Basisannahme

Basisannahme, also known as the axiom of choice, is a fundamental principle in set theory and mathematics. It states that for any set of non-empty sets, there exists a function that selects exactly one element from each set in the collection. In other words, it asserts the existence of a choice function for any family of non-empty sets.

The axiom of choice was first introduced by Ernst Zermelo in 1904 to prove the well-ordering theorem, which states that any set can be well-ordered. Since then, it has become a cornerstone of modern mathematics, with numerous applications in various fields such as topology, measure theory, and functional analysis.

Despite its widespread use, the axiom of choice is controversial due to its non-constructive nature. Critics argue that it leads to counterintuitive results, such as the Banach-Tarski paradox, which states that a solid ball can be decomposed into a finite number of disjoint pieces that can be reassembled into two identical copies of the original ball. However, proponents of the axiom of choice maintain that it is a natural and necessary principle for mathematical reasoning.

In 1963, Paul Cohen proved that the axiom of choice is independent of the other axioms of set theory, meaning that it cannot be derived from them and cannot be disproven without additional assumptions. This result has led to the development of various alternative set theories that reject the axiom of choice or modify its formulation. Despite these debates, the axiom of choice remains an essential tool for mathematicians and continues to be widely accepted in the mathematical community.