Why some mathematical theorems will always be unprovable
A statement can be true or false. But as Kurt Gödel demonstrated, there will always be mathematical assumptions that can neither be proven nor disproven
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My friends and colleagues often ask me to help with number-related questions. After all, I know a lot about math. Ironically, I’m actually quite bad at mental arithmetic.
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What many people don’t realize is that the academic subject of mathematics is not about doing quick sums and subtractions in your head. In fact, it wasn’t until I went to university that I understood what truly drives this abstract discipline. Mathematics is about creating worlds.
To do this, you establish a foundation from a few conclusive assumptions, so-called axioms, on which you gradually build. Increasingly complex interrelationships emerge, until you finally arrive at highly complex topics at the forefront of current mathematical research. In the process, you move up from elementary sets to numbers, from there to functions and finally to geometry, topology and more abstract areas.
Everything in mathematics therefore rests on the axioms, or basic building blocks, of the field. And it took until the beginning of the 20th century to come up with the axiom system we have today. That’s because its creation resembled a balancing act: On the one hand, you want to make as few assumptions as possible. On the other hand, these rules should provide enough flexibility to generate all modern mathematics. Moreover, the axioms should be intuitive. For example, it seems plausible to assume that an empty set exists.
Ultimately, most experts now agree on a framework called the Zermelo-Fraenkel set theory with the axiom of choice, or ZFC for short. It consists of nine basic assumptions.
All this mathematical world-building might lead you to think that mathematicians have it all figured out. But some of the most exciting and shocking findings in this field underscore the unknowability of certain truths, even within a system that has been carefully built from the ground up.
Gödel Lets the Dream Burst
In the 20th century, many mathematicians dreamed of finding a foundation for mathematics that was both complete (meaning all mathematical truths can be proven with it) and consistent (such that it did not lead to contradictions). But in 1931, a logician who was then just 25 years old, Kurt Gödel, destroyed these hopes.
His first incompleteness theorem states that there are necessarily unprovable statements in all sufficiently strong, contradiction-free systems. As if that were not enough, he added a second incompleteness theorem, according to which sufficiently strong contradiction-free systems cannot prove that they are contradiction-free.
That is, once you find a foundation powerful enough to produce the known correlations of modern mathematics, it necessarily contains statements that can neither be proven nor disproven. Moreover, the system itself cannot prove its own consistency.
As befits a logical proof, Gödel’s argumentation was very abstract and high-level. Therefore, his colleagues initially hoped that the young mathematician had found a purely academic oddity that would have no practical implications. But they were mistaken.
And the ZFC system has numerous examples of statements that cannot be proven—underscoring that Gödel was right. Probably the most famous is the so-called continuum hypothesis, which deals with the question of whether there is an infinity—or possibly several—whose size is between that of the infinity of all natural numbers and the provably larger infinity of all real numbers. Without extending the foundation of mathematics, we will never be able to get to the bottom of this question.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
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