Mathematical logic can basically be characterized as the search for an epistemology. We want to design some formal system which lets us prove all true facts: so then we can decide what to believe just based on what the formal system says. And of course the system needs to prove itself to be correct; because otherwise, how would we know that the system is correct?
That’s the basic, obvious, naive hope of mathematical logic. Basically, we want an epistemology: a formal, mathematically rigorous epistemology that can prove all true facts, including the true fact that it is the correct epistemology.
Now there’s already some magical thinking inherent in this idea. How do we know the epistemology is correct? Why, because the epistemology says so!
So that’s a circular argument; but what’s the alternative? How could we possibly know that our epistemology was correct, other than by using our epistemology?
There’s that problem, and it shows up even before Gödel’s theorem. But then Gödel’s theorem tells us that self-validating epistemologies aren’t even logically possible. It’s not just that if you make a self-validating epistemology, its proof of its own correctness doesn’t mean anything because it’s a circular argument. No, it’s worse: no coherent epistemology even contains a proof of its own coherence.
And we’re not even talking about the epistemology proving itself to be true. No, we’re talking about a much weaker condition: the epistemology proving itself to be consistent, i.e., not contradicting itself. An epistemology can’t even prove itself consistent — at least not if it really is consistent.
Oh, and then things get even worse. Along comes my inconsistency theorem, which says that all of our existing epistemologies do in fact contradict themselves. Damn!
So then we try to come up with an epistemology that doesn’t contradict itself. I came up with one recently. At any rate, my inconsistency theorem doesn’t work in this epistemology; and I can argue pretty forcefully that it is consistent.
Basically, the reason I can say that is this. In this theory, whether or not a statement is true is determined by the outcome of a certain computer algorithm. The computer algorithm always halts, and always outputs exactly one of true or false, no matter what statement you put in. So unless the world magically splits in two and the computer program says both yes and no, despite being programmed to say only one of those, the theory is clearly consistent.
But how do we prove it to be consistent? In math we don’t accept those sorts of vague verbal arguments, especially not for something as important as the internal coherence of the very foundation of our belief system. We want a formal proof, not a verbal argument.
Well, interestingly, Gödel’s theorem still applies to this theory, and it can’t prove itself consistent. But even if it could, what would that mean?
Logic is absolutely filled with consistency proofs. Take any well-known formal system, and you can find half a dozen different consistency proofs for it. But not only are all such consistency proofs circular and therefore meaningless — they’re wrong! Every well-known formal system is inconsistent. We can still prove them consistent in half a dozen different ways.
I hope that demonstrates the utter futility and fatuity of consistency proofs. We can prove any system to be consistent — whether or not it actually is! We can prove any system, consistent or inconsistent, to be consistent. Given that fact, proving that a system is consistent basically forms no argument for it actually being consistent.
So let’s return to the beginning. We want to adhere to an epistemology. But we need to justify that our epistemology is correct. If we do that using our epistemology, then we’re arguing in a circle. If we do that by stepping outside our epistemology, then we’ve just violated our epistemology. (Every epistemology is a dictum of the form, “thou shalt believe all and only the things determined by these rules.”) But then things get worse because we provably can’t construct a self-validating epistemology, most of our epistemologies contradict themselves, and every argument to the effect that any epistemology is coherent is demonstrably meaningless.
Long story short, skepticism wins. We can’t achieve an ironclad belief system, where everything is justified to the point of absolute certainty. It’s just obvious that we can’t do that with anything empirical; but now formal reasoning has taught us that we can’t even do it in the rarefied realm of pure math.
When you penetrate to the very heart of reason, when you gaze upon its hidden starry essence, you find not a perfect and indestructible core of truth, but a bug in the Matrix, a crack in the wall, a strange sort of Zen joke about epistemology. And you go home having learned that you can’t know anything for certain.