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
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
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
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
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
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.