Mathematician Explains Infinity in 5 Levels of Difficulty | WIRED
WIRED・2 minutes read
Mathematicians explore the concept of infinity, contrasting it with finiteness and showcasing its implications in arithmetic, set theory, and real numbers. The Axiom of Choice plays a crucial role in understanding different sizes of infinity and leads to paradoxes like Banach-Tarski, while Cantor's work introduces new cardinalities and the undecidable Continuum hypothesis.
Insights
- Mathematicians have devised precise methods to understand infinity, an endless concept that contrasts with countable finiteness, like the number of Skittles in a jar, allowing for the creation of infinite collections and larger infinities through repeated processes.
- The Axiom of Choice, exemplified by the well-ordering principle and Zorn's lemma, showcases the ability to select elements from infinite sets simultaneously, leading to profound implications in mathematics, including transfinite induction and the Banach-Tarski Paradox, which illustrates the surprising consequences of doubling a solid ball's volume through rigid motions.
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Recent questions
What is infinity in mathematics?
Infinity in mathematics refers to a concept that represents something that never ends. Mathematicians have developed precise ways to reason about infinity, contrasting it with finiteness, which pertains to countable quantities like the number of Skittles in a jar. Infinity can be considered a number in mathematics, akin to 13 or 10 million, representing the size of a set. Through processes like building infinite collections and establishing larger sizes of infinity, mathematicians explore the intricate nature of this boundless concept.
How do mathematicians reason about infinity?
Mathematicians reason about infinity through precise methods that contrast it with finiteness, which involves countable quantities. They can build infinite collections and explore larger sizes of infinity through repeated processes. Infinity is considered a number in mathematics, akin to 13 or 10 million, representing the size of a set. Concepts like Hilbert's Hotel illustrate the counterintuitive properties of arithmetic with infinity, showcasing how adding guests can still result in the same infinity.
What is the Axiom of Choice in mathematics?
The Axiom of Choice in mathematics allows for selecting an element from each set in an infinite or arbitrary family of sets where none are empty simultaneously. It has three equivalent forms: the well-ordering principle, Zorn's lemma, and the well-ordering of the real numbers. The Axiom of Choice implies the existence of a well-ordering of the real numbers, enabling transfinite induction and leading to remarkable results like covering three-dimensional Euclidean space with disjoint circles.
How does Cantor's work relate to infinity?
Georg Cantor's work in mathematics demonstrates that the natural numbers and integers have the same size of infinity through one-to-one correspondence. He also showed that rational numbers have the same size of infinity as natural numbers through a geometric counting method. Additionally, Cantor's diagonal argument proves that a set's power set has a larger cardinality, prompting considerations of a universe of all sets and the Continuum hypothesis.
What are the practical implications of understanding different sizes of infinity?
Understanding different sizes of infinity has practical implications in various fields, including computer science, algebraic geometry, and category theory in mathematics. Alan Turing's work on computability is one example of how insights into infinity can be applied. The Axiom of Choice, which allows for selecting elements from sets in infinite families, has led to remarkable results like covering three-dimensional Euclidean space with disjoint circles. Overall, grasping the nuances of infinity can have significant impacts on theoretical and applied mathematics.