Does there exist a totally-ordered-without-endpoints proper class $L$ such that every closed interval in $L$ does have the order type of a closed interval in the Conway's surreal numbers, but $L$ as a whole does not have the order type of Conway's surreal numbers?


In case it helps, the thinking that lead up to my posting this question:


I confess proper classes make me a bit uneasy. So for private intuition I use as a crutch set theory below a strongly inaccessible cardinal $\kappa$. If one considers, as a toy surreal numbers, just Conway's surreal numbers with birthdays less than $\kappa$, then it seems to me that one can then imitate the usual long line construction based on $\kappa^+$. 


That said, my understanding says one should find the surreal numbers and in particular the ordinals (as individuals) in ZFC but not the totality of surreal numbers. Perhaps that means one can't define any proper class that will play the role of $\kappa^+$ and fulfill my intuition. 



Does there exist a surreal long line?
There are many such structures. For example, simply form L = N x [0,1) in the dictionary order, with the least element deleted, where N is the set of natural numbers including 0, and [0,1) is the semi-open interval of surreals. Since No (i.e. the ordered class of surreals) does not have a cofinal subset and L clearly does, L is not order-isomorphic to No. Moreover, that every closed interval of L is order-isomorphic to a closed interval of No is straightforward.


That fact that No has no cofinal subset follows from Theorem 2 of my paper:“The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. The theorem states that (in NBG with global choice):


No is (up to isomorphism) the unique absolute linear continuum.


An ordered class A is said to be an absolute linear continuum if for all subsets X and Y of A where X < Y (every member of X precedes every member of Y), there is a z in A such that
X < {z} < Y. 


The absence of a cofinal subset in an absolute linear continuum is established by simply letting Y be the empty set. 


