Characterization and transformation of countryman lines and r embeddable coherent trees in ZFC

CHARACTERIZATION AND TRANSFORMATION OF COUNTRYMAN LINES AND R-EMBEDDABLE COHERENT TREES IN ZFC PENG YINHE (PhD, NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Peng Yinhe January 2013 Acknowledgements First of all, I would like to express my deepest gratitude to my supervisor Prof. Feng Qi who taught professional knowledge during my Ph.D study and provided guidance, discussions and suggestions all the time during my Ph.D research. I couldn’t carry out any research without his systematic training. I would like also to express my deepest gratitude to my co-supervisor Prof. Stevo Todorcevic who offered irreplaceable guidance and advice to my mathematical research including the research contained in this thesis. His patient guidance has helped me to understand the meaning of research better. Besides, I wish to thank the rest logicians in Singapore for their supports on logic study and research: Prof. Chong Chi Tat, Prof. Stephan Frank, Associate Professor Yang Yue and Associate Professor Wu Guohua. It is great to discuss logic with them. My sincere also goes to Prof. W. Hugh Woodin and Prof. Theodore Slaman who helped to organize the summer school held in Beijing and Singapore. They did help a lot. I wish to thank Prof. Franklin D. Tall and Lecturer Shi Xianghui who helped me iii Acknowledgements iv when I was visiting Toronto. I also wish to thank Associate Professor Wang Wei and Prof. Yu Liang for their help and discussion during summer school and their visiting Singapore. I am grateful to the staffs in NUS and math department for their supports in different ways. I would like to thank my friends who are also interested in the mathematical logic: Yang Sen, Wu Liuzhen, Li Yanfang, Shao Dongxu, Zhu Yizheng, Zhu Huiling, Shen Demin, Li Wei, Liu Yiqun, Cheng Yong, Fang Chengling, Liu Jiang, Wang Shenling. Lastly, I want to thank my family, my relatives and my friends for their moral support. Peng Yinhe Jan 2013 Contents Acknowledgements iii Summary vii Introduction 1.1 Countryman lines and coherent trees . . . . . . . . . . . . . . . . . 1.2 Transformations under M Aω1 . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Countryman lines and coherent trees and their connections 12 2.1 Partition trees and lexicographical orders . . . . . . . . . . . . . . . 12 2.2 Some results under M Aω1 . . . . . . . . . . . . . . . . . . . . . . . 19 From R-embeddable coherent trees to Countryman lines 3.1 26 R-embeddablity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 v Contents vi 3.2 An R-embeddable coherent tree may be not Countryman . . . . . . 30 3.3 An equivalent condition for coherence being Countryman . . . . . . 40 From Countryman lines to R-embeddable coherent trees 53 4.1 Special trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Basis for Countryman lines . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 A Appendix 87 Bibliography 96 Index 99 Summary Chapter will review the background of linear orders and tree orders. Chapter will review and generate the results on transformation under M Aω1 mainly coming from [1]. In particular, we prove that, under M Aω1 , coherent and Countryman are two equivalent conditions. Chapter will investigate the transformation from a R-embeddable coherent tree to a Countryman line. We first show that every Countryman line has a R-embeddable partition tree and explain the necessity of R-embeddability. We then show that for any countable linear order O that cannot be embedded into Z, it is consistent to have a R-embeddable coherent tree T ⊂ O[...]... Countryman lines and coherent trees and lines, and moreover, it should direct us towards a new direction for further research and application of Countryman lines and coherent trees 1.2 Transformations under M Aω1 As we know that every partition tree of a Countryman line is an Aronszajn tree However, Aronszajn is not enough to describe the Countryman property, since for example, Countryman line is preserved...1.1 Countryman lines and coherent trees consistent to have a 2-element basis for Countryman lines – a Countryman line C and its reverse C ∗ Shelah’s conjecture remained out of reach until S Todorcevic in [5] introduced his method of minimal walks on ordinals and used it to produce a number of concrete trees and lines that are both Countryman (with their lexicographical orders) and coherent (any... for Countryman may give a new view on different type of Countryman lines, i.e., how simple can a countable linear order O be such that the Countryman line can be partitioned into a R- embeddable O-ranging tree It is understood that besides R- embeddable coherent trees, there are some other kinds of interesting coherent trees, for example, Souslin coherent trees, which can be transformed into a Souslin... description between coherent trees and Countryman lines under M Aω1 , and gives a natural guessing that this may be true in ZFC However, the above results are still limited to some forcing axiom which is independent of ZFC and neither transformation is known in ZFC One major difficulty in studying the transformation between linear order and tree order is that the transformation needs the lexicographical... whether it is a consequence of ZFC or its negation is consistent The main aims of this study were: 1 to summarize and give a complete description of the transformation between Countryman lines and R- embeddable coherent trees under M Aω1 2 to investigate whether R- embeddable is necessary 3 to investigate the transformation between Countryman lines and R- embeddable coherent trees in ZFC 4 to investigate... several properties of Countryman line and coherent tree for better understanding and further research Chapter 2 proves the existence of the transformation under M Aω1 which completes aim 1 Chapter 3.1 proves the necessity of R- embeddability which complete aim 2 Chapter 3.2 constructs different models that provides different relations between Countryman property and coherent property and some model contains... Countryman lines or coherent trees that I know) use Countryman (or coherent) lines that are both Countryman and coherent Therefore, a nature question arises: are they the same? The question actually has two sides: 1 is every Countryman line coherent? 3 1.2 Transformations under M Aω1 2 is every coherent line1 Countryman? The answer to each question should give us a better description of both Countryman. .. transform a linear order into a tree and “lexicographical order” is used to transform a tree order into a line See [13] or [2] for more on partition tree and lexicographical order It is easy to see that for a tree with only tree order there are different ways to define lexicographical order on it But in this section we will see that the partition tree of a linear order is kind of “unique” up to take club restriction... Aronszajn lines are also coherent and in fact the coherence plays an important role in his proof Besides the solution to the basis problem for Aronszajn lines, Countryman lines and coherent trees are useful in other problems For example [8], J Moore has used Countryman lines to construct (under PFA) a universal Aronszajn line – every Aronszajn line can be embedded into it (an analogue to the role of Q in the... necessary 2 The transformation between Countryman lines and R- embeddable coherent trees under ZFC The transformation exists by assuming some additional forcing axiom, and this can decide the consistency of the transformation But the transformation without any additional axiom is still unclear So to see the whole picture of the transformation between Countryman lines and R- embeddable coherent trees, . investigate the transformation between Countryman lines and R-embeddable coherent trees in ZFC. 4. to investigate several properties of Countryman line and coherent tree for better understanding and further. charac- terizing the Countryman line using coherent trees. 1.1 Countryman lines and coherent trees Even after the method of partition trees and lexicographical orders was introduced, the structure of Aronszajn. CHARACTERIZATION AND TRANSFORMATION OF COUNTRYMAN LINES AND R-EMBEDDABLE COHERENT TREES IN ZFC PENG YINHE (PhD, NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF