# What uncountable ordinals live in the Long Line?

1. Apr 25, 2013

### lugita15

It is a relatively simple exercise to prove that a well-ordered set is order-isomorphic to a subset of R (under the usual ordering) if and only if it is countable. You can say that R is "too small" to contain any uncountable well-ordered sets.

So my question is, can you embed bigger well-ordered sets in the long line? For those who don't know, the long line can be constructed by taking the minimal uncountable well-ordered set (i.e. omega_1) and taking its Cartesian product with [0,1) under the dictionary order. So obviously omega_1 itself is emebeddable in the long line, just by taking the left endpoints of all the intervals [0,1). But can you embed bigger uncountable ordinals, and if so how big? I'm guessing that you may be able to embed all well-ordered sets with cardinality less than or equal to aleph_1, the cardinality of the set of countable ordinals.

Any help would be greatly appreciated.