How big is infinity? - Dennis Wildfogel

Summary

This script explores the concept of infinity and the surprising nature of different infinities. It discusses how even numbers and whole numbers have the same cardinality, despite seeming to be a subset. It also delves into Cantor's diagonal argument, showing the unlistable nature of real numbers, and touches on the continuum hypothesis, highlighting the existence of unanswerable questions in mathematics.

Takeaways

• 🤔 The concept of infinity can be counterintuitive; even numbers and whole numbers are both infinite but perceived differently.
• 🔍 The idea of matching elements one-to-one is fundamental in establishing that two sets have the same size, regardless of the method of counting.
• 🐑 Ancient people may have used one-to-one correspondence to keep track of items without the need for advanced counting methods.
• 🪑 The auditorium example illustrates that matching elements (chairs to people) can show sets are the same size without knowing the exact number.
• 🔢 The teacher's demonstration with whole numbers and their doubles visually shows a one-to-one correspondence between even numbers and all numbers.
• 📝 Georg Cantor's method of listing fractions in a grid and then diagonally sweeping to create a list shows a one-to-one match with whole numbers.
• 🈲 Cantor's diagonal argument proves it's impossible to list all real numbers (both rational and irrational) because a new number can always be constructed not on the list.
• 🔑 The existence of different sizes of infinity was a groundbreaking idea by Cantor, which showed there are larger infinities beyond that of whole numbers.
• 🌌 The analogy of rationals being like stars and irrationals like the blackness of the night sky illustrates the vastness of the set of irrational numbers.
• 🛑 Cantor's work on infinities was initially controversial and led to personal struggles, but his ideas are now fundamental in mathematics.
• 💡 The continuum hypothesis, the idea of finding an intermediate infinity between whole numbers and real numbers, was shown to be unprovable, indicating inherent limitations in mathematics.

Q & A

What is the initial confusion the speaker expresses about even and odd numbers?

The speaker initially believes that since even numbers are only a part of whole numbers, with odd numbers being the remainder, there should be more whole numbers than even numbers.

What concept does the speaker introduce to explain the idea of two sets being the same size?

The speaker introduces the concept of one-to-one correspondence, which means that the elements of two sets can be matched up one by one, indicating that the sets are the same size.

How did ancient people without numbers greater than three keep track of their sheep?

Ancient people would set aside a stone for each sheep as they went out to graze, and then put the stones back one by one when the sheep returned, allowing them to know if any were missing without counting.

What did the speaker's fourth-grade teacher demonstrate to show that there are as many even numbers as there are numbers?

The teacher laid out the whole numbers in a row and below each number placed its double, showing a one-to-one match between the numbers and the even numbers, thus proving there are as many even numbers as there are numbers.

What was Georg Cantor's method for listing all fractions?

Cantor's method involved placing all fractions into a grid and then creating a list by starting at the upper left and sweeping back and forth diagonally, skipping over fractions that represent the same number as ones already on the list.

Why can't we list all decimal numbers according to Cantor?

Cantor showed that it's impossible to list all decimal numbers because for any list one might claim to have created, he could construct a new decimal number that is not on the list by ensuring it differs from the numbers on the list at specific decimal places.

What is the difference between rational and irrational numbers?

Rational numbers are fractions, which are ratios of whole numbers. Irrational numbers, on the other hand, are non-repeating infinite decimals, like the square root of two and pi.

What is the significance of Cantor's discovery that there are different sizes of infinity?

Cantor's discovery challenges the traditional understanding of infinity as a singular concept, showing that there are actually an infinite number of infinities with different sizes, which has profound implications for mathematics and our understanding of the universe.

What is the continuum hypothesis proposed by Cantor?

The continuum hypothesis is Cantor's conjecture that there are no infinities of different sizes between the infinity of whole numbers and the infinity of real numbers (decimal numbers).

How did Gödel and Cohen's work impact the continuum hypothesis?

Gödel showed that one can never prove the continuum hypothesis is false, and Cohen showed that one can never prove it is true, indicating that there are unanswerable questions in mathematics.

What does the conclusion that mathematics has limitations imply about the nature of mathematical inquiry?

The conclusion that mathematics has limitations suggests that while it is a powerful tool for understanding the world, it is not all-encompassing, and there are questions that may remain forever beyond its reach.