The author, inventor and futurist Ray Kurzweil has written about the technological singularity, a time when he predicts things will change so rapidly that he likens it to a mathematical singularity. In particular, he postulates the invention of an artificial intelligence capable of re-designing itself, which will inevitably lead to ever-faster progress.

To back up his theory, Kurzweil likes to show exponential curves representing the ever-faster development of computer processor performance, the price of transistors, DNA sequencing costs and such like.

But just how mathematically rigorous is this theory? Is it right to speak of a singularity occurring some time in the middle of the 21^{st} century?

Students of cosmology know that genuine singularities are places where scary stuff happens. Especially naked singularities, which I cannot illustrate here for the sake of public decency. It is not a term to be treated lightly. In this article, I am going to prove that if Kurzweil’s prediction of strong AI is true, then a genuine singularity will occur, and in a surprising way, quite unlike popular thinking on the matter.

Let us suppose that there are an infinite number of inventions that people could ever invent. And by people I mean not just humans, but also alien civilisations, robots and any other class of inventive being. Let us number these inventions starting with 1 (a method for starting a fire, perhaps), then 2 (a method for avoiding burning your fingers), and so on.

Now let us assume, as Kurzweil’s exponential graphs indicate, that the time interval between inventions becomes ever smaller as more inventions are made. This is logical, because the more inventions that are available to an inventor, the easier it is to invent something new. Also, the smarter an AI becomes, the easier it is for it to invent an even-smarter version of itself. This is the key assumption in Kurzweil’s theory.

Now, here comes the tricky mathematical part. If the time between inventions becomes ever smaller as the number of inventions increases, then the total time taken to invent all possible inventions is finite. I will call this time T.

The proof of this is analogous to the resolution of Zeno’s paradox of Achilles and the tortoise. Zeno, the Greek philosopher, who believed that change is an illusion, outlined the following thought experiment. The great warrior Achilles is in a race against the tortoise and the tortoise is given a head start. If both runners start at the same time, then by the time Achilles reaches the starting point of the tortoise, the ponderous tortoise will have moved forward some distance less than Achilles, but will still be in front. By the time Achilles reaches the tortoise’s new position, the tortoise will have moved forward again. It will require an infinite number of steps for Achilles to catch up with the tortoise.

Of course, the paradox is easily resolved by realising that the time taken to complete each one of these infinite steps grows progressively shorter. Simple calculus shows that Achilles will overtake the tortoise after a finite time T.

The situation is precisely analogous to the question of invention. Although the number of possible inventions is infinite, if the time taken between inventions becomes progressively shorter, then after a time T, everything that can be invented will have been invented. This includes all possible books, all conceivable works of art, an infinite number of cat memes and even the flying car. And the time T is not infinitely far in the future, but is finite and in principle calculable.

Such a time has been prophesied by various cultures and religions throughout history. It has been called Ragnarok, The Twilight of the Gods and the End of Days, but in this article I will call it “T time”.

So, after T time there will be literally nothing to do, as everything interesting will already have been done. This is the true singularity, and it is not a time when things are changing ever more rapidly, but when they have changed so much that no further change is possible. In a neatly ironic way, it is a time when Zeno’s belief that change is impossible will become true.

Some people may think of this as a utopia, but it is really just time for taking a quiet nap after T.

**About the author:** Steve Morris studied Physics at the University of Oxford but discovered that writing about other people’s ideas is easier than having original ones yourself. He now writes about awesome technology at S21 and shares random thoughts at Blog Blogger Bloggest.