A question about cofinal totally ordered sets.

Question

Let $A$ be an uncountable set, and let $L$ be the poset consisting of all finite subsets of $A$ (the ordering on $L$ is inclusion). Show that $L$ does not have a totally ordered cofinal subset.

I am not really sure about the best way to go at this. I am not sure how to deal with uncountable sets; this is new to me. Thanks in advance.

Answer 1

HINT: Suppose that $C$ is linearly ordered and cofinal in $L$.

• Show that since $C$ is cofinal in $L$, it must be uncountable.
• Show that there is some $n\in\Bbb Z^+$ such that uncountably many members of $C$ have cardinality $n$.
• Derive a contradiction.