Consider the above statement. If it were true, then what it says is the case; but what it says is that it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.
Thus the truth value of the statement is undecidable. More embarrassingly, there is no method/process/algorithm/procedure to determine whether or not such statement are true or false.
This is called the liar’s paradox. I used it in the previous post. We couldn’t decide whether or not the new binary number we created was in the set of all infinitely long binary numbers. Since we could construct such a number but not count it, we were forced to accept that there are more infinitely long binary numbers than there are natural numbers.
(To be continued…)