What is the problem in this Proof

Amer

Active member
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

Bacterius

Well-known member
MHB Math Helper
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

[JUSTIFY]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 [/JUSTIFY]

Evgeny.Makarov

Well-known member
MHB Math Scholar
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

Well-known member
MHB Math Scholar
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