First of all, we have situational irony in the story. This is where there's a gap between what we expect to happen and what actually does happen. In By the Waters of Babylon we are surprised to find that the story is set in the future, given that everything seems so basic and primitive. Situational irony is epitomized by John's exploration of the ruins of what looks like an ancient city, but which is in fact New York.
Verbal irony, when someone says something contradictory to what they intend, can be observed in the warning from John's father not to let knowledge eat him up. What's ironic here is that this warning is accompanied by an encouragement to John to embark on a journey to the Place of the Gods. In other words, John's father warns him not to let knowledge eat him up, but in the same breath urges him to go to the Place of the Gods, which is precisely where he is likely to be "eaten up" by knowledge.
Finally, we have dramatic irony, where we have more information than the characters in the story. For instance, we know all about the technological innovations and scientific developments that are the stuff of John's dreams. But he doesn't understand the reality of them as we do.
Further Reading
To understand the answer to this question, one must understand the definitions of the three types of irony. Dramatic irony is a situation wherein the reader is keenly aware of a piece of information that the characters in the narrative are not. Situational irony is when a situation is in reality quite different than what it seems. Verbal irony is when a speaker has an expressive intention that is different from the literal meaning of the words that are spoken.
Most of what makes this story so effective owes to the dramatic irony that permeates the work. The reader becomes slowly aware that John and his people are not primitives in the traditional sense, but are actually futuristic human beings living after some sweeping and man-made near-extinction event, presumably a nuclear war. What john describes as a "city of the gods," we understand to be the ruins of a sprawling metropolis, most likely New York City.
Similarly, the situational irony of John's entire life and culture is slowly made clear to him over the course of his pilgrimage. Because he is willing to die for his curiosity, he slowly realizes that the denizens of the great dead city were not gods, but rather advanced humans who destroyed themselves.
Lastly, verbal irony does not have a strong presence in the story. Verbal irony is thought of as a type of language that is incredibly modern and often sarcastic, and more primitive human beings such as the type that are in the story are very sincere and clear with everything that they say and mean.
Dramatic irony is when the audience knows something that characters in the story do not. We as readers know...
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that John, the narrator, is describing New York City, and we know that in his visions he sees subways, airplanes, and nuclear war. John, coming from a far less technologically advanced culture, cannot understand as we do what he is seeing, and must describe it terms of his own context.
Situational irony arises when a situation is the opposite of what it seems. John and his people think gods built the big cities that are now called Dead Places, but we, as part of the civilization he is describing, know they were built by anything but gods. John will come to understand at the end that men built the cities, but he will not understand the full irony of the fact that the so-called "gods" destroyed themselves with nuclear weapons. Rather than be worshipped, our people should serve as a warning.
There is not much verbal irony in the story as John takes everything he sees with great seriousness and reverence, but there is some irony in George Washington's name being reduced to the letters ASHING, when his levelheadedness might have prevented a nuclear war and the ashes it produced.
There is definitely dramatic irony in "By the Waters of Babylon." Dramatic irony is a device that authors use to create situations where audiences know more about a situation than the characters do.
In "By the Waters of Babylon," this kind of irony begins appearing once John arrives in the Place of the Gods. We read about "god-roads," "SUBTREAS," and "ASHING." Those details allow readers to begin realizing that John is wandering around a fallen, post-apocalyptic city. Details continue to emerge about cold and hot water piping and the Grand Central Terminal. John is incredibly naive about what he is seeing, but readers eventually realize John is in is New York City. We are able to deduce that some kind of nuclear holocaust destroyed the human population, but John is completely unaware of all of this until the very end of the story. After he has his vision, then John realizes the gods were really regular people.