Gödel's Incompleteness Theorems and their Implications on the Human Condition

The Hilbert Program

David Hilbert is highly regarded as one of the most impactful thinkers and mathematicians of all time. In fact, if you were to look at the time he was alive, from 1862 to 1943, it would be difficult to find a mathematician with more influence than him. What makes Hilbert special is that he contributed to deep and complex topics and problems across a diverse set of disciplines. Hilbert spaces laid the foundation for future growth in quantum mechanics. He independently derived the equations of general relativity in the Hilbert action. His Zahlbericht reorganized the field of number theory into a collection of modern and abstract methods. Clearly, he was both a jack and master of all trades. To practically anyone else ever, this would be enough to retire as one of the greats. However, his greatest ambition was yet to come.

At the turn of the 20th century, mathematics was in an uncertain state. Paradoxes in set theory and foundational questions were shaking mathematicians' confidence in the field. In light of all of these developments and their shocks to the foundations of mathAs well as potentially a bit of hubris, Hilbert recognized the need for a modernized and solid foundation from which the field could advance, so he introduced the Hilbert Program. This program set out to prove that mathematics is perfect. To Hilbert, this could be proven by proving just three different ideas:

1. Completeness: Every true statement can be proven from a fixed set of axioms
2. Consistency: No contradiction can be derived from any statements
3. Decidability: There is a systematic procedure to determine if any statement is true or false

For nearly a decade, the questions remained open as mathematicians worked toward Hilbert's vision. Then, in 1931, an unexpected source delivered devastating news.

The First Incompleteness Theorem

Kurt Gödel was a 25-year-old Austrian logician who had just wrapped up his PhD at the University of Vienna. Even at this young age, Gödel published two groundbreaking papers that proved Hilbert's Program was impossible. In his first paper, he outlined the first incompleteness theorem, which says that any consistent formal system that's powerful enough to describe basic arithmetic contains true statements that cannot be proven true within that system. This, of course, attacked the first principle: completeness. In the context of this problem, a formal system is a set of axioms and rules for deriving new statements. Gödel proves this statement via the self-reference trick, essentially finding a statement that cannot be proven in a given system.

Imagine you had a rule book for an arbitrary game where rule x says, 'This rule cannot be verified using only this rule book'. Is this correct? It may seem like a silly example, but it gets to the heart of what Gödel is trying to say because this is unverifiable within the scope of our current systemThe rule book. However, Gödel didn't disappoint Hilbert with just a one-sentence proof; he needed a more robust way to deliver his thesis. His genius lies in discovering that mathematical statements could be encoded as numbers. This trick, known as Gödel numbering, allowed him to set up a self-referencing system where mathematical propositions could make statements about themselves.

Generally, let $G$ be a true statement that equates to 'Statement $G$ can't be proved in this system'. There are two cases to consider. If we suppose that $G$ is false, that means it is able to be proved since it is not true. However, this arrives at a contradiction as $G$ is said to be false, and proving a false statement to be true would result in the system being inconsistent. On the other hand, the trivial case is that if $G$ is true, it cannot be proven within the scope of the system. Note that this is NOT a paradox as $G$ is genuinely true, and we can see that from the outside, but it simply cannot be proven from within. Gödel argued that it must follow that any consistent formal system powerful enough for arithmeticThis matters because it allows support for the encoded statements as numbers. Simple systems that only express boolean logic are considered complete is incomplete as there will always be true statements it cannot prove.

The Second Incompleteness Theorem

The first incompleteness theorem killed Hilbert's dream of perfect mathematics, but Gödel wasn't done just yet. In his second published paper, Gödel started right where he left off. While the first theorem attacked completeness, the second theorem attacked consistency. As a reminder, for a system to be consistent, it cannot derive any contradictions. As a corollary to the first, the second theorem says that no consistent system can prove its own consistency. If we revisit the idea of the self-referential statement $G$, we determined that $G$ is true but cannot be proved within the system. But here's the problem: to prove consistency, you need to step outside to a stronger system. And how do you prove THAT system is consistent? Well, we need a larger system to verify that one. Guess what, we'll need a larger system to verify THAT one.

It's like trying to measure a ruler with itself: not possible. You need a longer ruler to verify the first one's length. But then how do you verify the longer ruler? You need an even longer one. And so on. This is an infinite regress with clearly no solution. Hilbert wanted to prove mathematics rests on a consistent foundation. Gödel showed this is impossible; there is no ultimate foundation. Gödel proved that consistency cannot be proven from within a system, since proving consistency requires reasoning stronger than what the system itself is capable of. The implications of Gödel's incompleteness theorems reach far beyond the context of mathematical systems. If you're like me, you start to wonder: if even mathematics, perhaps the most robust and logical system possible, is proven imperfect, what does that imply for everything else?

Epistemological Applications

I think one of the more optimistic interpretations of Gödel's work can be found in humanity's collective knowledge. For the sake of the argument, let's consider a system that contains everything humanity knows. The incompleteness theorems would tell us that we will never truly possess perfect and complete knowledge, but is this necessarily a bad thing? In his book The Beginning of Infinity, David Deutsch touches on this question. Deutsch, renowned in his own right as the 'father of quantum computing', covers a vast array of topics to ultimately deliver a major point home: our potential for knowledge is boundless. As we learn more and more and solve problems of the past, we face newer problems that were not apparent previously. This is a perfect parallel to Gödel's incompleteness: each new discovery reveals truths we couldn't access before, creating an infinite horizon of knowledge. But unlike the pessimistic reading of Gödel, Deutsch shows this is cause for optimism.

The problems top AI labs are trying to solve and optimize right now would have been inconceivable to researchers in the field a decade ago. New questions in gene sequencing would've seemed crazy if you had posed them before CRISPR. New answers illuminate the path forward to better questions which will in turn give us answers that point the way towards even better questions. This endless cycle is the foundation of scientific progress. Regardless of the scale of the system, whether it's research for a new drug or writing a book, this process encourages us to reevaluate the manner in which we learn and grow. It seems clear that with this framework, the pursuit of knowledge should not be to arrive at a final answer that doesn't exist, but to generate better explanations that both solve problems and open new ones. Thus, I think the overarching point here is that when it comes to understanding the world around us, the goal should be progress over completion. The next question is: does this also apply to the world within us?

Introspective Limits

We've examined the implications of Gödel's work on external knowledge, but what happens when we turn the lens inwards? What does it mean to have complete knowledge of oneself? From my perspective, having complete knowledge of oneself entails the ability to perfectly and objectively introspect, judge actions and thoughts, and just generally fully understand the inner motivations and thoughts that make you unique. Intuitively, it seems idealistic to believe that all of these could be true, but let's dive deeper than intuition.

Thoughtful introspection is undeniably one of the most important skills someone can cultivate throughout the course of their lifetime. I personally see it as the best way to individually refine skills, correct mistakes, and progress. However, it does require a degree of objectivity, as subjective introspection may actually be a harmful practice as it could reinforce bad habits. The natural question then is: "Is perfect objectivity possible?" This is where we start to see a parallel with the incompleteness theorems: to achieve perfect objectivity, you'd need to step outside your own perspective and observe your entire mental system, except there is nowhere to step to. This reduces to the self-referencing problem, just like the ruler example. Thus, when it comes to introspecting, or more generally evaluating our thoughts, we can't possibly be 100% objective. It is simply not possible to understand the way we think by thinking about how we think.

If perfect objectivity is impossible, what causes this fundamental limitation? The concept of a strange loop was pioneered by American computer and cognitive scientist Douglas Hofstadter. The premise of this concept is that there exists a hierarchy of levels that are linked via some arbitrary relationship. Moving through these levels eventually brings you back to where you started, creating a self-referential loopTrying to understand your own bias is like trying to see your own blind spot. The thing doing the checking is exactly what needs to be checked. Consciousness, Hofstadter argues, is a result of this strange loop structure forming in your brain as you develop. As you experience more and more, your brain systematically models the world. The model inherently needs a perspective or vantage point, so your brain is included in it. This process iterates continuously as it fuels the ego and builds your sense of self. After a certain point, your "I" evolves into consciousness as a result of this self-referencing entanglement. The idea that human consciousness is a direct result of an inescapable self-referential loop is profound.

This naturally plays well with Gödel's insight of proving consistency outside of the system. Perfect introspection requires you to step outside of yourself and your strange loop to get a reliable vantage point. However, since this is clearly not possible, we rely on external sources to do it for us. This is why therapy works, or why diversity is an indicator in team success. We need external systems to see our blindspots. So, given the fundamental impossibility of perfect self-objectivity, how do we make decisions about right and wrong?

Formalizing Morality

Can we create a perfect moral system? A system like this would provide consistent and verifiable answers to any ethical dilemma. Interestingly, frameworks for evaluating morality and ethics are fundamentally different from what we have discussed thus far because the focus is not so much on the provability of an ethical decision, but on generating a consistent system to handle any given scenario. Thus, when I say perfect moral system, I more specifically mean any set of rules or guidelines that can be leveraged to programmatically evaluate how "good" a decision is. While the majority of decisions can be cut and dry in terms of evaluation, there are also plenty of dilemmas where moral frameworks start to decompose.

Immanuel Kant's Categorical imperative attempts to generalize morality into the universal sense: what one ought to do regardless of circumstance. Utilitarianism generalizes morality into maximizing the overall well-being of a system. These two attempts at a robust system portray the limitations of a general ethical system since completeness and consistency often become mutually exclusive in edge case scenarios. Kant's CI tells you to always tell the truth, even in a circumstance where telling the truth may cause extreme harm to someone else. Utilitarianism gives a clear answer to the classic trolley problem, where you are able to divert the path of a train to save five people at the expense of one. But what if that one person was a loved one? The utilitarian approach still says that five lives are more valuable, but is that how it really feels?

Similar to how Gödel claims that powerful systems have unprovable truths, moral frameworks have unsolvable dilemmas. In general, there is always a breaking point for ethical systems. If you keep digging into edge cases, eventually there will be a point where principles will point in opposite directions. While this does not necessarily imply that there is no "truth" or correct decision in any given edge case, it is simply just not derivable within the inconsistent system.

But perhaps this is not a flaw. The incompleteness of mathematics ensures that we will always have something left to discover; the incompleteness of ethics ensures that we will always have room to grow. Every decision becomes an opportunity for refinement. We move forward not through finality, but through progression. And perhaps that's exactly what Gödel's theorems have been telling us all along.

Ad Infinitum

The general response to Gödel's incompleteness theorems, especially in the domain of mathematics, was very doom-and-gloom. Hilbert himself had a hard time accepting the fact that his most ambitious goal had turned out to be impossible. He questioned his life's work, and mathematicians everywhere did the same. The idea that their practice could not be perfected was truly fear-inducing. While this response is justified, advancement did not come to a standstill. Research continued, breakthroughs were made, and all of a sudden it didn't really matter whether mathematics were meant to be perfected, because there are always two sides to the same coin. The side that tells you that something may not be able to be complete flips over to the side that tells you that growth has no ceiling. This is what Deutsch understood, what Gödel figured out, and what underlies almost everything else. I truly believe that you can be the best at anything;With the required work of course. Though "best" is probably an incomplete idea to begin with potential is boundless.