- The goal for this is to show that there areinfinite sets that are uncountable
- By being uncountable, they can be shown to have a bijection with
- By being uncountable, they can be shown to have a bijection with
- For this proof by inversion, we assume that
is not uncountable. - Assume that
is not uncountable. So, must be countable then. Since is countable, consider an enumeration of .
Enumeration: Say you have some set
. To show that is is bijective to , you can list it in some infinite order , and . This is called an enumeration.
- Now consider the decimal expansion of the elements as follows, where
- Our goal is to find an element in
whose decimal representation differs from all - We need to find a
such that , , , etc.