TAILIEUCHUNG - Đề tài " A five element basis for the uncountable linear orders "

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements ∗ are X, ω1 , ω1 , C, C ∗ where X is any suborder of the reals of cardinality ℵ1 and C is any Countryman line. This confirms a longstanding conjecture. | Annals of Mathematics A five element basis for the uncountable linear orders By Justin Tatch Moore Annals of Mathematics 163 2006 669 688 A five element basis for the uncountable linear orders By Justin Tatch Moore Dedicated to Fennel Moore Abstract In this paper I will show that it is relatively consistent with the usual axioms of mathematics ZFC together with a strong form of the axiom of infinity the existence of a supercompact cardinal that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom a strong form of the Baire Category Theorem. The elements are X W1 Ò C C where X is any suborder of the reals of cardinality Ki and C is any Countryman line. This confirms a longstanding conjecture of Shelah. 1. Introduction Our focus in this paper will be to show that the Proper Forcing Axiom PFA implies that any uncountable linear order must contain an isomorphic copy of one of the following five orders X w1 ò C and C . Here X is any fixed set of reals of cardinality Ki and C is any fixed Countryman line. Such a list is called a basis. The simplest example of an uncountable linear order is R the real line. This object serves as the prototype for the class of linear orders and as the canonical example of an uncountable set. Early on in modern set theory Baumgartner proved the following deep result which suggested that it might be possible to prove more general classification results for uncountable linear orders. Revisions and updates to the paper were supported by NSF grant DMS-0401893. Travel support to present these results in Kobe Japan was provided by Grant-in-aid for Scientific Research C 2 15540120 Japanese Society for the Promotion of Science. 670 JUSTIN TATCH MOORE Theorem PFA 3 . If two sets of reals are K1-dense 1 then they are isomorphic. In particular if X is a set of reals of cardinality K1 then X serves as a single-element basis for the class of uncountable separable linear orders. .

• Thạc sĩ - Tiến sĩ - Cao học
• uncountable linear orders
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center