6.1.8. Conclusion

Try these first

1. 1.

   What is a countably infinite set? Provide an example.

   Hint

   Think of numbers you can count!

2. 2.

   What does it mean for a set to be uncountable?

   Hint

   Remember Cantor's work?

3. 3.

   What is true about uncountable sets?

   • They can be matched with natural numbers
   • They cannot be enumerated
   • They are finite

   Hint

   Think of sets that go beyond counting.

4. 4.

   Cantor's diagonalization argument is used to prove that which of the following is uncountable?

   • The set of integers
   • The set of all infinite binary strings
   • The set of rational numbers

   Hint

   Which example did we focus on when applying this argument?

5. 5.

   Prove that the set of all subsets of the natural numbers is uncountable using Cantor's diagonalization argument.

   Hint

   Consider how every subset can relate back to Cantor's argument.

6. 6.

   Discuss how Cantor’s argument implies the existence of different 'sizes' of infinity.

   Hint

   Think about the relationships between sets established.