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

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up vote 15 down vote accepted

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:

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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.)

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Well, not the single one: there are $\eta$ and $\eta+1$ (and maybe $\varnothing$ depending on the definition of "proper"). – Vladimir Reshetnikov May 6 '13 at 22:11
correct, thank you. – Goldstern May 6 '13 at 22:16

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