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 or such expansions there are three possibilities for the two one decimal expansion and one binary expansion e g sqrt two decimal expansions and one binary expansion e g and two decimal expansions and two binary expansions e g but i don't claim that each one has a finite binary expansion which is essentially what you're claiming t c