TITLE: Is there a linear order with this property
QUESTION [7 upvotes]: I was trying and failing to construct a linear order L each of whose uncountable subsets contains an uncountable well ordered subset but L is not a countable union of well ordered subsets. Is this possible?
Thanks for any ideas!

REPLY [2 votes]: For each countable limit ordinal $\alpha$ choose some $\langle \alpha_n : n < \omega \rangle$ increasing and cofinal in $\alpha$. Then it can be checked that the set $L$ of these sequences ordered lexicographically satisfies your requirements.