6.1.1. Introduction

Try these first

1. 1.

   What is a countably infinite set?

   Hint

   Think about how you can list elements.

2. 2.

   Give an example of an uncountable set.

   Hint

   Consider numbers that include decimals.

3. 3.

   What is the primary purpose of Cantor's diagonalization argument?

   • To prove all sets are countable
   • To show uncountable sets exist
   • To list all real numbers

   Hint

   Consider the result of the argument.

4. 4.

   True or False: The set of all binary strings of infinite length is countable.

   • True
   • False

   Hint

   Reflect on the properties of infinite sequences.

5. 5.

   Explain the implications of Cantor's diagonalization argument on the understanding of mathematics and infinity.

   Hint

   Reflect on the nature of different sizes of infinity.

6. 6.

   Imagine a new list of infinite binary strings. Construct a diagonal string and demonstrate how it differs from all strings in your list.

   Hint

   Use the flipping method on the diagonal.