`  :You're still missing the point.  It is not that formalizing set theory into first-order logic creates a paradox  that makes it sound as though first-order logic is too strong, whereas what is actually going on here is that first-order logic is too weak to capture all of the intended meaning of the notion of ``uncountable``.  The only way to get a paradox here (in the sense of something contrary to intuition) is if you have an intuition that first-order logic ought to be able to capture all of that semantic content.  But that intuition is just simply wrong, and the resolution is to get a better intuition, after which there is no more paradox.   `