Intitial Segments, Posets

selenne431 · **Joined:** Sun, 15 Mar 2009 08:04:48 UTC **Posts:** 6

Prove that for all $\alpha, \beta$ in $\omega_1$ ,

$\text{seg}_{\omega_1}(\alpha) +_o \text{seg}_{\omega_1}(\beta) <_o \omega_1$

and

$\text{seg}_{\omega_1}(\alpha) \cdot_o \text{seg}_{\omega_1}(\beta) <_o \omega_1$ .

Notation: $\omega_1$ denotes the first uncountable ordinal. $\text{seg}$ above denotes the initial segment, $\cdot_o$ denotes the product of posets, +_o

denotes the sum of posets. $U <_o V \Leftrightarrow (\exists x \in V ) [U =_o \text{seg}_V(x)]$ . =_o

denotes order isomorphic.

S.O.S. Mathematics CyberBoard

Intitial Segments, Posets

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