TITLE: How to tell a paradox from a "paradox"?
QUESTION [6 upvotes]: Russell's paradox showed that naive set theory leads to a contradiction. This was something that was taken seriously and caused a lot of work.
Now, Banach–Tarski paradox is arises from a result that a ball can be decomposed into finite amount of pieces and the pieces can be used to built two identical copies of the decomposed ball. Banach-Tarski paradox is often treated as a "paradox", basicly meaning that, yes, it is counter intuitive but yet there is no problem - mathematics just occasionally is counter intuitive.
To be honest, I have never understood why Banach-Tarski is not a "real" paradox but not being expert of measure theory I chose to accept the common view.
Is there some high level explanation on how to tell a paradox from a "paradox"? What is it that makes a counter intuitive result to a "real mathematical paradox" that we should start worrying about?

REPLY [3 votes]: What you're describing as a "true paradox" is sometimes called an "antimony" and it means an actual logical inconsistency in the underlying theory.  The Burali-Forti paradox is another example and it means there can't be a set of all ordinals (the ordinals are a proper class).  By contrast, the Banach-Tarski theorem is consistent; it's just counterintuitive.  The reason we don't hear much about "true paradoxes" (antimonies) these days is that logicians in the 1920's got the earlier inconsistencies under control, and (in all likelihood) we're not dealing with any actual inconsistent systems today, at least in everyday mathematics.