Why some mathematical theorems will always be unprovable

Mathematics is not primarily about quick mental calculations. It is about creating structured “worlds” by starting from a few conclusive assumptions called axioms. From these axioms, increasingly complex interrelationships are built, leading to advanced topics in current research. The development moves from elementary sets to numbers, then to functions, and onward to geometry, topology, and other abstract areas. All of mathematics depends on the axioms as basic building blocks. A modern axiom system took shape only in the early 20th century, balancing minimal assumptions with sufficient flexibility to generate modern mathematics. The axioms also aim to be intuitive, such as the plausibility of an empty set. Many experts use Zermelo-Fraenkel set theory with the axiom of choice (ZFC), which includes nine basic assumptions.

"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."

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