Gödel’s Incompleteness Theorems

Gödel’s Incompleteness Theorems

Two of the the greatest intellectual achievements of modern times might surprise you. Both were developed by Austrian mathematician and logician Kurt Gödel in 1931. They are called simply Gödel’s incompleteness theorems and apply to all of mathematics, formal logic, and even philosophy (epistemology in particular). The implications turned out to be deeply profound and have thrown all of mathematics, logic, and even philosophy into disarray ever since. Despite almost a century of attempts, no one has been able to disprove them. In fact, almost all attempts end up supporting, and even reinforcing and expanding them. They now are accepted as almost certainly true.

The theorems sound simple enough at first blush. The first incompleteness theorem states that in any consistent formal system (mathematics, logic, physics, etc) in which a certain amount of arithmetic can be carried out, there are statements of the language of which can neither be proved nor disproved in that language. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).

What is so shocking about these two simple theorem? They prove something devastating: that mathematics and logic is not complete. There will always be truths in reality that the system cannot prove. It means that some problems can NEVER be solved in some kinds of mathematics or logic. You can even try making new systems of math/logic (Algebra, Calculus, etc) but they ALL will have things they cannot prove. It meant that you might work on a mathematical, physics, or logic problem your whole life, and none of the systems we know about might be able to solve it – even though it might have a solution. There might even be some problems that if we make infinite numbers of logical or mathematical systems, we might STILL not be able to find a solution.

Veritasium did an absolutely fabulous video on the topic that’s worth a listen.

It blew my mind when I learned about Godel’s incompleteness theorems in college. Knowing that our tools are limited is frightening at first. It completely unseats our certainty that known mathematics or science as we have today is sufficient. In fact, we know it is NOT sufficient. In fact, we know that we’ll almost certainly have to make more logical systems for the rest of eternity. We can never have a grand unified theory of everything. There is no ‘bottom’ to reach.

Yet this opens the reality that there will ALWAYS be something new to learn and know. There will be countless other models that might work for problem we have but we haven’t found yet – even though each one will be flawed and incomplete in their own way.

Many purists find this knowledge to be disastrous. It rips the rug out from anyone that asserts we can know everything. Others were excited by the fact there will always be new developments. Others are left in awe that even our very universe/reality itself lacks the limits we have. Still others have taken this as proof of the infinite. I know at least one mathematician that believed it gave us proof of God.

I do believe in God – without question. Many people forget that the vast majority of modern science was developed by believers in God that saw no conflict with discovery of properties of the physical world. The idea that faith and science are incompatible is a very modern and absolutely incorrect train of thought.

Instead, I see this reality as much like ourselves. None of us are perfect, yet each of us has a uniqueness that might just express a great truth no one else in history has seen or could see. This is why life is so infinitely precious and a tragedy to all when even one life is lost. This is why it is a crime to all humanity when we decide suffering is reason to end a life or that a disadvantage life is a life not worth living when we have such contrary examples and saw exactly where that idea led too in the early 20th century during WW 2.

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