What ? Russell’s proof ? No. He precisely developed formal logic to circumvent i...

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What ? Russell’s proof ? No. He precisely developed formal logic to circumvent imprecisions of human language ;-)

But if I try to explain the first line (the following are the formal demonstration that this line is logically true), it might be translated to English as “let’s define alpha and beta as two sets that contain each 1 item, such that the intersection of alpha and beta is empty (i.e the two items are not the same) then the union of the two sets contains 2 items”

The formalism here allows to put limits to the cases where addition is allowed, or rather, clearly defined. For example this forbids to “add a thing to itself” : when you add 2 oranges, they must be different : you cannot add an orange to itself with this rule (you might, if you define the addition differently in this case as I explained in the answer). When you add 1+1, Russell says that you have to consider the “1”s as different : in formal logic they are sets that contain each 1 different item (orange). This is important when you add things that are not simple numbers, such as infinites…