TITLE: Is it possible to reconstruct an order type from its initial segments?
QUESTION [17 upvotes]: Suppose

$T$ is a totally ordered set without a maximal element,
$\tau$ is the order type of $T$,
$S$ is the set of order types of all proper initial segments (downward closed subsets) of $T$. 

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

REPLY [15 votes]: 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, there are $2^{\aleph_1}$ distinct order types with this property: https://math.stackexchange.com/a/174404/19661

REPLY [14 votes]: No. Take $\omega_1$, with each element replaced by a copy of $\mathbb Q$. Then $S$ will contain a single order type.  (The rest is left as an exercise.)