Find a Mistake in Proof: 0,1,2,3 are All Even

[Demonoid]

Basically I need to find a mistake in this "proof".

I claim that 0,1,2,3...are all even.

I will use induction to prove that 'n is even' for n = 0,1,2,3...
Base case is n = 0, which is true, 0 is even. I assume that the statement is true for
n = 0,1,2,3...,k and consider n = k+1. By assumption, 1 and k are both even, and thus k+1 is even as well. This means that n = 0,1,2,3... are all even.

I can't seem to find a hole in the proof. I know that 1 is not even and when we add 1 to an even number, we get an odd number. But, by assumption 1 is even, so, what do I do know ?

 

[LCKurtz]

Try your argument when getting from k = 0 to k = 1.

 

[disregardthat]

If k is even, then k+1 is odd...

 

[Suk-Sci]

There is nothing wrong in you proof...wat you are doing is dat you are assigning a subscript for each even number so A0=0, A1=2...so by induction you r getting this peculiar result which is for the subscript...

 

[spamiam]

As LCKurtz said, the problem is between 0 and 1. You seem to be tacitly assuming that k is greater than 1 when you say that 1 is even by the inductive hypothesis. But for k=0, this isn't true since k+1=1.

 

[LCKurtz]

As LCKurtz said, the problem is between 0 and 1. You seem to be tacitly assuming that k is greater than 1 when you say that 1 is even by the inductive hypothesis. But for k=0, this isn't true since k+1=1.

More accurately, his argument fails because he can't choose 1 and k > 1 to add together in the induction step because there is no such k.

The real lesson in this example is that in induction arguments, the induction step must be independent of the value of k, which it isn't in this argument.