I Am a Strange Loop

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.