No, for one very subtle reason and one very unsubtle one.
 Without an additional axiom (the "continuum hypothesis") it is not determined whether the cardinality of omega_1 is that of the real numbers. Consistent models of set theory with and without this axiom exist, and much of set theory can be done without regard to this question.
A much more familiar example of this sort of situation is found in geometry, where both Euclidean geometry (with the parallel axiom) and nonEuclidean geometry (without it) have consistent models, with a significant common fragment known as "absolute geometry" which can be proved without reference to the parallel postulate and whose results are valid in both geometries.
 The real numbers are not a wellordered set [not every subset has a least element] and thus their order type is not an ordinal.
Good Hunting!
RD
Justin,
In your question you say "Omega 1, the order structure of the real numbers", but omega 1 is a well ordered set and the real numbers, in their usual order, is not a well ordered set. If you are asking "Can the real numbers be reordered to have the order structure of omega 1?" then the answer is RD's point 1.
Harley
