Degrees Of Infinity

In the late 19th century, mathematician Georg Cantor showed that not all infinite sets were of the same size. For example, while there are the same amount of odd integers as there are integers (because you can match them one to one forever), there are many more points on a line, or real numbers, than there are integers. In fact the ratio of integers to real numbers is effectively zero. The ratio of all integers to real numbers between zero and one is similarly zero.

The implication of this discovery is that not all infinities are equal, i.e., some infinities are more infinite than others. This is a hard concept around which to wrap one’s mind, but it’s another, perhaps simpler way of illustrating my “big idea” that AI will never tame complex systems, than using Graham’s Number and TREE(3).

My contention is that AI is climbing the integer ladder, while complex systems are expanding in the realm of real numbers. (If you’re unclear on real numbers, they are comprised of both rational numbers — those that can be expressed as fractions like 1/2 or 2/1 — and irrational numbers like the square-root of two or pi that cannot be expressed fractionally.) Real numbers comprise every discreet point on a line, whereas rational numbers (and obviously integers) leave gaps.

Hence, no matter how powerful AI becomes, it will never be able to solve the stock market, central-plan an economy, create forests, coral reefs or complex new life forms. The more juice we give it, the farther up the integer ladder it climbs, but no matter what heights it reaches, it will never cover even an infinitesimal step to the next rung on the real number one. AI is just a machine, perhaps a powerful and impressive one, but no match for the Tao, operating, as it were, in a higher dimension of infinity.