Up and down Cantor’s ladder

I understand nothing in mathematics, and know it. Therefore, excuse me for wadding so frivolously in its waters. People who know nothing always believe that the sea is knee-deep. If there is a complete ignorant over a topic, that’s me on math. So, I’m allowed to write everything stupid that may or may not come into a human head.

This is an attempt to explain the transfinite numbers to, say, a blonde, and then to get beyond what even she would tolerate.

It’s easy with the ordinary numbers. 1 + 1 = 2. 6 / 2 = 3. But what happens if you get very big numbers? So big that they are actually infinite? Infinity + infinity = infinity (not 2 infinities, as you might expect). Infinity * 5 = infinity (not 5 infinities). Does math stop being valid there?

It doesn’t, tells us a German mathematician named Georg Cantor. Infinity is a special number. It is like 1, but of a higher order than all finite numbers. We would say that all finite numbers are of the order aleph0, and infinity is 1 of the order aleph1. (Mathematicians call these orders “powers”.)

This way, math continues to be valid even there. 1(aleph1) + 1(aleph1) = 2 (aleph1). Etc. Simple and elegant. Enjoy, blondes all over the world, regardless of your hair color.

What happens with infinity(aleph1)? It is 1(aleph2). On a need, aleph3 can be conjured, etc. Be calm. The justice is restored, Clint Eastwood will gun down the bad guys and get the gold.

I personally found a deep satisfaction in this. (Remember, I’m complete twit in maths. Maybe that’s completely wrong, and they teach the right thing in the elementary school.) What plagued me from my school years was that you may not divide by zero. Maths rules say that if A / B = C, then A = B * C. And if B = 0 (and A is not 0), no C on this earth could satisfy the rules.

It appears, however, that a C from the heaven (that is, aleph1) would do nicely – at least, I think so. A number from the aleph1 order could be multiplied by zero, and to still give a nonzero aleph0 number. Hoooray! Find a rest already, my thick and empty head!

But a question rang in there instead. If C * B = A, and C is aleph1, and A and B are aleph0, then something feels wrong. Something gets lost somewhere. Right? Let’s go on a quest to find it.

To start with, we may like to view all numbers we work with as complex numbers. Complex is a number that has two or more parts, which don’t mix. For example, 1(aleph1) is a complex number – as we saw, 1 (an ordinary number) cannot be expressed directly in aleph classes. In other words, we will have eg. 1(aleph0) = 0(aleph0) * 1(aleph1). 1 = 0 * 1 would not embarass me too strongly – as said above, the aleph1 power makes it possible. However, the alephs part of the equation doesn’t look well. Something is wrong there.

If we lived on the first heaven, in the aleph1 realm, 1(aleph1) would be for us simply 1. The same with any other aleph1 number. But what would be for us 1(aleph0)? Any number of the aleph0 realm, for that matter?… I can find only one logical answer – it would be zero.

Back to our 1(aleph0) = 0(aleph0) * 1(aleph1). What is 0(aleph0) is actually eg. 1(aleph-1)? (That is the power aleph minus 1.) Then we get

1(aleph0) = 1(aleph-1) * 1(aleph1)

I don’t know why, but this appears to me much more logical. Makes sense to my dumb head. Looks… beautiful, somehow. Provides a very elegant explanation of why, for God’s sake, you can’t divide by zero – and, even better, a direction how you can do it, and get results that don’t contradict the math rules.

It appears that the Cantor’s ladder goes not only up, but down, too. That we have not only aleph1, aleph2 etc, but also aleph-1, aleph-2, etc.

Which is sad. We always thought (or at least hoped) that we know where everything starts from. That we have a firm, eternal, reliable ground under our legs. If not anywhere else, then at least here, in the mathematics. Now we see that we are staying in the middle of nowhere, on a ladder that continues forever in both directions.

But it is also full with joy. Because, until now, we had only one direction to go to, and discover new and wonderful things there. Now, we have a second one… I don’t know what this may mean, and what it will bring. Maybe nothing. Or maybe much.