I'm looking for a countable set S with a partial order < that has an uncoubtable number of maximal chains. I had many ideas but non of then is correct (for example- S= natural numbers, "<" is division- a < b iff a|b. But I think that in this case the numebr of maximal chains is countable).
Thanks!
Take $S$ to be the set of all finite binary strings, and say $a\lt b$ iff $a$ is a prefix of $b$. Then every infinite binary string defines a maximal chain in $S$, and there are uncountably many infinite binary strings.
You can picture this same example as an infinite binary tree, which has countably many nodes but uncountably many branches; here $a\lt b$ iff $a$ is an ancestor of $b$.