sounding facts are consequences of each other! In other words, KG is unprovable not only although it is true, but worse yet, because it is true.
This weird situation is utterly unprecedented and profoundly perverse. It flies in the face of the Mathematician’s Credo, which asserts that truth and provability are just two sides of the same coin — that they always go together, because they entail each other. Instead, we’ve just encountered a case where, astoundingly, truth and unprovability entail each other. Now isn’t that a fine how-do-you-do?
Incompleteness Derives from Strength
The fact that there exists a truth of number theory that is unprovable in PM means, as you may recall from Chapter 9, that PM is incomplete. It has holes in it. (So far we’ve seen just one hole — KG — but it turns out there are plenty more — an infinity of them, in fact.) Some statements of number theory that should be provable escape from PM’s vast net of proof — they slip through its mesh. Clearly, this is another kind of nightmare — perhaps not quite as devastating as Bertrand Russell’s worst nightmare, but somehow even more insidious and troubling.
Such a state of affairs is certainly not what the mathematicians and logicians of 1931 expected. Nothing in the air suggested that the axioms and rules of inference of Principia Mathematica were weak or deficient in any way. They seemed, quite the contrary, to imply virtually everything that anyone might have thought was true about numbers. The opening lines of Godel’s 1931 article, quoted in Chapter 10, state this clearly. If you’ll recall, he wrote, speaking of Principia Mathematica and Zermelo-Fraenkel set theory: “These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e., reduced to a few axioms and rules of inference.”
What Godel articulates here was virtually a universal credo at the time, and so his revelation of PM’s incompleteness, in the twenty-five pages that followed, came like a sudden thunderbolt from the bluest of skies.
To add insult to injury, Godel’s conclusion sprang not from a weakness in PM but from a strength. That strength is the fact that numbers are so flexible or “chameleonic” that their patterns can mimic patterns of reasoning. Godel exploited the simple but marvelous fact that the familiar whole numbers can dance in just the same way as the unfamiliar symbolpatterns of PM dance. More specifically, the prim numbers that he invented act indistinguishably from provable strings, and one of PM’s natural strengths is that it is able to talk about prim numbers. For this reason, it is able to talk about itself (in code). In a word, PM’s expressive power is what gives rise to its incompleteness. What a fantastic irony!
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 Godelian 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, Godel 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 Godel 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 Godel, 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 Godel’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 Godel-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; Godel’s recipe will instantly produce a brand-new cousin to KG. Once again, this new self- referential Godelian 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)”. Godel’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.
To recap it just one last time, what is it about KG (or any of its cousins) that makes it not provable? In a word, it is its self-referential meaning: if KG were provable, its loopy meaning would flip around and make it unprovable, and so PM would be inconsistent, which we know it is not.
But notice that we have not made any detailed analysis of the nature of derivations that would try to make KG appear as their bottom line. In fact, we have totally ignored the Russellian meaning of KG (what I’ve been calling its primary meaning), which is the claim that the gargantuan
