Infinity is bigger than you think - Numberphile

Summary

In this thought-provoking video, James Grime introduces the concept of infinity, explaining its surprising complexities. He explores the idea of countable infinity, or 'listable' numbers, such as whole numbers, integers, and fractions, which can be arranged in an infinite list. However, he delves deeper into the notion of uncountable infinity, revealing that some infinities, like real numbers, cannot be listed—demonstrating this through a fascinating diagonal argument. Grime highlights Georg Cantor's revolutionary discovery of different sizes of infinity, once ridiculed but now widely accepted, leaving viewers with a profound understanding of infinity's true nature.

Takeaways

• 😀 Infinity is not a number, but a concept that represents something endless, going on forever.
• 😀 The idea of infinity is something that even children encounter early on when they start counting.
• 😀 There are different types of infinity, and some infinities are actually bigger than others.
• 😀 The first type of infinity is called countable (or listable), where numbers can be listed, like whole numbers and integers.
• 😀 Whole numbers (1, 2, 3, ...) and integers (positive and negative numbers) are examples of countable infinities.
• 😀 Surprisingly, fractions can also be listed, though you need a more clever approach to do so.
• 😀 To list fractions, you can use a diagonal method, ensuring all fractions are eventually covered.
• 😀 Real numbers, such as decimals, can't be listed in a similar way because they are uncountable, meaning there's no way to list them all.
• 😀 A demonstration of the uncountability of real numbers shows that by creating a new number based on diagonals, one can always generate a number not on the original list.
• 😀 This idea of different sizes of infinity was proposed by the mathematician Georg Cantor, who faced significant ridicule but was eventually recognized for his work.
• 😀 Cantor's work on different types of infinity paved the way for understanding mathematical concepts involving uncountable infinities.

Q & A

What is infinity, and how is it different from a number?

Infinity is not a number, but rather a concept or idea. It represents something that goes on endlessly, without limit, and cannot be fully counted or measured.

Why does James Grime prefer the term 'listable' over 'countable' for some infinities?

James prefers 'listable' because, while infinity cannot be counted, some sets of numbers can be listed in an ordered way, such as whole numbers or integers. He believes 'listable' more accurately reflects this concept.

What is the significance of the difference between countable and uncountable infinities?

Countable infinities are those that can be listed (like whole numbers or fractions), while uncountable infinities cannot be listed in any systematic way. The difference lies in the size and nature of the infinity, with uncountable infinities being 'larger'.

How does James Grime illustrate the idea of countable infinity with fractions?

James illustrates countable infinity with fractions by arranging them in a grid (a rectangle) and listing them diagonally. This method ensures every fraction will eventually appear in the list, showing that fractions, despite their infinite nature, are countable.

What problem arises when attempting to list all decimal numbers (real numbers)?

The problem with listing all decimal numbers is that no matter how many you list, there will always be new decimal numbers that aren't included, which means they cannot be listed in a complete way.

What technique does James Grime use to prove that real numbers cannot be listed?

James uses the diagonal argument: by constructing a new number from the diagonals of a supposed list of decimals, he ensures that this new number is different from every number in the list, showing that it's impossible to list all real numbers.

What is the main difference between countable and uncountable infinities in terms of their size?

Uncountable infinities are 'larger' than countable infinities. Even though both are infinite, uncountable sets (like the real numbers) are so large that they cannot be matched one-to-one with the natural numbers, unlike countable sets.

Who is Georg Cantor, and what was his contribution to the understanding of infinity?

Georg Cantor was a German mathematician who developed the theory of different sizes of infinity. He showed that some infinities are larger than others and proved that real numbers form an uncountable infinity. His work was initially ridiculed, but later became foundational in mathematics.

What was the initial reaction to Cantor's theory of different sizes of infinity?

Cantor's theory was initially met with ridicule and harsh criticism. He was called a charlatan, and his ideas were dismissed as nonsense by many of his contemporaries.

How did Cantor's work eventually gain recognition?

Though initially rejected, Cantor's ideas were eventually recognized as groundbreaking. By the end of his life, his work on different sizes of infinity became widely accepted, and he gained the recognition he deserved, even being featured on Numberphile.