In the Wikipedia account of the Cantor tertiary set, it is remarked that the cardinality of the set is uncountable. Can this can be strengthened to it needing to be as large as the continuum? (This is not necessarily the same thing if we assume that the continuum hypothesis is false, so that uncountable could also be aleph-1, below the continuity of the continuum.) It would seem by the proof given there that it would be, but I am not certain. If one started out with a subset of the interval with only aleph-one points in it, would the construction go through?