I am alive, 1 is not equal to 0 (sometimes), and the continuum hypothesis

Here is evidence that I have not been abducted by some aliens. While taking an afternoon nap just now, I had the most ridiculous dream about UFO's in the shape of skulls made of some beige-coloured material. These were really tiny UFO's. They could fit into your palm! And so I was swatting one particular UFO with my rusty squash strokes, and caught the cracked object with my hand. Then the skull spoke to me before it took it's last breath.... "take a look at what we are made of...". And I saw that it was made of silicone.

Haha. "You meant silicon right?"

Well, I don't know because I woke up immediately after. It's a little bit bothersome that I cannot rule out the possibility that I had been beamed up by these little critters made of the material that enters your favourite breast implants, with the intention that I spread the message that the future of civilization is in silicone-based lifeforms. Utterly bewildering.

That possibility aside, I should be in the version of the universe where I am freezing my butt off in my poorly-heated room, trying to revive a blog that has neither direction nor pictures of my house (those will come eventually, maybe in the summer). Well, suffice to say, I have been trying my darndest to wrap my head around these algebraic topology concepts that are truly quite intimidating. That should be enough evidence that all is well with me.

On to the mandatory geeky stuff. I often hear people describe mathematicians as people who spend their time "proving that 1 is equal to 0". This is completely false. Firstly, it is ludicrously simple to "prove that 1 is equal to 0". All you have to do is to define a set of objects containing "1" and "0" and declare them to be one and the same thing! Or, if you are not satisfied with this trivial manner of construction, you can also define, abstractly, a set endowed with certain properties that ultimately require "1=0" to hold. The simplest example would be the "zero ring".

The point is, in most algebraic structures of any interest to anyone operating in the real world, "1=0" does not hold. There is no point in trying to prove it, because it is false, although there is some romantic notion in proving absurdity. This is not simply because a vertical line looks different from an oval. The question is not whether the symbols "1" and "0" are equal, but whether the abstract object they represent are really one and the same thing (what the hell is "same-ness" anyway??). The arabic symbol "1", the chinese character "--" and the roman character "I" are all just things you write on the blackboard to refer to "unity", which is this object that stubbornly refuses to do anything to every other fellow object even if you leave them in a room overnight and force them to multiply. "Unity" does not belong to you or me, or even on the planet earth. It's obstinateness is much further-reaching. Skull-shaped UFO's share the same "unity" as us, for example, although they may have curly fingers which led them to represent "unity" with "S" instead.

Now, this is getting scary. It seems that the concept of "1" is so compelling as to permeate the entire existential universe. Just to be clear, the "1" and "0" that I am referring to are "integers", the kind of stuff we were all force-fed with when we were in primary school, and not things in the "zero-ring" (where they are really Jekyll and Hyde). Is that reasonable? Leopold Kronecker famously said:

"God made the integers, all else is the work of man."

Maybe you believe that 1,2,3.... are God-given. Fair enough, Kronecker agrees. Well, he probably would have said the same thing about numbers and fractions. How about the real numbers? Now, this is the bomb. Everyone who has ever used any calculus has implicitly declared that these creatures exist. But, wait. What is the problem? Can't you just invent anything in mathematics? Just like you pretended that integers existed, you can also pretend that real numbers exist!

Here's the problem. With integers, there was already the pesky question of "what is the largest integer?", which already leads us headlong into the realm of infinity. This sort of infinity is not so bad, since we can count our way there. But the real numbers are terrible. There are so many of them, that mathematicians had to invent a new sort of infinity to create them. A kind of monster that led us to also create calculus, Fourier analysis, and all sorts of associated appendages. All sorts of important mathematical tricks require us to be able to handle such humongous infinities, and you will have grounds to question whether we can handle that. In typical narcissistic style, we invoke the Axiom of Choice to endow us with that power. (So God gave us the integers, and free will to create and play with real numbers...).

It requires, quite literally, a much bigger leap of faith to embrace the real numbers, even though we use it routinely in our everyday earth-bound lives. The problem does not even end there. It turns out that we are not clever enough to create a set with more objects than the integers, but fewer than the real numbers. Which is a little silly don't you think? David Hilbert regarded this anomaly, commonly called the "continuum hypothesis", as such an important question that it is the first of his famous "Twenty-three problems". This problem has already been solved, in some sense. The answer is that, we can never know whether or not we are clever enough to create such an intermediate set; even invoking the Axiom of Choice does not help at all. Pretty smart eh?

Unfortunately, the alien died before I could ask him whether he used calculus. It appears that even in my wildest dreams, the answer to the above questions continues to elude me.