I made a mistake in the definition of f_0, here is the correct version: f_0(A) = Mi if and only if |A| ∈ N f_0(A) = Ma if and only if |A'| ∈ N f_0(A) = Un if

I made a mistake in the definition of f_0, here is the correct version: f_0(A) = Mi if and only if |A| ∈ N f_0(A) = Ma if and only if |A'| ∈ N f_0(A) = Un if and only if |A| = ∞ ∧ |A'| = ∞ I mixed up the concept of infinite and cofinite in my previous comment, oops. Actually I just realized that an even simpler example of a function satisfying the conditions would be f_1(A) = Un for all A, which is also the global minimum of the entire set.