Douglas Hofstadter - I Am a Strange Loop

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.

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 number that I called ‘g’ possesses a rather arcane and recherché number-theoretical property that I called “sauciness” or “non-primness” . You’ll note that in the last couple of pages, not one word has appeared about prim numbers or non-prim numbers and their number-theoretical properties, nor has the number g been mentioned at all. We finessed all such numerical issues by looking only at KG’s secondary meaning, the meaning that Bertrand Russell never quite got. A few lines of purely non-numerical reasoning (the second section of this chapter) convinced us that this statement (which is about numbers) could not conceivably be a theorem of PM.

Consistency Condemns a Towering Peak to Unscalability

Imagine that a team of satellite-borne explorers has just discovered an unsuspected Himalayan mountain peak (let’s call it “KJ”) and imagine that they proclaim, both instantly and with total confidence, that thanks to a special, most unusual property of the summit alone, there is no conceivable route leading up to it. Merely from looking at a single photo shot vertically downwards from 250 miles up, the team declares KJ an unclimbable peak, and they reach this dramatic conclusion without giving any thought to the peak’s properties as seen from a conventional mountaineering perspective, let alone getting their hands dirty and actually trying out any of the countless potential approaches leading up the steep slopes towards it. “Nope, none of them will work!”, they cheerfully assert. “No need to bother trying any of them out — you’ll fail every time!”

Were such an odd event to transpire, it would be remarkably different from how all previous conclusions about the scalability of mountains had been reached. Heretofore, climbers always had to attempt many routes — indeed, to attempt them many times, with many types of equipment and in diverse weather conditions — and even thousands of failures in a row would not constitute an ironclad proof that the given peak was forever unscalable; all one could conclude would be that it had so far resisted scaling. Indeed, the very idea of a “proof of unscalability” would be most alien to the activity of mountaineering.

By contrast, our team of explorers has concluded from some novel property of KJ, without once thinking about (let alone actually trying out) a single one of the infinitely many conceivable routes leading up to its summit, that by its very nature it is unscalable. And yet their conclusion, they claim, is not merely probable or extremely likely, but dead certain.

This amounts to an unprecedented, upside-down, top-down kind of alpinistic causality. What kind of property might account for the peculiar peak’s unscalability? Traditional climbing experts would be bewildered at a blanket claim that for every conceivable route, climbers will inevitably encounter some fatal obstacle along the way. They might more modestly conclude that the distant peak would be extremely difficult to scale by looking upwards at it and trying to take into account all the imaginable routes that one might take in order to reach it. But our intrepid team, by contrast, has looked solely at KJ’s tippy-top and concluded downwards that there simply could be no route that would ever reach it from below.

When pressed very hard, the team of explorers finally explains how they reached their shattering conclusions. It turns out that the photograph taken of KJ from above was made not with ordinary light, which would reveal nothing special at all, but with the newly discovered “Gödel rays”. When KJ is perceived through this novel medium, a deeply hidden set of fatal structures is revealed.

The problem stems from the consistency of the rock base underlying the glaciers at the very top; it is so delicate that, were any climber to come within striking distance of the peak, the act of setting the slightest weight on it (even a grain of salt; even a baby bumblebee’s eyelash!) would instantly trigger a thunderous earthquake, and the whole mountain would come tumbling down in rubble. So the peak’s inaccessibility turns out to have nothing to do with how anyone might try to get up to it; it has to do with an inherent instability belonging to the summit itself, and moreover, a type of instability that only Gödel rays can reveal. Quite a silly fantasy, is it not?