The opening of the film presents a simple thought experiment in a not-so-simple and enjoyably quirky way:
What do you think? If you're anything like me, the first answer that comes to your mind is an obvious "no". The axe has been completely re-constructed, so the original parts that were used to behead the now-rotting, corpsified zombie-dude are no longer in your possession and are most likely just adding to the mass of waste at some local landfill.
However, that's not the reason that I think the answer of "no" is astoundingly obvious (you might not have caught the primary reason for the answer being "no" on your first watch of that video; if so, watch it again. Good to go? Awesome). Hopefully you saw that the primary reason that the answer to the question is "no" is because what slew Swastika-Tongue in the first place was one, some, or all of the eight bullets that you had shot him with before using the axe to remove his head (like I said, it's a simple riddle). However, if you take away the obvious answer and just allow yourself to assume that the real question of the riddle is whether or not the axe you now hold in your hand in the presence of Zombie-Swastika-Tongue is the same one that you had used the previous winter to remove the head of his former self, then you have another riddle that is really a re-hash of a much older thought experiment: The Ship of Theseus.
The Ship of Theseus is a thought experiment proposed by Plutarch in the first century C.E. It goes something like this: the ship in which the hero Theseus and the young Athenian men returned from Crete (see the myth of Theseus and the Minotaur) was honored by Athenians and kept in good repair in the harbor of Athens for many centuries. Over time, as parts of the ship would degrade, they were slowly replaced, so eventually there were not many of the original parts of the ship remaining. The question then became "Is the ship, after replacing part for part over time, still the Ship of Theseus?"
The thought experiment, as it stands, really questions the value an object has based upon its parts. It's a question of philosophical identity.
There are other versions of this thought experiment. Some of them predate Plutarch's Ship of Theseus. For instance, there is a version in which Socrates and Plato each slowly exchanged the parts of their carriages such that the parts that once were in Plato's carriage have been completely replaced with parts from Socrates' carriage and vice versa and then the question is posited as to whether Plato is now using Socrates' carriage or if he's still in his own.
Other variants of the thought experiment have come since the time of Plutarch (with some interesting additions). The version that appears most similar to the prologue from John Dies at the End is the one known as "My Grandfather's Axe": my grandfather had an axe which he gave to my father. My father replaced the haft before giving the axe to me. I had to replace the head. Do I still have my grandfather's axe?
One of the more interesting variants of this thought experiment was proposed by Thomas Hobbes, the 1700's English philosopher and author of Leviathan. Hobbes' addition to the thought experiment works this way: you have the Ship of Theseus. You slowly take one piece of the ship off and replace it with a new piece. The old piece you keep. You continue in this manner, replacing pieces of the ship and saving the removed pieces. As this is happening over time, you take the pieces that had been removed and use those pieces to build a new ship, of exactly the same structure and design. By the time you have replaced the final original piece of the ship, you now have two identical ships. Which one is the Ship of Theseus?
Here's a fantastic breakdown of the original Ship of Theseus thought experiment and the Hobbes version from Wireless Philosophy:
The thesis of Joseph Butler, as reviewed in that video and suggesting that "objects persist in only a loose and popular sense", seems like a nice way to shrug off the problem as not being a problem in the first place. This is usually a fun approach to a lot of philosophy problems since a lot of the time it seems like there's no resolution to a lot of philosophy problems.
The reason I like this thought experiment, be it after replacing axe parts following your unexplained need to slay and behead some dude with a swastika tattooed on his tongue or replacing pieces of Theseus' ship, is because it questions identity. We are constantly shedding cells and gaining new ones, so are we ever identical with who we were previously? Darth Vader was almost fully replaced by mechanical parts, so was he still Anakin Skywalker? The philosopher Wittgenstein might have thought these questions were balderdash ("Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing") and, if that were the case, he might have been right.
It would be interesting if we could just say that something is such because people agree to call it such. Maybe the Ship of Theseus is really just whatever anyone decides to call "the Ship of Theseus". Maybe Darth Vader is Anakin Skywalker because someone calls him so. If that were the case, then the answer to the thought experiment as proposed in John Dies at the End might be that the axe you're holding in your hand is the same axe as the one that beheaded Swastika-Tongue because his zombie has now said that it is the same axe (it might be a good conclusion since chances are you should be more worried about dealing with said zombie before considering philosophical puzzles anyway).
However, I still feel like the answer would be "no". Even if you had killed the dude with an axe in the first place (and not with one, some, or all of those eight bullets), the original axe has been completely replaced. The answer feels like "no" because none of the original axe remains and there are only two major parts of the axe to replace. When the problem is introduced as in the case of the Ship of Theseus, where the object is replaced a small amount at a time, that's when it gets harder to decide when to even consider the ship to no longer be the original.
Maybe one of the more interesting answers comes from those who like to add the temporal dimension to the consideration, such as in the Worm Theory as presented in the video above. When we question the temporal aspect of an object along with it's identity, we start hinting at a possible answer to the question (see Temporal Parts at Stanford Encyclopedia of Philosophy). We can say that any one thing is only ever fully identical to itself at one point in time, but then at other points in time it can only be similar to itself. Then using a name to define something falls back to the "loose and popular" context that Butler suggested. That sounds just about right, honestly. The answer that would suggest then is that the Ship of Theseus was only the same ship in the sense that it bore similarity to itself over time and that people still called it the Ship of Theseus is the only thing that made it the Ship of Theseus. In that case, the axe that hewed the head laden with a swastika-marked tongue and the axe that you now have to defend yourself against the zombie at the door are only similar, and maybe you would call it a different axe since you know you've replaced the parts but the zombie calls it the same axe since it looks similar to the original. Not a very rewarding answer, but an answer nonetheless (and now you can get on with hacking down the zombie as he is more than likely about to come at you).
There's rich food for thought there. Maybe Wittgenstein is right and it's nonsense to even worry about two things being identical. That seems to fit well with the answer that considers the temporal aspect to mean that an object has unique temporal parts during its existence (look up perdurantism). Whatever anyone's consideration of this little thought experiment may be, I think we can all agree that it's a lucky thing we don't truly live in a world where a guy can get shot 8 times and have his head chopped off with an axe but then still find a way to come back from death and then sew his head back on before coming to find us with the likely intent of exacting revenge.