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Introduction to theories without the independence property
"... We present an updated exposition of the classical theory of complete first order theories ..."
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We present an updated exposition of the classical theory of complete first order theories
Hausdorff's Theorem for posets that satisfy the finite antichain property
 Fund. Math
, 1999
"... Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satis ..."
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Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC).
What is a Structure Theory
 Bulletin of the London Mathematical Society
, 1987
"... into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consis ..."
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into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consisted of equalities and inequalities between uncountable cardinals. In my experience most mathematicians find uncountable cardinals depressing, if they have any reaction to them at all. In fact Shelah was quite right to be so happy, but not because of his cardinal inequalities. He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic. That is what I want to discuss here. In brief, Shelah's work is about the notion of a class of structures which has a good structure theory. We all have a rough intuitive notion of what counts as a good structure theory. For example the structure theory of finitely generated abelian groups
On Fraïssé’s conjecture for linear orders of finite Hausdorff rank
, 2007
"... We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered ..."
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We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is wellordered”.
A partition theorem for scattered order types
 Combinatorics Probability and Computing
, 2003
"... 796 revision:20020613 modified:20020616 If ϕ is a scattered order type, µ a cardinal, then there exists a scattered order type ψ such that ψ → [ϕ] 1 µ,ℵ0 holds. In this note we prove a Ramsey type statement on scattered order types. A trivial fact on ordinals implies the following statement. If ..."
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796 revision:20020613 modified:20020616 If ϕ is a scattered order type, µ a cardinal, then there exists a scattered order type ψ such that ψ → [ϕ] 1 µ,ℵ0 holds. In this note we prove a Ramsey type statement on scattered order types. A trivial fact on ordinals implies the following statement. If µ is an infinite cardinal, then µ+ → (µ+) 1 µ. It is less trivial but still easy to show that if ϕ is an order type, µ a cardinal then there is some order type ψ that ψ → (ϕ) 1 µ holds. One can say that these results show that the classes of ordinals and order types are both Ramsey classes in the natural sense; given a target element and a cardinal for the number of colors, there is another element of the class, which, when colored with the required number of colors, always has a monocolored copy of the target. One can wonder which other classes have similar Ramsey properties. A natural, and well investigated, class in between is the class of scattered order types. For this class, the Ramsey property fails for the following well known and simple reason. There is some scattered order type ψ that for every scattered ϕ one has ϕ ̸ → [ψ] 1 ω. See Lemma 1. In this paper we show that this is the most in the negative direction, that is, for every scattered order type ϕ and cardinal µ there exists a scattered order type ψ such that ψ → [ϕ] 1 µ,ω holds. Notation. We use the standard axiomatic set theory notation. If ϕ, ψ are order types, then ϕ ≤ ψ denotes that there is an order preserving embedding of ϕ into ψ, that is, every ordered set of order type ψ has a subset of order type ϕ. If ϕ is an order type, then ϕ ∗ denotes the reverse order type, that is, if ϕ is the order type of (S, <), then ϕ ∗ is the order type of (S,>). ω is the ordinal of the set of natural numbers, (N, <). η is the order type of the set of rational numbers, (Q, <). If ϕ, ψ are order types, µ is a cardinal, ϕ → (ψ) 1 µ denotes the following statement. If (S, <) is an ordered set of order type ϕ and f: S → µ then for some i < µ the subset f −1 (i) contains a subset of order type ψ. That is, if a
Milner: Some remarks on simple tournaments
 Alg. Universalis
, 1972
"... algebra universalis Vol. 2, fasc. 2, 1972 BIRKH4USER VERLAG BASEL pages 238245 ..."
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algebra universalis Vol. 2, fasc. 2, 1972 BIRKH4USER VERLAG BASEL pages 238245
Linear Orders in the Pushdown Hierarchy
"... Abstract. We investigate the linear orders belonging to the pushdown hierarchy. Our results are based on the characterization of the pushdown hierarchy by graph transformations due to Caucal and do not make any use of higherorder pushdown automata machinery. Our main results show that ordinals belo ..."
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Abstract. We investigate the linear orders belonging to the pushdown hierarchy. Our results are based on the characterization of the pushdown hierarchy by graph transformations due to Caucal and do not make any use of higherorder pushdown automata machinery. Our main results show that ordinals belonging to the nth level are exactly those strictly smaller than the tower of ω of height n +1. More generally the Hausdorff rank of scattered linear orders on the nth level is strictly smaller than the tower of ω of height n. As a corollary the CantorBendixson rank of the tree solutions of safe recursion schemes of order n is smaller than the tower of ω of height n. As a spinoff result, we show that the ωwords belonging to the second level of the pushdown hierarchy are exactly the morphic words. 1
EQUIMORPHISM INVARIANTS FOR SCATTERED LINEAR ORDERINGS
"... Two linear ordering are equimorphic if they can be embedded in each other. We define invariants for scattered linear orderings which classify them up to equimorphism. Essentially, these invariants are finite sequences of finite trees with ordinal labels. Also, for each ordinal α, we explicitly desc ..."
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Two linear ordering are equimorphic if they can be embedded in each other. We define invariants for scattered linear orderings which classify them up to equimorphism. Essentially, these invariants are finite sequences of finite trees with ordinal labels. Also, for each ordinal α, we explicitly describe the finite set of minimal scattered equimorphism types of Hausdorff rank α. We compute the invariants of each of these minimal types.