You haven t given an upper estimate because I just constructed a list larger than yours which isn t nearly complete The step where you say that you have an upper bound is unjustified and where the argument goes wrong As for Do you claim that not every certainly I agree that every real number has a binary expansion and a decimal expansion in fact each has either 1 or 2 such expansions There are three possibilities for the two one decimal expansion and one binary expansion e g sqrt 2 two decimal expansions and one binary expansion e g 1 5 and two decimal expansions and two binary expansions e g 1 5 But I don t claim that each one has a finite binary expansion which is essentially what you re claiming t c