Puzzle #236: Number Categories

The numbers one, two, five, eight, and nine have a certain property. The numbers zero, three, four, six, and seven do not have this property. What is this property? [Hint: It has to do with the English spelling of each of the words.]

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, December 13, 2017 at noon Pacific Time. I will post the answer shortly after.