Well of course 1+1 != 0 in classical arithmetic but here is a proof for the contrary

This is false. There is a special "proof" that people have come up with, but it sneakily utilizes a false rule. It surreptitiously divides by zero. When you divide by zero, any thing's possible. Here's the proof.

We'll start with 2 variables: A and B
Let A=1,
B=1
A=B // Now multiply by 1... both A and B are one.
A^2=AB // Now subtract B^2
A^2-B^2=AB-B^2 // Now factor both sides.
(A-B)(A+B)=B(A-B) // Now divide by A-B****
A+B=B // Now sub in 1 for A and 1 for B
1+1=1
2=1 // subtract one from both sides
1=0 // multiply both sides by 2
2=0
1+1=0
****The problem is that we divide by (A-B)... but A-B=1-1=0...uh oh... we divided by zero. And when we divide by zero... anything's possible, but completely wrong!