Suppose


* $T$ is a [totally ordered set](http://en.wikipedia.org/wiki/Total_order) without a [maximal element](http://en.wikipedia.org/wiki/Maximal_element),
* $\tau$ is the [order type](http://en.wikipedia.org/wiki/Order_type) of $T$,
* $S$ is the set of order types of all proper [initial segments](http://en.wikipedia.org/wiki/Initial_segment) (downward closed subsets) of $T$.


Is it always possible to unambiguously reconstruct $\tau$ from $S$?



Is it possible to reconstruct an order type from its initial segments?
The answer is **No**.


Let $\eta$ be the order type of $\mathbb{Q}$, and $\omega\_1$ - the order type of the set of countable ordinals. The order types $\eta$ and $\eta \cdot \omega\_1$ are different (because they have different cardinality), but the set of order types of all proper initial segments of some instances of $\eta$ and $\eta \cdot \omega\_1$ is the same. Actually, as proved by [Joel David Hamkins](https://mathoverflow.net/users/1946/joel-david-hamkins), there are $2^{\aleph\_1}$ distinct order types with this property: <https://math.stackexchange.com/a/174404/19661>


