What is the problem in this Proof

[Amer]

In your point of view what is the problem in this Proof
Claim any two natural a,b are equal
By induction
Let m= max{a,b}
if m=1 then a=b=1 since a,b natural
suppose it is hold for m=k
if
max{a,b} = k then a=b
test if
max{a,b} = k+1 , sub 1
max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

I saw it in facebook

 

[Nono713]

In your point of view what is the problem in this Proof
Claim any two natural a,b are equal
By induction
Let m= max{a,b}
if m=1 then a=b=1 since a,b natural
suppose it is hold for m=k
if
max{a,b} = k then a=b
test if
max{a,b} = k+1 , sub 1
max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

I saw it in facebook

I saw it on math.stackexchange too. The problem is that $\max{(a, b)} = k ~ \implies ~ a = b$ is clearly wrong, and the proof was designed to hide this fact. For instance, $\max{(5, 7)} = 7$, but last time I checked we had $5 \ne 7$.

In essence, this "proof" doesn't show that all natural numbers are equal, it shows that any two equal natural numbers are equal ;)​

 

[Evgeny.Makarov]

The problem is that $\max{(a, b)} = k ~ \implies ~ a = b$ is clearly wrong, and the proof was designed to hide this fact. For instance, $\max{(5, 7)} = 7$, but last time I checked we had $5 \ne 7$.

But this does not explain which proof step in particular is wrong. Of course the implication max(a, b) = k ⇒ a = b is false, just like the original claim that a = b for all a, b. But the proof claims to show just that, and the question is where the mistake in the proof is located.

 

[caffeinemachine]

In your point of view what is the problem in this Proof
Claim any two natural a,b are equal
By induction
Let m= max{a,b}
if m=1 then a=b=1 since a,b natural
suppose it is hold for m=k
if
max{a,b} = k then a=b
test if
max{a,b} = k+1 , sub 1
max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

I saw it in facebook

The proof goes wrong in the last two sentences.

We have max{a,b}=k+1

Now we have max{a-1,b-1}=k.

We now want to apply the induction hypothesis here to have a-1=b-1 and thus a=b. But we can't do this. This is because we are not sure if a-1 and b-1 are natural numbers. We can very well have one of a-1 and b-1 equal to 0.

So this is the problem in the proof.