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Abstract:
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This thesis is the first systematic study of trees and ramified partially ordered sets and of their close relationship to linear orderings. It was the source of many crucial notions and problems in this area as, for example, the notions of Aronszajn and Souslin tree. The problem whether inaccessible cardinals have the tree property i. e., whether they satisfy the analogue of Koning’s infinity lemma is considered in this thesis for the first time. The thesis consists of Chapter I (the subchapters t1-t8), Chapter II (the subchapters t9-t12), and an appendix ("Complément"). In t8A11 trees are classified as "large", "étrioit" and "ambigu" according to their heights and widths. In the Theorem 5bis the following property is presented: the very thin and tall trees ("étrioit") always have cofinal branches i.e., chains intersecting every level. This result was a source of the problem whether the same fact is true about the class of slightly wider trees ("ambigu") i.e., the trees of height equal to some cardinal Θ and whose levels are now only assumed to be of size less than Θ. This is the problem known today as the problem whether Θ has the tree property. In t10.2 the important notion σE is defined, where E is a linearly (or partially) ordered set. Namely, σE is a tree of all nonempty bounded and well-ordered subsets of E with the end-extension as the tree ordering. The problem whether inaccessible cardinals have the tree property appeared in t10.3. In t10.4 two following problems are mentioned: whether every Aronszajn tree is a subtree of σQ, and if every two uniformly branching Aronszajn trees are isomorphic. A question related to previous one, whether there is a homogeneous Aronszajn tree is also mentioned. The property that every two countable infinitely branching trees of the same height are isomorphic is proved in t10.5 (Theorem 1). Appendix contains a proof that Souslin’s problem is equivalent to the statement that every uncountable tree contains an uncountable chain or untichain i.e., that three are no Souslin trees. |