Can randomness be defined? Since I believe that the word means the absence of order, the proving of randomness involves proving a negative. What, then is the meaning of the terms random number and random sequence? Does the designation of any events e.g. radioactive decay, as random mean anything other than "uninfluenceable by any known agency" and/or "not showing any regularity discernible by humans"?

Someone who knows more about the physics than I do would need to answer the latter question, but I believe that certain physical phenomena are supposed to be random in a much stronger sense that just "unintelligible to humans". There is supposed to be nothing other than probabilistic facts about when and how the phenomenon might occur. Even God cannot know in advance when and what is to happen, for there is simply no such fact to be known. There is a sophisticated mathematical theory of randomness. See the Wikipedia entry on Kolmogorov complexity . It describes a way, in terms of informational complexity, to characterize when a sequence of 0s and 1s is random.

What if a person could indefinitely predict the outcome of a coin flip? I understand that's not much of a question; but I want to know what that would mean in terms of either that single person, or the universe in general. If it happened tomorrow, what happens next?

I'm not sure I understand the question: Is it this? Let's say we have a fair coin, C, and that it is going to be flipped once every minute starting at noon tomorrow. Now let's imagine a person Fred. Fred is about to have an amazing streak of luck. Each minute, he is going to call "heads" or "tails", and Fred is going to get it right every time, and he will continue to do so for as long as he keeps going. I don't see anything impossible about that. Maybe it's unlikely, but it's clearly possible, and nothing much seems to follow. Now we don't have Fred making an infinite prediction here. So maybe we should change the story. Let's say that Fred says that the first flip will be even, every even flip will be followed by an odd one, and every odd one by an even one. And now let's suppose that, for as long as we keep flipping the coin, Fred turns out to be right. We can even suppose that we flip the coin forever, and Fred is always right. Again, that's clearly possible, if very unlikely, but nothing much...