I Am a Strange Loop

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?

Downward Causality in Mathematics

Indeed it is. But Kurt Gödel’s bombshell, though just as fantastic, was not a fantasy. It was rigorous and precise. It revealed the stunning fact that a formula’s hidden meaning may have a peculiar kind of “downward” causal power, determining the formula’s truth or falsity (or its derivability or nonderivability inside PM or any other sufficiently rich axiomatic system). Merely from knowing the formula’s meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically “upwards” from the axioms.

This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false (or provable or unprovable).

For instance, if I tell you, “There are infinitely many perfect numbers” (numbers such as 6, 28, and 496, whose factors add up to the number itself), you will not know if my claim — call it ‘Imp’ — is true or not, and merely staring for a long time at the written-out statement of Imp (whether it’s expressed in English words or in some prickly formal notation such as that of PM) will not help you in the least. You will have to try out various approaches to this peak. Thus you might discover that 8128 is the next perfect number after 496; you might note that none of the perfect numbers you come up with is odd, which is somewhat odd; you might observe that each one you find has the form p(p+1)/2, where p is an odd prime (such as 3, 7, or 31) and p+1 is also a power of 2 (such as 4, 8, or 32); and so forth.

After a while, perhaps a long series of failures to prove Imp would gradually bring you around to suspecting that it is false. In that case, you might decide to switch goals and try out various approaches to the nearby rival peak — namely, Imp’s negation ∼ Imp — which is the statement “There are not infinitely many perfect numbers”, which is tantamount to asserting that there is a largest perfect number (reminiscent of our old friend P, allegedly the largest prime number in the world).

But suppose that through a stunning stroke of genius you discovered a new kind of “Gödel ray” (i.e., some clever new Gödel numbering, including all of the standard Gödel machinery that makes prim numbers dance in perfect synchrony with provable strings) that allowed you to see through to a hidden second level of meaning belonging to Imp — a hidden meaning that proclaimed, to those lucky few who knew how to decipher it, “The integer i is not prim”, where i happened to be the Gödel number of Imp itself. Well, dear reader, I suspect it wouldn’t take you long to recognize this scenario. You would quickly realize that Imp, just like KG, asserts of itself via your new Gödel code, “Imp has no proof in PM.”

In that most delightful though most unlikely of scenarios, you could immediately conclude, without any further search through the world of whole numbers and their factors, or through the world of rigorous proofs, that Imp was both true and unprovable. In other words, you would conclude that the statement “There are infinitely many perfect numbers” is true, and you would also conclude that it has no proof using PM’s axioms and rules of inference, and last of all (twisting the knife of irony), you would conclude that Imp’s lack of proof in PM is a direct consequence of its truth.

You may think the scenario I’ve just painted is nonsensical, but it is exactly analogous to what Gödel did. It’s just that instead of starting with an a priori well-known and interesting statement about numbers and then fortuitously bumping into a very strange alternate meaning hidden inside it, Gödel carefully concocted a statement about numbers and revealed that, because of how he had designed it, it had a very strange alternate meaning. Other than that, though, the two scenarios are identical.