First, let's look at what it means for two sets to the 'same size' as sets. One way (the standard way, in fact) to define this is that two sets have the same size if they can be paired off with each other: each element of one set corresponding to one element of the other set. We say the two sets have the same 'cardinality'.Not according to my math gurus it ain't.

As it turns out? In math, it's possible to show that certain types of infinity are "more" than other types-- provably so (within math, of course).

So, at least according to math, an infinity of infinities is a greater infinity than just a singular infinity.

If that makes sense to you? No? Me neither, but my math superiors assure me that it does-- in **math**.

<grin>

So, for example, the set {1,2,3,4,5} can be paired off with the set {2,4,6,8,10} by pairing 1<->2, 2<->4, 3<->6, 4<->8, and 5<->10, so they have the same size as sets (five elemts each).

But when we apply thise definition to infinite sets, we can get some counter-intuitive results. For example, the set of positive integers {1,2,3,4,5,...} and the set of even integers {2,4,6,8,10,...} can be paired off in this way (as above), even though the second has 'more' things in it than the first. So these sets have the same cardinality even though the second is a proper subset of the first. This can only happen for infinite sets and is even sometimes used as the *definition* of infinite for sets.

Now, sets that can be paired with the set of positive integers {1,2,3,4,5,..} are called 'countably infinite'. But it turns out there are sets that are infinite and not countably infinite. The set of all decimal numbers is such a set. This set is an infinite set *larger* than the infinite set of positive integers. In fact, it turns out that there is an infinite heierarchy of 'sizes' of infinite sets.

Now, the collection of decimal numbers between 0 and 1 is infinite: examples are .1,.01,.001,.0001, etc. But the set is bounded. What is happening is a playoff between two different ideas of size: cardinality as above, and length. The interval between 0 and 1 is infinite in cardinality and finite in length. Because of things like this, it is crucial to be clear which of many different ideas of size are used in any given context.

Now, when we define addition and multiplication of infinite sizes, we generalize off the finite definitions. So, to add two infinite sizes, we take sets of those sizes that do not overlap and put them together. The sum of the sizes is the size of the resulting set. For multiplication, we have a copy of one of the sets for each element of the other, etc. I tturns out that addition and multiplcation of cardinalities is easy: the sum or product of two infinite cardinalities is just the larger of the two (so it will be infinite also).