I have a question about the following passage:
How is this $F$ injective? For example, $0.0111\sim = 0.1000\sim$. I don't understand how "discarding the nonstandard representations" is defined. The domain of $F$ clearly contains $g$ and $g'$ such that $F(g) = 0.0111\sim = 0.1000\sim = F(g')$. Thanks for help.