What appears below is copied from Section 2. at this link…
https://plato.stanford.edu/entries/axiom-choice/
…closely related to the idea of an infinitesimal…less than any rational number, but not equal to 0.
“…a countable collection of pairs of sets of real numbers fails to have a choice function”…….Let’s think about that…
Cosets in R/Q+ determined by a pair of complements…a pair of complementary infinite binary strings and the continued fraction representation they correspond to via the Stern-Brocot tree. Now there’s a pair of sets of real numbers…and all together they constitute a set of disjoint pairs in R and are countable sets….and we can most certainly put together a countable collection of such pairs (…assume there exists…)…it is “consistent with the standard axioms of set theory to assume” that such a collection has no choice function. Now that would make the construction of a Cantor type list/table for the purpose of diagonalization rather difficult and a “certainty” argument built on it impossible, wouldn’t it.
Holes all the way down…
21st Century Paradox 