TED-Ed: How big is infinity? - Dennis Wildfogel

The video explores the concept of infinity, starting with the idea that there are as many even numbers as whole numbers. It explains the fundamental concept of matching elements one-to-one to determine set sizes. Cantor's method for listing fractions is introduced, followed by a discussion on irrational numbers and their infinite nature. The video highlights Cantor's contributions to mathematics, including the notion of different sizes of infinity. It concludes with the continuum hypothesis, an unsolved problem in mathematics, and its implications for the limits of mathematical reasoning.

What is the fundamental idea behind determining if two sets are the same size?

• Matching elements one-to-one

• Counting the elements in each set

• Comparing the largest element in each set

• Using a mathematical formula

How did Georg Cantor demonstrate that fractions can be listed?

• By using a grid and diagonal sweeping

• By using a computer algorithm

• By counting them directly

• By arranging them in a circle

What distinguishes irrational numbers from rational numbers?

• They are always larger than rational numbers

• They are finite decimals

• They are infinite, non-repeating decimals

• They can be expressed as a fraction

What did Cantor prove about the list of decimal numbers?

• All decimals can be listed

• Decimals represent a smaller infinity

• Decimals are finite

• Decimals cannot be fully listed

What is the continuum hypothesis concerned with?

• The size of finite sets

• The existence of infinities between whole numbers and decimals

• The number of rational numbers

• The properties of even numbers

What did Gödel and Cohen's work reveal about the continuum hypothesis?

• It is irrelevant to mathematics

• It can be proven true

• It can be proven false

• It is undecidable

What was the impact of Cantor's ideas on mathematics?

• They were initially rejected but later accepted

• They were proven incorrect

• They were never accepted

• They were immediately accepted