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Why We Say "Mystery Babylon"
The Greek of Revelation 17:5 probably reads "a mystery — Babylon the Great," not "Mystery Babylon" as a compound title. But as shorthand for distinguishing the Revelation 17 Babylon from every other Babylon in Scripture, the label works.
Is "Mystery Babylon" actually her name? Probably not.
The KJV renders Revelation 17:5 like this:
And upon her forehead was a name written, MYSTERY, BABYLON THE GREAT, THE MOTHER OF HARLOTS AND ABOMINATIONS OF THE EARTH.
— Revelation 17:5 (KJV)
It looks like "Mystery" is part of the title. But the Greek μυστήριον more likely functions as a descriptor of what follows — "a mystery: Babylon the Great." John is being told that what he sees is a mystery, and the name on the forehead is "Babylon the Great." Paul uses the word the same way. At Ephesians 3:3–5 the mystery is what "was not made known to the sons of men in other generations as it has now been revealed"; at Colossians 1:26 it is "the mystery hidden for ages and generations but now revealed to his saints"; at Romans 16:25 it is "the mystery that was kept secret for long ages" (ESV). In none of those places is Paul naming something. He is flagging something previously hidden that is now being disclosed. The word marks a category of revelation, not a proper noun.
So the more precise rendering is: "a mystery — Babylon the Great." That is how the URV reads it: "and on her forehead was a name written, a mystery, Babylon The Great, The Mother Of Whores And Abominations Of The Earth" (Revelation 17:5, URV).
That said, this study uses "Mystery Babylon" freely. Revelation mentions Babylon more than once, and not every Babylon is the same entity. Calling the harlot of Revelation 17 "Mystery Babylon" keeps her distinct from the Jerusalem-Babylon of 11:8 and 16:19. It is a label of convenience that happens to capture what the Greek is doing — just in the wrong grammatical slot. The woman's name is Babylon the Great. The fact that she is a mystery is how John tells you to approach her. We merged the two into a title because it works.