My argument aws: The constructible numbers are not "complete". You can always find cauchy sequences of constructible numbers that do not converge to a construct

My argument aws: The constructible numbers are not "complete". You can always find cauchy sequences of constructible numbers that do not converge to a constructible number. If you add the requirement that the sequence itself must be constructible things _might_ change. It depends on how you define constructible sequence. I thought Cantors argument would show that there are uncountable constructible sequences: - Assume there are countable many C[1], C[2] - Construct a new one D = C[1][1] + 1, C[2][2] + 1, ... - D in constructible, hence it's on the list: C[j] = D - But D[j] =/= C[j][j] - Contradiction. This proof is however not correct. The sequence D does not need to be constructible.