I’ve spent a lot of time in the past describing the paradoxes that arise when we try to involve infinities into our reasoning. Another way to get paradoxes aplenty is by invoking self-reference. Below are three of the best paradoxes of self-reference for you to sort out.
In each case, I want you to avoid the temptation to just say “Ah, it’s just self-reference that’s causing the problem” and feel that the paradox is thus resolved. After all, there are plenty of benign cases of self-reference. Self-modifying code, flip-flops in computing hardware, breaking the fourth wall in movies, and feeding a function as input to itself are all examples. Self-referential definitions in mathematics are also often unobjectionable: as an example, we can define the min function by saying y = min(X) iff y is in X and for all elements x of X, y ≤ x (the definition quantifies over a group of objects that includes the thing being defined). So if we accept that self-reference is not necessarily paradoxical (just as infinity is sometimes entirely benign), then we must do more to resolve the below paradoxes than just say “self-reference.”
1. Berry’s Paradox
Consider the set of integers definable in an English sentence of under eighty letters. This set is finite, because there are only a finite number of possible strings of English characters of under eighty letters. So since this set is finite and there are an infinity of integers, there must be a smallest integer that’s not in the set.
But hold on: “The smallest positive integer not definable in under eighty letters” appears to now define this integer, and it does so with only 67 letters! So now it appears that there is no smallest positive integer not definable in under eighty letters. And that means that our set cannot be finite! But of course, the cardinality of the set of strings cannot be less than the cardinality of the set of numbers those strings describe. So what’s going on here?
2. Curry’s paradox
“If this sentence is true, then time is infinite.”
Curry’s paradox tells us that just from the existence of this sentence (assuming nothing about its truth value), we can prove that time is infinite.
Let’s call this sentence P. We can then rewrite P as “If P is true, then time is infinite.” Now, let’s suppose that the sentence P is true. That means the following:
Under the supposition that P is true, it’s true that “If P is true, then time is infinite.”
And under the supposition that P is true, P is true.
So under the supposition that P is true, time is infinite (by modus ponens within the supposition).
But this conclusion we’ve just reached is just the same thing as P itself! So we’ve proven that P is true.
And therefore, since P is true and “If P is true, then time is infinite” is true, time must be infinite!
If you’re suspicious of this proof, here’s another:
If P is false, then it’s false that “If P is true then time is infinite.” But the only way that sentence can be false is if the antecedent is true and the consequent false, i.e. P is true and time is finite. So from P’s falsity, we’ve concluded P’s truth. Contradiction, so P must be true.
Now, if P is true, then it’s true that “If P is true, then time is infinite”. But then by modus ponens, time must be infinite.
Nothing in our argument relied on time being infinite or finite, so we could just as easily substitute “every number is prime” for “time is infinite”, or anything we wanted. And so it appears that we’ve found a way to prove the truth any sentence! Importantly, our conclusion doesn’t rest on the assumption of the truth of the sentence we started with! All it requires is the *existence of the sentence*. Is this a proof of the inconsistency of propositional logic? And if not, then where exactly have we gone wrong?
3. Curry’s paradox, Variant
Consider the following two sentences:
1) At least one of these two sentences is false.
2) Not all numbers are prime.
Suppose that (1) is false. Well then at least one of the two sentences is false, which makes (1) true! This is a contradiction, so (1) must be true.
Since (1) is true, at least one of the two sentences must be false. But since we already know that (1) is true, (2) must be false. Which means that all numbers are prime!
Just like last time, the structure of the argument is identical no matter what we put in place of premise 2, so we’ve found a way to disprove any statement! And again, we didn’t need to start out by assuming anything about the truth values of sentences (1) and (2), besides that they have truth values.
Perhaps the right thing to say, then, is that we cannot always be sure that self-referential statements actually have truth values. But then we have to answer the question of how we are to distinguish between self-referential statements that are truth-apt and those that are not! And that seems very non-trivial. Consider the following slight modification:
1) Both of these two sentences are true.
2) Not all numbers are prime.
Now we can just say that both (1) and (2) are true, and there’s no problem! And this seems quite reasonable; (1) is certainly a meaningful sentence, and it seems clear what the conditions for its truth would be. So what’s the difference in the case of our original example?