Banach-Tarski Paradox

Definition

The Banach-Tarski Paradox is a theorem in set-theoretic geometry that states it is possible to take a solid ball in 3-dimensional space, divide it into a finite number of non-overlapping pieces, and then reassemble those pieces into two solid balls identical to the original. This paradox illustrates the counterintuitive nature of infinity and challenges our understanding of volume and measure, connecting deeply to the ideas of amenable groups and Følner sequences.

5 Must Know Facts For Your Next Test

1. The Banach-Tarski Paradox relies heavily on the Axiom of Choice, allowing for the selection of points from infinitely many sets.
2. The paradox demonstrates how conventional notions of volume break down when dealing with infinite sets, challenging intuitions about size and quantity.
3. In practical terms, the Banach-Tarski Paradox is not applicable in the physical world since it requires the use of non-measurable sets.
4. The pieces used in the paradox are highly non-constructive and cannot be described explicitly; they are purely mathematical abstractions.
5. The connection between the Banach-Tarski Paradox and amenable groups lies in understanding how certain groups can behave under infinite transformations and approximations.

Review Questions

• How does the Axiom of Choice play a role in the formulation of the Banach-Tarski Paradox?
  • The Axiom of Choice is essential for the Banach-Tarski Paradox as it allows for the selection of points from an infinite collection of disjoint sets. Without this axiom, it would be impossible to create the non-overlapping pieces necessary for the paradox. This reliance on the Axiom highlights how foundational concepts in set theory influence geometric results, further complicating our understanding of volume.
• Discuss the implications of the Banach-Tarski Paradox on traditional concepts of volume and measure in mathematics.
  • The Banach-Tarski Paradox challenges traditional notions of volume by demonstrating that one can create two identical copies of an object from a single original object, which contradicts basic principles of measure theory. This paradox illustrates that our intuitive understanding of size and quantity does not hold in the realm of infinite sets. It raises important questions about what it means for a set to have a 'size' and prompts mathematicians to reconsider how measures are defined within different contexts.
• Evaluate how the Banach-Tarski Paradox can be related to concepts within ergodic theory, particularly regarding amenable groups.
  • The Banach-Tarski Paradox reveals deep connections between set theory and ergodic theory, especially through amenable groups. Amenable groups possess invariant means that enable them to behave nicely under limits and approximations. The paradox's implications for volume can help understand how these groups manage infinite processes while preserving certain properties. Analyzing such relationships enhances our grasp of both mathematical frameworks, showing how abstract results like Banach-Tarski can inform our understanding of dynamical systems.

Related terms

Axiom of Choice: A principle in set theory stating that given a collection of non-empty sets, there exists at least one choice function selecting an element from each set.

Measure Theory: A branch of mathematics that studies ways to assign a size or measure to sets, focusing on properties such as volume and integration.

Amenable Group: A type of group that has an invariant mean, which means it can be approximated by finite sets in a way that preserves certain properties, often leading to interesting results in ergodic theory.