The Mathematical Incompleteness of Ethics
How complete can our arguments be?
Can we ever fully prove what is right or wrong? Philosophers and ethicists have long debated whether moral principles can be established as objective truths. But as it turns out, mathematics—specifically Gödel’s Incompleteness Theorems—may have something important to say about this problem.
Kurt Gödel, a German mathematician of the 20th century, revolutionized our understanding of formal reasoning. His work shattered the optimistic belief that every truth could be logically proven from a set of axioms. This idea, known as provability, is central not only to mathematics but also to ethical discussions. Many people hope that moral questions can be settled definitively, that we can prove what is good and what is bad. But Gödel’s theorems suggest that no system—including ethics—can be fully self-contained and complete.
Gödel’s Incompleteness Theorems, Explained Simply
To understand the connection between mathematics and ethics, let’s first break down Gödel’s theorems into simple terms.
A formal system (such as arithmetic, logic, or even ethical principles) is said to be:
- Complete if every possible statement within the system can be classified as either true or false.
- Consistent if it never contains contradictions (i.e., no statement is both true and false).
Gödel’s First Incompleteness Theorem states that in any consistent system, there will always be true statements that cannot be proven within the system itself. In other words, some truths require reasoning from outside the system.
His Second Incompleteness Theorem goes even further: no system can prove its own consistency without relying on something external.
A simple way to understand this is by imagining a child asking “Why?” over and over again. No matter how many answers you give, the questioning never ends—each answer requires another explanation, and eventually, you might run out of answers entirely. That’s the essence of incompleteness.
A Train Station Thought Experiment
To visualize this concept, imagine you’re sitting in a train at a station. Another train is right next to you, also waiting to depart. Suddenly, you feel movement. But for a few seconds, you’re unsure—is your train moving, or is the other train moving?
At that moment, your understanding is incomplete. You don’t have enough information to determine the truth just by looking at the train next to you. Only when you glance outside at the platform or the landscape do you gain an external reference point that confirms your movement.
This is Gödel’s First Theorem in action. Sometimes, we can’t determine the truth of something by looking only within a system. We need outside information. And this is exactly what happens in ethical debates.
What Does This Mean for Ethics?
Now, let’s apply this idea to moral reasoning. Imagine ethics as a formal system, where statements like:
✅ “One should help others” (true)
❌ “One should harm others” (false)
… are the equivalent of logical truths and falsehoods.
But if Gödel’s theorems hold, then ethical systems are incomplete—there will always be moral truths that we cannot fully justify using ethics alone. We need something outside the system.
This realization points to a deep human intuition: throughout history, people have sought external validation for their moral beliefs. Many have turned to religion, which provides a system where moral statements are backed by divine authority. For believers, the words of God serve as axioms—absolute truths that do not require proof. In this way, religious morality offers a kind of theoretical completeness.
But what about those who reject religious explanations?
Modern Ethics and the Illusion of Completeness
For many, religion is no longer a satisfying answer to ethical questions—it relies on faith rather than empirical evidence. So, modern thinkers have tried to build rational systems of morality, using reason and logic to justify ethical principles.
The problem? Rational morality also requires an external foundation.
For example, some ethical systems rely on determinism—the idea that actions can be judged based on their predictable consequences. But this too requires faith, just in a different form: faith in reason itself. No matter how carefully we build a moral system, it will always be resting on something outside itself—something that we must accept as a given.
Gödel’s theorems reveal a fundamental problem: we may never have a final, provable answer to moral questions.
The Search for Meaning in an Incomplete World
So, does this mean we should give up on ethics? Should we stop trying to determine right from wrong? Not at all.
Gödel’s work does not destroy our ability to reason—it only reminds us that our reasoning will always have limits. Ethics may never be provably complete, but that doesn’t mean we shouldn’t try to understand it.
At the end of the day, we still have to make choices—whether or not we can prove them. In the limited time we have, we rely on a mix of logic, intuition, and experience to determine what feels right or wrong.
Perhaps the real lesson from Gödel’s theorems is not that we should stop searching for moral truth, but that we must accept the uncertainty that comes with it. Ethics may be incomplete, but our actions still matter. And in the end, whether you choose to act or not—you are still making a choice.