 14 '12 at 21:42
    
Thanks, I get it from Clive Newstead answer. It is so clear, i'm confused i didn't see that. –  Mike Sep 14 '12 at 21:47
    
Three Points Theorem: Through any three points in the plane one can draw a straight line.* The extension to N points is left to the reader. This result is attributed to the famous Greek mathematician Testicles in the Nth century BC. It is easily demonstrated on an old-fashioned chalkboard by turning the chalk sideways. * Provided the pencil is thick enough. –  user63541 Feb 22 '13 at 15:16
    
As a remark: this is a variant of the "All horses have the same colour" paradox often attributed to Pólya –  Willie Wong Feb 22 '13 at 15:54
add comment

3 Answers 3

up vote 5 down vote accepted

Since you've tagged this as homework I won't just give you the answer, but think about this: your stage 3. relies on $P_2 \ne P_k$. When might this not happen?

Edit (in response to comment): Let $\phi$ be a statement about natural numbers; we say $\phi(n)$ if $\phi$ holds for some particular $n$. Let $n_0 \in \mathbb{N}$. The principle of mathematical induction states that $\phi(n)$ holds for all $n \ge n_0$ if and only if both of the following conditions are satisfied:

  • $\phi(n_0)$ holds; and
  • For each $n \ge n_0$, if $\phi(k)$ holds then $\phi(k+1)$ holds.

Here we can set $n_0=2$. But the second condition is not satisfied, since if it were to be satisfied we'd need $\phi(2) \Rightarrow \phi(3)$, but as shown this is not the case since the argument that would make it so would require $P_2 \ne P_2$.

share|improve this answer
    
I need an example of strictly mathematical explanation. And i tagged this question as homework, just for more attantion. We learned a lot of such fake-paradox (by math.induction) and for every them, our teacher says like all this answers, hints; it is clear that step from n=2 doesn't work to n=3. I just want to see one strictly full explanation. Thanks. –  Mike Sep 14 '12 at 21:26
    
@Mike: Okay, I've edited my answer. –  Clive Newstead Sep 14 '12 at 21:32
    
Just so simple? I'm confused. –  Mike Sep 14 '12 at 21:35
    
@Mike: Sorry, I don't know what you're asking for. –  Clive Newstead Sep 14 '12 at 21:52
    
I mean that I'm ashamed. It was very clear to understand by myself. –  Mike Sep 14 '12 at 21:55
add comment

The argument in the induction step doesn’t work in going from $n=2$ to $n=3$: the two lines aren’t forced to be the same line. Try drawing a picture of such an example.

share|improve this answer
add comment

Hint: We know that through any 2 points in the plane we can draw a straight line, but that is not true for any 3 points. Thus, you should focus your attention on the argument that the claim is true for $n=2$ implies that it is true for $n=3$. Does the argument in 3. (Fake Pair Of Ducks) work for $k=2$ (3 points)?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.