All mathematical facts are unmovable and absolute. The only way an appearance of change may come is from redefining concepts. For example, one may redefine plus so that x + y is always 1, regardless of x and y. Properly speaking, this is an abuse of notation because we did not bother to come up with a new symbol and instead reuse the same +, which has a different meaning in a different context. This is why it may seem that 1 + 1 = 2 is not true anymore.

To make more mathematical sense, we can consider the set {0, 1} instead of all natural numbers and define x + y = 0 if x = y = 0 or x = y = 1; otherwise, x + y = 1, by definition. This newly defined operation has all familiar properties of +, including 0 + x = x, x + y = y + x, (x + y) + z = x + (y + z), etc.