Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose

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

share|improve this question
add comment

2 Answers

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: http://math.stackexchange.com/a/174404/19661

share|improve this answer
add comment

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.)

share|improve this answer
5  
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.