I'm studying for my logic final and I cant seem to find an answer for this practice problem: (Using < as proper subset since I dont have the right type) T_1 < T_2 < T_3... be a strictly increasing sequence of satisfiable L-Theories. a) Show that the union of T_n is satisfiable (over all n in the natural numbers). b) Show that the union of T_n is not finitely axiomatizable. A) Pretty simple, every finite subset is satisfiable since the largets T_n is satisfiable thus by compactness their entire union is satisfiable. B) This is were I have some problems. I'm not sure how to go about showing this. I want to show that any finite amount of sentances can only axiomatize up to T_n and then we can simply show that T_n+1 is not axiomatized. Any ideas on this? EDIT: I posted this in the wrong forum apparently, if anyone could move it to the logic section I would appreciate it.