Aleph-One

 Aleph-one is the first uncountable number. What does that mean? It means that any set with at least this many elements can not be put into one-to-one correspondence with the set of positive integers. Put another way, it's a number so large that even if you attempted to list it's arguments 1st,2nd,3rd, ... and continued this list forever, you still wouldn't be able to account for all it's arguments!! This number is larger than infinity to its own power, larger than a power tower of infinities infinitely high ... it simply defies description. It can't be compared with aleph-null. At first it seems impossible, nightmarish. Yet it's existence seems inescapable. Cantor did not set out trying to prove there was a larger infinity. He stumbled upon it by accident. He assumed that all infinities were the same. But when he tried to set up a one-to-one correspondence between the positive integers and the real numbers he discovered he couldn't. Eventually he realized that the reason he was having so much difficult had to be because no such correspondence actually existed. So he then set out to prove there were more reals than positive integers. Here he succeeded. It was soon also discovered that the power set of aleph-null had to be greater than aleph-null. Cantor tried to prove that the power set of aleph-null was the same as the cardinality of the real numbers, but he was unable to do this. Some say it was this question that drove him mad. Later on it was proved that the question itself was undecidable, meaning no proof existed within Cantors framework that would show that the power set of aleph-null was also the cardinality of the reals. It's at this point that we learn something disturbing about studying infinity: that certain "truths" about them are unknowable and must simply be assumed one way or another. Some mathematicians have assumed Cantor's conjecture to be correct, or at least a useful way of working with infinities, and have taken it as an axiom, where others have explored other possible systems. But this begs the question: how can infinities defined in radically different systems be compared. The most important feature of any large number competition is that eligible entries must be well-ordered. That is, for any two distinct members, a larger member can be determined, at least in principle. When that breaks down we no longer have a competition, and for me personally, that is the essence of googology: a competition with a verifiable champion.