  Turing's famous 'machine' framework provides an intuitively clear conception
of 'computing with real numbers'. A recursive counterexample to a theorem shows
that the theorem does not hold when restricted to computable objects. These
counterexamples are often crucial in establishing reversals in the Reverse
Mathematics program. All the previous is essentially limited to a language that
can only express countable mathematics directly. The aim of this paper is to
show that reversals and recursive counterexamples, countable in nature as they
might be, directly yield new and interesting results about uncountable
mathematics with little-to-no modification. We shall treat the following
topics/theorems: the monotone convergence theorem/Specker sequences, compact
and closed sets in metric spaces, the Rado selection lemma, the ordering and
algebraic closures of fields, and ideals of rings. The higher-order
generalisation of sequence is of course provided by nets (aka Moore-Smith
sequences ).
