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Up until here your are fine. This is wrong. An uncountable set cannot be bijectively mapped onto a countable set (i.e. one-to-one and onto, with one-to-one being the thing that cannot be done). But I can definitely map an uncountable set to a countable set: map all of the real numbers to 0, for...

Right here you hit a few problems. First, it's not uncountable, it's incomplete you are looking for, that's the first incompleteness theorem. Completeness basically means that any statement able to be formalized is either provably true or false from the set of axioms. Second, there's a major...