The space to valuation systems of Lukasiewicz logic is a dense set?

Question

Let $n$ be a integer number such that $n \geq 2$, define the set $L_n = \Big\{ \frac{i}{n-1} \mid 0 \leq i \leq n-1 \Big\}$.

Now define the space to valuation systems of Lukasiewicz logic:

$L = \displaystyle\bigcup_{i=2}^\infty L_i$

My question is: "L is dense set"?

Answer 1

$L$ is the set of all rationals between $0$ and $1$ (inclusive) - basically, $a\over b$ is the $a$th element of $L_b$ - so it's dense in $[0,1]$. (Of course, it's not dense in $\mathbb{R}$, but I suspect that's not what you're interested in.)