Here I want to present a novel version of a paradox first formulated by José Bernardete in the 1960s – one that makes its connections to the Yablo paradox explicit by building in the latter puzzle as a ‘part’. This is not the first time connections between Yablo’s and Bernardete’s puzzles have been noted (in fact, Yablo himself has discussed such links). But the version given below makes these connections particularly explicit.

First, we should look at Bernardete’s original. Imagine that Alice is walking towards a point – call it A – and will continue walking past A unless something prevents her from progressing further. There is also an infinite series of gods, which we shall call G_{1}, G_{2}, G_{3}, and so on. Each god in the series intends to erect a magical barrier preventing Alice from progressing further if Alice reaches a certain point (and each god will do nothing otherwise):

(1) G_{1} will erect a barrier at exactly ½ meter past A if Alice reaches that point.

(2) G_{2} will erect a barrier at exactly ¼ meter past A if Alice reaches that point.

(3) G_{3} will erect a barrier at exactly ^{1}/_{8} meter past A if Alice reaches that point.

And so on.

Note that the possible barriers get arbitrarily close to A. Now, what happens when Alice approaches A?

Alice’s forward progress will be mysteriously halted at A, but no barriers will have been erected by any of the gods, and so there is no explanation for Alice’s inability to move forward (other than the un-acted-on intentions of the gods, which isn’t much of an explanation). Proof: Imagine that Alice did travel past A. Then she would have had to go some finite distance past A. But, for any such distance, there is a god far enough along in the list who would have thrown up a barrier before Alice reached that point. So Alice can’t reach that point after all. Thus, Alice has to halt at A. But since Alice doesn’t travel past A, none of the gods actually do anything.

Now let’s change the puzzle a bit. Imagine that Alice is an expert logician enjoying her morning walk (which, as usual, passes through point A). Alice will continue walking unless she hears someone utter a paradoxical sentence or set of sentences. Hearing a paradox is paralyzing to Alice, however. Upon hearing such a thing, she will instantly stop in her tracks (and she is able to detect paradoxes instantaneously, the minute they are uttered). Finally, Alice walks in total silence, never uttering a word.

As before, we also have an infinite series of gods G_{1}, G_{2}, G_{3}, … and each god intends to act in a particular way if Alice reaches a certain point on the path past A. But now they are not erecting barriers, but are instead merely making utterances:

(1) G_{1} will say:

“Everything the other gods have said so far is false.”

if Alice makes it ½ meter past A.

(2) G_{2} will say:

“Everything the other gods have said so far is false.”

if Alice makes it ¼ meter past A.

(3) G_{3} will say:

“Everything the other gods have said so far is false.”

if Alice makes it ^{1}/_{8} meter past A.

And so on.

In short, each god in the series will accuse all of the other gods who have already spoken of being liars, if Alice makes it far enough. Now, what happens when Alice approaches A?

Again, Alice’s forward progress will be halted at A: Imagine that Alice did travel past A. Then she would have had to go some finite distance past A. But, for any such distance, there is a god far enough along in the list (in fact, infinitely many of them) who would have said:

“Everything the other gods have said so far is false.”

before Alice reached that point. Let G_{m} be any one of the gods whose point Alice has passed. Notice that if Alice passed god G_{m}, then she also passed all of G_{m+1}, G_{m+2}, G_{m+3}… Now, G_{m} uttered:

“Everything the other gods have said so far is false.”

when Alice passed the appropriate point (that is, when Alice has reached ^{1}/(_{2}^{m}) meters past A). But before that each of the gods whose number is greater than m (i.e. G_{m+1}, G_{m+2}, G_{m+3},…) will have already said the same thing about the gods who spoke before them. As a result, G_{m}’s utterance can be neither true nor false.

Assume that G_{m}’s utterance is true. G_{m}’s utterance amounts to his saying that each of G_{m+1}, G_{m+2}, G_{m+3},… was lying when they made their respective utterances. So each of G_{m+1}, G_{m+2}, G_{m+3},… must in fact be lying. But then each of G_{m+2}, G_{m+3}, G_{m+4},… must be lying. But G_{m+1}’s assertion that:

“Everything the other gods have said so far is false.”

is equivalent to saying that each of G_{m+2}, G_{m+3}, G_{m+4},… is lying. So G_{m+1} is telling the truth. Contradiction, so G_{m} cannot be telling the truth.

Thus, G_{m} utterance must be false. But we can run the same argument given in the previous paragraph on G_{m+1}, G_{m+2}, G_{m+3},… just as easily as on G_{m} (after all, if Alice passed the point at which G_{m} makes his utterance, then she also passed all the points corresponding to G_{m+1}, G_{m+2}, G_{m+3},…). Thus, all of G_{m+1}, G_{m+2}, G_{m+3},… are lying as well. But then G_{m}’s assertion that:

“Everything the other gods have said so far is false.”

is true after all. Contradiction again.

Note: The reader who finds the previous two paragraphs difficult may want to consult my previous discussion of the Yablo paradox.

Thus, Alice cannot walk any distance past A, no matter how short, since doing so would mean she would have to pass a point at which a paradox had already been uttered. So she halts when she reaches A. But, since she doesn’t pass A, no one (neither Alice nor any of the gods) has said anything. So what, exactly, stopped Alice?

It seems as though we’re discounting the existence of a barrier at A, but why?

If I have two points A=0 and B=1 meter away, and I’m a ball starting at A who has a god, g1, who moves me forward 9/10 meter to A, putting me at 0.9 meters, and a god, g2, who moves me forward an additional 9/100 of a meter to 0.99 meters, then given an infinite number of gods, labeled g_i for the ith god, each one moving me an additional 9/10^i meters, have I reached point B after all infinite gods have had their way? Yes. 0.999… = 1, and I’m exactly at B.

The reason the Bernardete paradox seems like a paradox is the given of infinite gods. Like above, the action of an infinite number of entities is not necessarily intuitive, though the above example I gave is easier to digest due to the construction of the situation. The action of all gods implies that there erects a barrier exactly at A to prevent me from moving further. It’s hard to elaborate in a comment, but to think it over, consider the aspects of how this situation must manifest. At any given point in time or any given position Alice occupies, every single god must parse their simple conditional of “is Alice at this point, then construct a barrier” to which after an infinite number have done so, her existence at A creates the last barrier at the “infinitesimally” small location which happens to be exactly at A.

I just wanted to add to the above that Yablo’s paradox is legitimately a paradox, but one in a manner that isn’t inherently due to its infinite nature but rather its similarity to the liar paradox and sentences predicating the truth of other sentences, whereas the paradox perceived in the Bernardete Paradox is based on not understanding infinite processes.

I’m undecided on the married version you’ve put forward here. :)

[…] Inspired by Roy Cook’s brief but excellent article: https://blog.oup.com/2015/07/yablo-bernardete-paradox/ […]

The Bernardete Paradox is the result of believing, without justification, that Alice can only be stopped by a wall. Sure, a wall stops Alice, but it does not stand to reason that only a wall stops Alice. In this case, we showed that Alice can be stopped by an infinite collection of hypothetical walls as well — in fact, a proof to this effect.

If this seems strange, it is only because it instantaneous wall erection and stopping are both totally unphysical to begin with and already strange. A realistic model of causal wall erection and stopping would necessarily require an epsilon distance between Alice and the erection of a wall, to allow Alice to “stop” before already having moved past the point of the wall. With such an epsilon distance, there is clearly no strangeness, as there would have been walls erected by the time Alice reaches A.