a trinity of infinities

Explanations of the Triune God often resort to analogies. These analogies tend to suggest that each of the three parts must be less than the whole. And a triangular division into three also tends to suggest that the whole must be finite: else how could something be a third of infinity?

Yet a mathematical analogy for infinity is available. Suppose that someone counts, and never stops counting. The resulting collection of numbers is "all the multiples of 1". These are usually called the natural or whole numbers. The entire collection is limitless, which is provable by a straightforward test. Any time you think you've reached the largest number, count one more time. The next number is greater than any of the prior numbers, so it's a new number (i.e. not equal to any of the others). Therefore the set is produced by a well-understood procedure, and nevertheless infinite or unbound.

Now consider some different procedures that result in different sets. The numbers 3, 4, and 5 have no factors in common; none is a multiple of the other two. 3, 4, and 5 are quite distinct. Now, for each number in the set from before, i.e. all the multiples of one, multiply that number by 3. If it helps, think of two tedious people standing side by side. After the first person counts, the second person immediately multiplies the new number by three. This set is "all the multiples of 3". Like all the multiples of 1, it's produced by a well-understood procedure, and nevertheless infinite or unbound.

We can do the same for 4 and 5. Finally, there's four infinite sets: the multiples of 1, 3, 4, 5. Since the procedure to construct the set of multiples of 3 consisted of multiplying each multiple of 1 by 3, all the multiples of 3 are also multiples of 1. The same applies to all the multiples of 4 or 5. These sets are provably infinite and still included in the "whole" original set, the multiples of 1. We have a trinity of infinities that are still contained in a infinite whole.

Although the member-numbers of the three infinite sets all belong to the set of natural numbers, the three sets aren't completely distinct. Presently, there are numbers that are multiples of both 3 and 4 (the multiples of 12), or 3 and 5 (the multiples of 15), or 4 and 5 (the multiples of 20). If it's more satisfying to consider three distinct infinite sets, then alternative procedures could accomplish that goal. However, the cost is greater complication and the exclusion of many former multiples. For the "3" set, instead of multiplying each natural number just by 3, 1) multiply by 3 then 2) multiply by 4. The result is a multiple of both 3 and 4, simply because it's equal to 3 multiplied by "other stuff" (i.e. the original natural number) and equal to 4 multiplied by "other stuff" (i.e. the original number multiplied by 3 a moment ago). Similarly, 3) multiply that by 5 to achieve the end goal of a number that's a multiple of 3 or 4 or 5. Now take that number and 4) add 3. This final number is a multiple of 3, because it's 3 added to a multiple of 3. On the other hand, it isn't a multiple of 4, because it's 3 added to a multiple of 4, which is 1 too few to "reach" the next multiple of 4. Neither is it a multiple of 5, because 3 is 2 too few to "reach" the next multiple of 5. Presto! The procedure can start with any (all) natural number and construct a number that's a multiple of 3 but not 4 or 5. Procedures that are broadly similar can produce an infinite set of numbers that are multiples of 4 but not 3 or 5, and an infinite set of numbers that are multiples of 5 but not 3 or 4. Due to the logical fact that a particular number can't both be a multiple and not be a multiple, it's a logical impossibility for a number that belongs to one of these sets to belong to either of the others. The first few members of the "3" set are 63, 123, 183, 243. The "4" set, 64, 124, 184, 244. The "5" set, 65, 125, 185, 245. (Naturally, the predictable gaps of 60 between adjacent numbers are due to 3 times 4 times 5. If you graphed these procedures as mathematical functions, with the result as the vertical axis and the starting number as the horizontal axis, you would see a set of closely space parallel lines.)

All different, all infinite, all drawing from the same source.