Is there a way to prove this wrong?

Question

We have two statements:

1. 0 = 1
2. Both statements are false

If the second statement is true, then it is false. We've come to a contradiction.

If the second statement is false, then at least one of the statements is true. The second one is false, therefore, the first one is true. We've proven that 0 = 1.

How is that possible and is there a way to prove this is wrong?

Answer 1

You can simplify the example to a single "this statement is not true". It just shows you that you cannot expect statements to talk about themselves and still have meaningful truth values.

That's why one has to be so careful in constructing Set Theory, and the bare naive approach doesn't work.

Answer 2

A statement is not allowed to be self-referencing. See for example this Wikipedia article:

Another statement that people often ask about is "This statement is false". Again this is not a "valid" statement that you can call true or false because it references itself.