Cantor: The Uncountable Continuum, part 2

Peering between the cracks

There are lots of whole numbers, but does that include every magnitude imaginable? No. How about 4⅓ or $5.17? These are fractions. The first number is four plus one cut into three parts, and the second is five dollars plus 17 cents, where a cent is a hundredth of a dollar (just like a century is a hundred years). Now, let’s think about how many fractions there are. I’ve lined them up here for you, in an orderly way:

So we have a row of unities, a row of halves, a row of thirds, a row of fourths, and so on. Each row has an infinite number of numbers in it, and there are an infinite number of rows. So does that give us a “size” for the collection of fractions of infinity×infinity, or infinity2? What would that even mean? It certainly seems to be a larger infinite, and it fills in so many holes between the whole numbers. Just between one and two, there are a whole infinitude of fractions, and the same between two and three, and so on. In fact, between any two fractions, we can always find more — an endless density of numbers. But, remember what we saw earlier about infinites depending on how you count them. The fractions have a trick up their sleeves:

By this diagonal, snaking means of traversing the fractions, we will reach them all in a definite way. Think of any fraction in your head. Now, won’t this snake eventually reach it? So what if we label the fractions by the order that we reached them in this snaking path?

Wow! Now, just like with the whole and even numbers, or the upper- and lowercase letters, each blue fraction has a red whole number associated with it. Every fraction has an order it was reached, and every distance along the snake corresponds to a fraction. So again, we can't say that there are actually any more fractions than whole numbers.

As an aside, note that the fractions here are like locations along a grid. If you had a three-dimensional grid, you could reach every point of the grid with a similar, three-dimensional snake. That means that such a grid, although it appears three-dimensional, is actually of the same size as (it maps directly onto) the one-dimensional line of whole numbers — so from this standpoint, there’s no difference in the size of two dimensions compared to one! How does this fit with our earlier examples of the number of dimensions of action of the machine tools? Is something missing, or was the analogy of increasing dimensions an erroneous one?

Finding more cracks to peer between

This is getting to be a bit disconcerting. Are the differences in qualitative infinites discussed earlier, nowhere to be found? But, then again, we haven’t finished looking at all numbers yet, have we? Consider this demonstration of why the square root of two is not a fraction.

Even in this huge infinite of fractions, the square root of two cannot be found. It is like the jet engine turbine that couldn’t be manufactured with a three-axis mill! Although there are an infinite number of fractions, we don't see the limit that prevents them from being all-encompassing, until we go beyond that limit. Nothing seems to be amiss, until we get a sense of something that lies beyond the old limits. (Think of the “finite but unbounded” or “finite but self-bounded” universe, along the lines of LaRouche’s reading of Einstein.) This demonstration of the square root of two shows a new infinite in two respects. First, the repeated fraction expression for the square root of two is itself infinitely long when written out, and its end seems infinitely far away. Secondly, the density of the fractions, which seem to fill all spaces between numbers, include gaps — gaps which you are unaware of until you find something in one of them.

Let’s try to conceptualize all possible numbers one more time, and see if we’ve finally found the larger infinite that must exist. Here I have a list of all possible numbers that could ever exist, written as a series of digits that could potentially each go on forever past the decimal point. Let’s go ahead and label each of our numbers with one of the red whole numbers:

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So, I have a list of all possible numbers. But now, I am creating a new number on the bottom, and I'm carefully making it in such a way that it isn’t anywhere in the list (even though the list supposedly contains every possible number). Here’s the technique: For the first digit of my new number, all I have to do is make sure it isn't the same as the first digit of the first number in the list. That digit is a 3, so I'll choose 1. Next, I’ll make sure the second digit differs from the second digit of the second number. So, instead of the 7 I find in the second number, I’ll again choose 1. For the third digit, you guessed it, I'll make sure it differs from the third digit of the third number. So, instead of 1, I’ll choose 2. If I continue in this way, I’ll create a blue number that was nowhere in my original list, but which should have been. Since that list was infinitely long (where infinite is understood as the red whole-number, “countable” infinite), we’ve gone beyond that infinity. We have made a blue number without a corresponding red partner. To repeat the reasoning: our new number isn’t the first number in the list, because the first digit differs, and it isn’t the second either, for the same reason, and so on, infinitely. We don’t actually create the number with this technique, but show how it could be made. Since we have a solitary blue number (and could have made many, many more), we can say that the number of all possible numbers is certainly larger than the number of whole numbers. We can now distinguish between the countable infinite, and the uncountable infinite (or the “continuum”).

For those of you who read through the above aside, we can now resolve our problem of the diagonal snaking technique making two- or three-dimensional grids actually one-dimensional. What was missing in those examples was continuity! Now that we have continuous change (rather than discretely different magnitudes), and have a concept of filling in the gaps, the problem of lacking true dimensionality vanishes. Even though mathematicians have shown that it is possible to create curves that fill all of space, there is no continuous one-to-one mapping. (Look up “space-filling curves” to learn more.) With continuity, we now have dimensions!

Infinites as connections

This example from Cantor has shown us how the greater size of infinite isn't larger by our normal idea of size, but in density. Think back to the different milling machines: with a greater number of independent axes of motion, the 5-axis mill had a more powerful domain of potential three-dimensional shapes it was capable of creating, but it still created three-dimensional shapes. The extra power isn't seen in the size of the produced parts, but in the power brought to bear in the productive process.

As another example of this, take Archytas's construction for doubling the cube. Like the example of the side of a doubled square, Archytas's physical creation of the magnitude capable of doubling the volume of a cube, creates a magnitude lying between 1 and 2 that is remarkable not in its quantitative length, but in the quality of "power" required to create it. This one-dimensional length actually exists only as a projection from three dimensions. Three dimensions of action are required to make the length.

The distance from the middle of the torus, to point P, the intersection of the cone, cylinder, and torus, is the side of a cube double the volume of that built on the diameter of the torus. If you have the Wolfram CDF Player installed, you can see an interactive version below.

Click here to download the animation.


Next: Bernhard Riemann