ω_1 and -ω_1 May Be the Only Minimal Uncountable Linear Orders

摘要

In 1971 Laver proved the following result, confirming a long-standing conjecture of Fraisse.rnTheorem 1.1 [10]. IfLi(i ＜ω) is a sequence of σ-scattered linear orders, then there exist i < j such that L_i- is embeddable into L_j. In particular, the a -scattered orders are well-founded when given the quasi-order of embeddability.rnHere a linear order is scattered if it does not contain a copy of the rationals; a linear order is σ-scattered if it is a countable union of scattered suborders.rnAround the same time, Baumgartner proved the following theorem. (As usual, ZFC is used to denote "Zermelo-Fraenkel set theory with the axiom of choice" and MA to denote "Martin's axiom".)