Unmöglichkeitstheorem

Unmöglichkeitstheorem translates from German to "impossibility theorem." In mathematics and computer science, an impossibility theorem is a statement that proves that a certain type of object, system, or solution cannot exist or be constructed under a given set of rules or axioms. These theorems often arise in fields dealing with formal systems and logical constraints.

A well-known example is Gödel's incompleteness theorems, which demonstrate inherent limitations in formal axiomatic systems, implying that no consistent, sufficiently powerful system can prove all true statements within its domain. Another is the impossibility of trisecting an arbitrary angle using only a compass and straightedge, a result stemming from Galois theory. In computer science, the Halting Problem, proven by Alan Turing, shows that no general algorithm can determine whether an arbitrary program will eventually halt or run forever. These theorems highlight fundamental boundaries in what can be known, computed, or constructed within specific theoretical frameworks. They are crucial for understanding the scope and limitations of various disciplines.