Yablo's paradox
Yablo's paradox is like a game where you start with a list of sentences numbered 1, 2, 3 and so on, and each sentence is either true or false.
Here's how the game works. You start by looking at sentence 1. If it's true, you write down "sentence 1 is true" on a new list. If it's false, you don't write it down. Then you move on to sentence 2: if it's true, you write down "sentence 2 is true" on the new list; if it's false, you don't write anything down.
You keep going like this, looking at each sentence in turn and writing down "sentence N is true" if it's true, and not writing anything down if it's false.
But here's the tricky part: Yablo's paradox asks whether it's possible for the new list to be empty, even though every sentence is either true or false.
It turns out that the answer to this question is "yes" – it is indeed possible for the new list to be empty. Here's why: suppose that sentence 1 is true, sentence 2 is false, sentence 3 is true, sentence 4 is false, and so on, with the truth values alternating in this way. In this case, the new list will be empty, because every second sentence will be false (and so won't be written down), and the others will be true (but these are "blocked" by the false sentences that came before them).
In other words, Yablo's paradox shows that even though every sentence is either true or false, it's not always possible to say something true about each sentence, because the truth of one sentence may depend on the truth of another sentence. This is a bit like a game of dominos, where the fall of one piece depends on the fall of others.
Here's how the game works. You start by looking at sentence 1. If it's true, you write down "sentence 1 is true" on a new list. If it's false, you don't write it down. Then you move on to sentence 2: if it's true, you write down "sentence 2 is true" on the new list; if it's false, you don't write anything down.
You keep going like this, looking at each sentence in turn and writing down "sentence N is true" if it's true, and not writing anything down if it's false.
But here's the tricky part: Yablo's paradox asks whether it's possible for the new list to be empty, even though every sentence is either true or false.
It turns out that the answer to this question is "yes" – it is indeed possible for the new list to be empty. Here's why: suppose that sentence 1 is true, sentence 2 is false, sentence 3 is true, sentence 4 is false, and so on, with the truth values alternating in this way. In this case, the new list will be empty, because every second sentence will be false (and so won't be written down), and the others will be true (but these are "blocked" by the false sentences that came before them).
In other words, Yablo's paradox shows that even though every sentence is either true or false, it's not always possible to say something true about each sentence, because the truth of one sentence may depend on the truth of another sentence. This is a bit like a game of dominos, where the fall of one piece depends on the fall of others.