So my brain is frazzled which is probably why this seems like a big deal to me right now, but I just can't get over this reasoning:
Suppose you have $$F = \{\text{all } 1\text{-}1, \text{increasing functions } \Bbb N \to \Bbb N\}$$
$1$-$1$: means that every value of the domain maps to some unique value of the range and every value of the range is equal to $f(n)$ for some $n$. Hence, since $f$ is $1$-$1$ and increasing, the only function that exists is the trivial function, $f(n) = n$.
Please tell me why that is wrong.
(ftr, this isn't the homework question. I am proving uncountability of $F$, which is why my brain-fart is bothering me more)
Answer: @Prometheus left this as a comment and then deleted it, probably because he didn't wish to be associated with stupidity as great as mine. The error is that I'm assuming one-to-one => onto, which it clearly doesn't. huddles in a ball and cries