Mathematician Explains Infinity in 5 Levels of Difficulty | WIRED
WIRED・23 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.
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Summary
00:00
Understanding Infinity: Mathematics Beyond Countable Limits
- Mathematicians have developed precise ways to reason about the concept of infinity, which is something that never ends.
- Infinity is contrasted with finiteness, which refers to something countable, like the number of Skittles in a jar.
- Mathematicians can build infinite collections and even larger sizes of infinity through repeated processes.
- Infinity is considered a number in mathematics, like 13 or 10 million, representing the size of a set.
- Hilbert's Hotel illustrates the counterintuitive properties of arithmetic with infinity, showing how adding guests can still result in the same infinity.
- Functions in mathematics define one-to-one correspondences between sets, crucial for reasoning about infinity.
- Georg Cantor's work demonstrates that the natural numbers and integers have the same size of infinity through one-to-one correspondence.
- Rational numbers are shown to have the same size of infinity as natural numbers through a geometric counting method.
- The real numbers, forming the continuum, are proven to have an uncountable infinity, distinct from the countable infinity of natural numbers.
- Practical implications of understanding different sizes of infinity include Alan Turing's work on computability and applications in fields like computer science, algebraic geometry, and category theory in mathematics.
15:09
Axiom of Choice and Set Theory Consequences
- The Axiom of Choice allows for selecting an element from each set in an infinite or arbitrary family of sets where none are empty simultaneously.
- Three equivalent forms of the Axiom of Choice are 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 proving remarkable results like covering three-dimensional Euclidean space with disjoint circles.
- The Banach-Tarski Paradox demonstrates that using rigid motions, one can double a solid ball's volume, showcasing the Axiom of Choice's unusual consequences.
- In set theory, the concept of inaccessible cardinals and Grothendieck Universes are introduced to establish boundaries for sets' sizes in categories.
- Cantor's diagonal argument proves that a set's power set has a larger cardinality, leading to the consideration of a universe of all sets and the Continuum hypothesis.
- The Continuum hypothesis remains undecidable within standard mathematical axioms, highlighting the need for transparency in assumptions and the potential impact of changing foundational theories on established theorems.
