Determine which of the following statements is true.
- Exactly one of these statements is false.
- Exactly two of these statements are false.
- Exactly three of these statements are false.
- Exactly four of these statements are false.
- Exactly five of these statements are false.
The first step to a solution is to notice that these statements are incompatible with each other, so at most one of them can be true. Then, if we feel like being methodical, we can work through the statements in turn: if the first statement is true then exactly four statements are true (which is impossible); if the second statement is true… and so on, until we conclude that the fourth statement is true, and “exactly four of these statements are false”.
That immediately suggests a quicker solution: rewrite the statements as claims about truth rather than falsehood.
- Exactly four of these statements are true.
- Exactly three of these statements are true.
- Exactly two of these statements are true.
- Exactly one of these statements is true.
- None of these statements is true.
It’s immediately clear that the fourth statement is the only one that can consistently be true. It’s also clear that in the obvious generalisation to n statements, it is the (n-1)th statement, “Exactly one of these statements is true”, that will be true.
If we take the case , we run into difficulties. We now have a single statement: “None of these statements is true”, or more succinctly, “This statement is false”. This is the famous Liar Paradox, which has been worrying logicians for a couple of millennia — because although it looks like a silly wee novelty in itself, if we want to construct a language in which to perform logic we can’t be quite content if that language can express statements like the liar paradox. That’s a road which leads us from Saint Jerome via Bertrand Russell to Alfred Tarski and beyond…
How about the limit ? There is certainly no problem constructing the set of statements: statement i is “Exactly i of these statements are false.” But now we can’t turn these into statements about truth, because any finite number of false statements still leaves an infinite number of true statements… no two of which can both be true. Joyner’s problem therefore has a well-defined solution for any finite set of statements, but not for an infinite set.
In fact, there are whole families of unexpected results that come about when we try to go from finite to infinite problems. Perhaps the most notorious is Hilbert’s Hotel: if you haven’t heard of it then a few minutes spent following the link will give your credulity a nice work-out and your brain a nice introduction to the concept of countable sets…