Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the order types that can be formed in this way more restricted than that?



Order types of positive reals
Yes, one can have any countable ordering. Indeed any countable totally
ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as
$ \lbrace a\_1,a\_2,\ldots \rbrace $
and define the embedding recursively: once you have placed $a\_1,\ldots,a\_{n-1}$
there will always be an interval to slot $a\_n$ into.


