my dear, dear friend:
beautiful article. a good subject, which is explained very well. when I read it, however, some thoughts arose which I will bring here for everyone to judge.
let me begin by saying that there are 2 types of infinity that we as humans can think of (although not necessarily understand). the type that you and the quote you brought from Principia Cybernetica are talking about, the one that is achieved by setting x=1 and adding 1 to x unto infinity, I will call "negative infinity". "negative", because it is reached not by a positive assertion of infinity (finding a number that is infinite), but by negating the possibility of finiteness, by showing that nomatter what x is, x+1 will always exist. the other type, or "positive infinity", would mean to find a number that is infinite unto itself, without need of the function.
what I was thinking when i read the article, is that the second, or positive kind of infinity, must necessarily exist in order for the first kind to be possible. the function is dynamic, it is constantly changing and adding 1 to itself. now the nature of change is such that a changing thing must be in contact with something else. it can't be the only thing there IS, because if something is the only thing there is (in other words, if it is everything) then it is impossible for it to change, since it already contains everything and whatever changed in it was already there. that situation can only happen in a positive type of infinity. now, if the function you suggested, or any other infinite function, is in fact in contact with something other than itself, then it's infinite nature must be dependant of something other than itself. for example, a geometric line that isn't the only thing that exists (meaning that other geometrical figures exist) must exist in SPACE, and so its infinite nature is dependant on the infinite nature of space. for a line in space to be infinite, space must be infinite, for a dynamic function in time to be infinite, time must be infinite, for an idea to be infinite, thought must be infinite, etc. and if time and space themselves are a function of the form x=x+1, they must also be dependant on something else that is infinite and so must that something, unto infinity (heh, heh). so we either get an impossible function within function that we can never become since we can never understand (unlike x=x+1), or we must accept that SOMEWHERE under all these layers there is a positively infinite thing that all other infinities are dependant on.
tell me what you think..
