| > What makes it okay to identify the naming of a large number with producing that large number of things (e.g. 1's)? Nice catch! I glossed over this but this is pretty important. The issue is that for any _specific number_, it takes a constant amount of time to write a number larger than that. Related: solving chess is technically O(1) since there are a constant number of positions we have to check. What we need instead for this concept is a series of numbers, those can be compared with the question "Does one eventually get to a point where it will always be larger than the second?". We can parametrize the "biggest number" by the number of seconds given, say we can write down one character per second. Then a person's largest number function might be something like "f(n) = 9 ^ n", which is easy to see _is_ a computable function, and thus smaller than BB(n) for the same number of seconds, eventually. In practice, just something like BB(100) - or BB(BB(100)), even can't really be written out by any computable method in a finite amount of time. > which was: what characterizes the Busy Beaver function? The proper one is just the number of 1s, but I talked about the shift function because it's easier to see why it's uncomputable. Tibor Rado proved that the Busy beaver function is also uncomputable in his 1962 paper, if you're interested you could check out his reduction. |