Problem $5$: Computability $(20$ Points$)$

Let $P$ be the set of all computer programs that can be written in your favorite programming language.

Argue that $($hard drive capacity limitations aside$),P$ is at most countably infinite.

In most programming languages, we can write programs to compute values as their output. If we consider programs that run forever in an infinite loop, we can even compute complicated real numbers. In the previous problem above, we are essentially writing a simple program that, in a loop, computes $\sqrt[\phantom{2}]{2}.$ As another example, infinite loops can be written that output the digits of $$ forever $($since $$ does not terminate$).$

Argue, however, that no matter how hard we try, most real numbers cannot be computed as the output of any program.

Argue that this is the case regardless of what programming language we use.