The Infinite, part 4. Difficulties, order.
January 16, 2010 4 Comments
There are many orders of infinity. With no humor intended, there are infinitely many such orders. But when dealing with large collections of things, usually these are very abstract things, you can get into trouble.
For example: in a number of philosophical arguments, people have at least implicitly made reference to the collection of “everything” for comparative purposes perhaps. This idea is incoherent, at least if we try to apply the previous devices. If A is the collection of everything, then Cantor’s theorem says P(A) is bigger than A. Not very likely mate.
Or consider the collection of all collections that do not belong to themselves. We give this the name C. This seems like a reasonable thing, doesn’t it? But does C belong to itself? If it does, then it cannot, and if it does not, then it must. This sort of thing was unsettling to early theorists and made intuitions about large collections seem suspect. Along the way, the whole thing seems to have contributed oddly enough to the stomping death of empiricism as well as the paradise of unfettered infinity.
So dealing with “large” things can be troublesome and one must be careful to restrict the size of objects, or allow only those collections which are somehow “better” or more precisely defined. Not that infinite objects cannot be dealt with, but infinities that are in some sense “underived” like the collection C can be the source of paradox, and if not that, discomfort on some level. Another way to deal with the situation is to restrict how really big collections can be used. Make them second class citizens in a way. Generally, if an infinite collection is generated in some well-defined way, like P(N) for example, then it’s allowed. Being more precise here gets us into details that require considerable study. On the other hand, perhaps this knowledge leads to a certain caution about using the word infinite and its correlates (eternal, forever, etc.) in our religious life. Does the “uncautious” use of these terms infuse our speech with lurking contradiction? For me at least, this is an important question.
Abstraction can lead to infinity in the strangest places. For example, imagine your home or apartment. You think of the items contained in it. Would you allow that groups of these things qualify as things? What about groups of groups? Groups of groups of groups? If you get that far, then continuing is not much of a stretch. There are infinitely many things in your house! <grins>
One bit of useful terminology. Objects that are the same size as N, are called, for obvious reasons, “countably infinite.” Objects, like P(N), or the decimal set in part 3 which are larger than N, are called “uncountable.” (You can’t count them by matching with N.)
Aside from size, there are other properties of sets that find use in describing concepts that appear in theological contexts. One of these is order.
Collections of numbers are naturally ordered. We normally think of time as naturally ordered, past, present, future. In scientific descriptions of reality, at least in the models that are used in those descriptions, we often think of time as “infinitely” divisible. No matter how small an interval of time may be, we can think of a smaller one. In relativistic quantum theory, the formalism engages the idea of infinitely divisible space and time. They are treated on an equal footing.
Next: Infinite Mormonism.
 Of course the Heisenberg uncertainty business creates an interesting caveat. Is it meaningful to deal in extremely small time intervals? The answer is yes, but not in a way that impacts the question of the nature of “reality” perhaps. For example, a vacuum may randomly produce energy so long as that energy doesn’t stay around long enough to be observed.