Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What uncountable ordinals live in the Long Line?

  1. Apr 25, 2013 #1
    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.

    Thank You in Advance.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted