I'm trying to solve a problem, using the diagonal lemma (I know how to solve it with a different method).
For some function f ∈ N → (N → N) I need to find a set E ⊆ N → N, with cardinality |E| = ℵ, such that E ∩ Imf = ∅.
I thought about something like g(n) = (f(n))(n) + k, k∈N. But then |E| = ℵ0, which is not good enough.
any help will be appreciated, thanks!
HINT: for any $n$, as long as $g(n)=(f(n))(n)+k$ for $k$ positive, $g\not=f$. So you have some freedom in picking $k$. In particular, it can be a different $k$ for different $n$s! Do you see a way, given an infinite sequence $(a_i)_{i\in\mathbb{N}}$ of zeroes and ones (there's continuum many of these), to come up with a sequence $(k_i)_{i\in\mathbb{N}}$ of positive integers ("things to add") based on that sequence? Do you see how to use this to define such an $E$, and prove that it has the desired properties?