Let be the minimal uncountable well-ordered set.

(a) show that has no largest element.

Is it enough to say that if there was a largest element in and we were to remove it, we would still be left with an uncountable set and an ordinal set which will then contradict "minimality".

(b) Show that for every , the subset is uncountable.

Can I say: Let and let be a countable set. Then . Since is uncountable (defined in he beg. of the problem), then either S or T has to be uncountable and seeing as T is already defined as a countable set it follows that S must not be countable. Thus, the set is uncountable.

Note: I apologize in advance for my latex. The S_Omega is supposed to be S (subscript Cap Omega).