How big is infinity? - Dennis Wildfogel

TL;DR

Different types of infinity exist; rational vs. irrational numbers, subsets, and unanswerable questions.

Transcript

When I was in fourth grade, my teacher said to us one day: "There are as many even numbers as there are numbers." "Really?", I thought. Well, yeah, there are infinitely many of both, so I suppose there are the same number of them. But even numbers are only part of the whole numbers, all the odd numbers are left over, so there's got to be more whole... Read More

Key Insights

• ♾️ The concept of infinity in mathematics entails different sizes of infinities, such as rational and irrational numbers.
• 😫 Georg Cantor's work revolutionized the understanding of infinity, showcasing the uncountable nature of certain sets.
• 💦 The continuum hypothesis, a fundamental mathematical problem, remained unsolved until Gödel and Cohen's groundbreaking work.

Questions & Answers

Q: How do even numbers have the same quantity as all numbers?

Even numbers and whole numbers are shown to have the same quantity as each can be matched up one to one, proving that both sets have the same number of elements.

Q: How did Georg Cantor create a list of all fractions?

Cantor arranged fractions in a grid and created a list by sweeping diagonally, skipping repeated fractions, demonstrating a one-to-one correspondence between fractions and whole numbers.

Q: Why can't all real numbers be put on a list?

Cantor's diagonal argument illustrates that not all decimal numbers can be listed as producing a new number not on any list, proving the existence of an uncountable infinity beyond whole numbers.

Q: What does the continuum hypothesis state?

The continuum hypothesis suggests an infinite set of real numbers between whole and decimal numbers, unresolved until Gödel and Cohen showed its unprovability.

Summary & Key Takeaways

• The concept of infinity in mathematics is explored, highlighting how even numbers have the same quantity as all numbers due to matching elements.

• The idea of forming a list of all fractions is discussed through a clever grid method by Georg Cantor.

• Cantor's groundbreaking work on the continuum hypothesis and the discovery of unanswerable questions in mathematics is presented.