“This sentence is false.”
If this statement is true, then what it states must be true. It states that it is false. So, if we assume the statement is true we conclude it must be false. If the statement is false, then what it states should be false, but it correctly states that it is false. So, it is true. Thus, if we assume the statement is false, we conclude that it must be true. The statement appears to be neither true nor false and yet it must be either true or false. That is a paradox.
Now consider another statement: “This statement is true.” Is this true or false? Could it be both? Be careful with self-referential statements, they are tricky and can prove anything. The following statement can never be false, “I’m the hottest Brazilian in the world or this statement is false.” Think about it. I have just proved I’m the hottest Brazilian alive!