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 P: 110 @ramsey2879 Basically, my intention is to clarify all the elements by translating to the first order theory language. As you know any (mathematical) statement can be translated to a statement such as "$\forall x\in S \forall y \in K \exists b\dots$ as the fist order theory is implemented set theoretic elements. In my level of mathematical maturity, I don't directly see this first order language structure of the Hardy's, i.e., I can't directly translate this Hardy's statement into the pure first order theory sentence in my mind, therby the meaning is not clarified to perfection but rather is possessed within a some intuition level vague to some extent. It'd be burdensome to translate the statement into the pure first order one, but I think it might be that the translation can be shortend if some set theoretic definitions are properly applied. Anyway I can't do it myself. Especially, the concpet of 'rearrangement' and 'product(multiplication)', I can't dare to think of the first order language structure of them. Note: obviosuly translation into the pure first order language will be really long, but as you know it can be shortened by using proper definitions (summariziation of some sentences or parts of a sentence). Right?