Is it wrong to say "I work at a hospital" if I mean that I work in a specific hospital unknown to the person I am talking to? How is it different from the more common "I work at the hospital"?
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It's not wrong. It's the same with this "a" or "the" as it is with any other. "I work at a hospital" means you are not volunteering the specific hospital at which you work. "I work at the hospital" says you are referring to a specific hospital, and that there is only one hospital to which the listener could reasonably assume you are referring.
Hopefully you will consider my response the answer, and not just an answer... :-)