If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$.


I am trying to prove the following:


* If $(M,+,.,0,1)$ is a model of open induction, (or equivalently, the set of positive elements of an integer part of a real closed field) then $s(M,<)$ where $x < y$ is defined by $\exists z(z \neq 0 \wedge x + z = y)$ is regular.


I can prove this when no element of $M$ has a cofinal sequence of powers using a real closed field having $M \cup (-M)$ as an integer part. However, I wonder if my proof misses the point.


I am trying to find some literature about $s(E,<)$ where it could have been studied and where its regularity may have been questioned. Does anyone know where to find such study?



Least ordinal not embedded in a total order
Your claim isn't true.


For a counterexample, let's construct a $\kappa$-like model. A model of arithmetic (or indeed any ordered structure) is $\kappa$-like, if it has size $\kappa$, but every proper initial segment of it has size less than $\kappa$. 


One can construct a $\kappa$-like model $M\models\text{PA}$ for any uncountable $\kappa$, including singular $\kappa$, by iteratively using the MacDowell-Specker theorem, which asserts that every model of arithmetic has a proper elementary end-extension (and so it has one of the same cardinality). Specifically, let $M\_0\models\text{PA}$ be any countable model of arithmetic, and construct a tower of end-extensions 
$$M\_0\prec\_e M\_1\prec\_e\dots\prec\_e M\_\alpha\prec\_e M\_{\alpha+1}\prec\_e\dots\prec\_e M$$
where at successor stages we apply the MacDowell-Specker theorem, choosing $M\_{\alpha+1}$ to have the same size as $M\_\alpha$, and at limit stages take unions. Let $M=\bigcup\_{\alpha<\kappa}M\_\alpha$ be the model at stage $\kappa$, which will be $\kappa$-like, since the initial segments are contained in the various $M\_\alpha$, which have size $|\alpha\cdot\omega|$, which is less than $\kappa$. 


It follows that $\kappa$ order-embeds into $\langle M,<\rangle$, since we added new points on top at each stage, but $\kappa+1$ does not embed, since no bounded segment of $M$ has size $\kappa$. So your characteristic $s(M,<)$ will be $\kappa+1$, which is not a regular cardinal. So this is a counterexample.


You asked for literature, and there seems to be a lot of work on $\kappa$-like models, such as:


* R. Kaye, Notre Dame Journal of Formal Logic, [The theory of $\kappa$-like models of arithmetic](https://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1040136915), vol. 36, no. 4, 1995.


