I'm just thinking aloud on this one. So let's imagine a hypothetical function S_r that's the real analogue of S_n (the successor function over the naturals). Now let's say a, b in R are defined as follows: b=S_r(a); b is the real successor of a. Now of course if such a function existed then b-a must equal zero. If it didn't, then we can always find a c in R that's between a and b which would then make c a new candidate for a's successor. But if a-b is equal to zero, then a must necessarily equal b and so b can't be its successor. Such an argument seems to show that consecutive reals cannot exist, or at least can't be expressed using real numbers themselves. On the other hand, if we are working with the reals under the assumption that consecutive reals do not exist, then there is no intuitive way to refer to the "next" real number. Certainly there's no way to define it using reals themselves as above. But if one says that there actually and certainly is no next real number then how are we moving along the field at all? How do you move between one and two if there's no conceptual way of referring to the real number that comes immediately after the number 1 in the set of reals. If b-a must equal zero then you can add all the zeroes you want in the world to a but it won't change a thing unless you give infinity some mystical power to turn those zeroes into real values. I'm somewhat familiar with Cantor's diagonal argument now and I understand that that is precisely the whole point: that there exists no bijection between the reals and the naturals, and so no consecutive real by definition. But while I submit that such a thing can't be formalized in anyway using current language, doesn't the above reasoning at least justify the existence of consecutive real numbers in some as of yet to-be-defined form?