I Am a Strange Loop

Bertrand Russell’s Second-worst Nightmare

Any enrichment of PM (say, a system having more axioms or more rules of inference, or both) would have to be just as expressive of the flexibility of numbers as was PM (otherwise it would be weaker, not stronger), and so the same Gödelian trap would succeed in catching it — it would be just as readily hoist on its own petard.

Let me spell this out more concretely. Strings provable in the larger and allegedly superior system Super-PM would be isomorphically imitated by a richer set of numbers than the prim numbers (hence let’s call them “super-prim numbers”). At this point, just as he did for PM, Gödel would promptly create a new formula KH for Super-PM that said, “The number h is not a super-prim number”, and of course he would do it in such a way that h would be the Gödel number of KH itself. (Doing this for Super-PM is a cinch once you’ve done it for PM.) The exact same pattern of reasoning that we just stepped through for PM would go through once again, and the supposedly more powerful system would succumb to incompleteness in just the same way, and for just the same reasons, as PM did. The old proverb puts it succinctly: “The bigger they are, the harder they fall.”

In other words, the hole in PM (and in any other axiomatic system as rich as PM) is not due to some careless oversight by Russell and Whitehead but is simply an inevitable property of any system that is flexible enough to capture the chameleonic quality of whole numbers. PM is rich enough to be able to turn around and point at itself, like a television camera pointing at the screen to which it is sending its image. If you make a good enough TV system, this looping-back ability is inevitable. And the higher the system’s resolution is, the more faithful the image is.

As in judo, your opponent’s power is the source of their vulnerability. Kurt Gödel, maneuvering like a black belt, used PM’s power to bring it crashing down. Not as catastrophically as with inconsistency, mind you, but in a wholly unanticipated fashion — crashing down with incompleteness. The fact that you can’t get around Gödel’s black-belt trickery by enriching or enlarging PM in any fashion is called “essential incompleteness” — Bertrand Russell’s second-worst nightmare. But unlike his worst nightmare, which is just a bad dream, this nightmare takes place outside of dreamland.

An Endless Succession of Monsters

Not only does extending PM fail to save the boat from sinking, but worse, KG is far from being the only hole in PM. There are infinitely many ways of Gödel-numbering any given axiomatic system, and each one produces its own cousin to KG. They’re all different, but they’re so similar they are like clones. If you set out to save the sinking boat, you are free to toss KG or any of its clones as a new axiom into PM (for that matter, feel free to toss them all in at once!), but your heroic act will do little good; Gödel’s recipe will instantly produce a brand-new cousin to KG. Once again, this new self-referential Gödelian string will be “just like” KG and its passel of clones, but it won’t be identical to any of them. And you can toss that one in as well, and you’ll get yet another cousin! It seems that holes are popping up inside the struggling boat of PM as plentifully as daisies and violets pop up in the springtime. You can see why I call this nightmare more insidious and troubling than Russell’s worst one.

Not only Bertrand Russell was blindsided by this amazingly perverse and yet stunningly beautiful maneuver; virtually every mathematical thinker was, including the great German mathematician David Hilbert, one of whose major goals in life had been to rigorously ground all of mathematics in an axiomatic framework (this was called “the Hilbert Program”). Up till the Great Thunderclap of 1931, it was universally believed that this noble goal had been reached by Whitehead and Russell.

To put it another way, the mathematicians of that time universally believed in what I earlier called the “Mathematician’s Credo (Principia Mathematica version)”. Gödel’s shocking revelation that the pedestal upon which they had quite reasonably placed their faith was fundamentally and irreparably flawed followed from two things. One is our kindly assumption that the pedestal is consistent (i.e., we will never find any falsity lurking among the theorems of PM); the other is the nonprovability in PM of KG and all its infinitely many cousins, which we just showed is a consequence flowing from their self-referentiality, taking PM’s consistency into account.