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Suppose I'm told that for a particular function $V$ from sets to real numbers and any two sets $X, Y$ such that $V(X) < V(Y)$, I can name a set $Z$ such that $V(X) < V(Z) < V(Y)$.

I'm trying to show that given two sets $X, Y$ and a number $w$ such that $V(X) < w < V(Y)$, I can find a set $W$ such that $V(W) = w$.

However, I'm having an incredibly difficult time showing this - any guidance would be appreciated.

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    A function that takes sets to real numbers - I'll update the question. – 2017-01-11
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    Your question is incredibly vague. – 2017-01-11
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    You’re having a hard time because the result is false (and I see that Hagen has provided a counterexample). – 2017-01-11
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    Were there any assumptions about the function $V$ that you forgot to mention? – 2017-01-11

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Let $\iota\colon \Bbb N\to \Bbb Q$ be a bijection. If $S$ is any set, let $$V(S)=\begin{cases}\iota(\min(S\cap \Bbb N))&\text{if }S\cap\Bbb N\ne \emptyset\\ \pi&\text{otherwise}\end{cases} $$ Then $V$ has the desired denseness property, but is not onto $\Bbb R$.