Applying Zorn's lemma to Cantor set

Question

Let $C_n$ be the set obtained in the $n$-th step for constructing the Cantor set. It is not hard to see that $C_i \subseteq C_j$ if $i>j$ and we know that the Cantor set is $\cap_{n=0}^{\infty} C_n$. However, recently, while doing some exercise concerning number theory, I noticed that we may apply Zorn's lemma to $C_n$. Obviously, each $C_n$ is non-empty and is a subset of $[0,1]$, thus there is a partial order defined by inclusion. If we consider the chain $$C_0 \supseteq C_1 \supseteq ...\supseteq C_n \supseteq ...$$ By Zorn's lemma, it seems to me that this becomes stationary after $n$ steps(everything is trivially lower-bounded by $\varnothing$), which contradicts what we know about Cantor set. Could anyone tell me where I have made a mistake?

Answer 1

It is important to note that you are working with a slightly different version, with "maximal" and "upper" replaced by "minimal" and "lower" respectively. Keeping that in mind, your problem arose because you have not made it completely clear what the relevant poset is. If the poset is $$\{C_0,C_1,C_2,...\}$$ with inclusion, then Zorn's lemma tells us nothing, since the chain $$\{C_0,C_1,C_2...\}$$ has no lower bound in this poset.

If the poset is $$P([0,1])$$ Then indeed Zorn's lemma guarantees that there is a minimal element, but this is trivial since $$\emptyset\in P([0,1])$$

I suspect that it is one of these that you intended, if not, once you explicitly write down the poset, and actually verify that the conditions of the lemma hold, then all confusion will disappear