A generalization of Tennenbaum's theorem on effectively finite recursive linear orderings

1984 ◽  
Vol 49 (2) ◽  
pp. 563-569 ◽  
Author(s):  
Richard Watnick
Keyword(s):  
Ordered Set ◽  
Order Type ◽  
Test Case ◽  
Linear Orderings ◽  
Order Types ◽  
Concrete Model ◽  
Linearly Ordered Set ◽  
Effective Translation ◽  
Recursive Set

An effective translation of the fact that any infinite ordered set contains an infinite ascending or descending sequence is that any infinite recursive set A ⊆ Q has a recursive subset with order type ω or ω*. Tennenbaum's theorem states that this translation is false, and there is a counterexample A with order type ω + ω*. Tennenbaum suggested that this counterexample is an infinite recursive linearly ordered set which is effectively finite, and that the collection of all such counterexamples could provide a concrete model of nonstandard arithmetic. The purpose of this paper is to determine the collection of order types for which there is a counterexample.It is readily seen that any counterexample to the effective translation must have order type ω + Z · α + ω* for some α [2], [3]. Let be the collection of order types α for which there is a counterexample. As a test case, we have previously shown that contains the constructive scattered orderings [3], [4]. In this paper we determine exactly which order types are in . We easily show that if ω+ Z · α + ω* is recursive, then α is . The main result is that . Consequently, .

scholarly journals Rigidity in order-types

1977 ◽  
Vol 24 (2) ◽  
pp. 203-215 ◽  
Author(s):  
J. L. Hickman
Keyword(s):  
Ordered Set ◽  
Characterization Theorem ◽  
Decomposition Theorem ◽  
Order Type ◽  
Partial Ordering ◽  
Limit Laws ◽  
Order Types ◽  
Totally Ordered Set ◽  
The Axiom Of Choice ◽  
Increasing Sequences

AbstractA totally ordered set (and corresponding order-type) is said to be rigid if it is not similar to any proper initial segment of itself. The class of rigid ordertypes is closed under addition and multiplication, satisfies both cancellation laws from the left, and admits a partial ordering that is an extension of the ordering of the ordinals. Under this ordering, limits of increasing sequences of rigid order-types are well defined, rigid and satisfy the usual limit laws concerning addition and multiplication. A decomposition theorem is obtained, and is used to prove a characterization theorem on rigid order-types that are additively prime. Wherever possible, use of the Axiom of Choice is eschewed, and theorems whose proofs depend upon Choice are marked.

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

Dense, uniform and densely subuniform chains

1977 ◽  
Vol 23 (1) ◽  
pp. 1-8 ◽  
Author(s):  
C. J. Ash
Keyword(s):  
Ordered Set ◽  
Order Type ◽  
Inverse Semigroups ◽  
Independent Interest ◽  
A Chain ◽  
Linearly Ordered Set

AbstractA chain, or linearly ordered set, is densely subuniform if it is dense and for every order type the elements whose corresponding initial sections have this order type, if any, are dense in the chain. It is uniform if all intial sections are isomorphic. This paper gives constructions for densely subuniform chains which are not uniform. The question arises from the study of congruence-free inverse semigroups, but may also have independent interest.

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

A Note on a Fully Ordered Ring

1967 ◽  
Vol 10 (5) ◽  
pp. 757-758 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Associative Ring ◽  
Ordered Set ◽  
Additive Subgroup ◽  
Ordered Ring ◽  
Linearly Ordered Set

A ring R (associative ring) is said to be fully ordered provided that R is a linearly ordered set under a relation such that for any a, b and c in R, implies that and if c ε 0 then and . We say a subset K of R is convex provided that if a, b ε K such that then the interval [a, b] is a subset of K. Obviously an additive subgroup K of R is convex if and only if b ε K and b > 0 implies [a, b] ⊆ K.

MULTIPLE TRANSITIVITY AND MIN-WISE INDEPENDENCE IN PERMUTATION GROUPS

2004 ◽  
Vol 03 (04) ◽  
pp. 427-435
Author(s):  
C. FRANCHI
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Ordered Set ◽  
Permutation Groups ◽  
Linearly Ordered Set

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if [Formula: see text] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property that for each subset X⊆Ω with |X|≤k-1, the stabilizer G{X} of X in G is transitive on Ω\X. Using this fact, we determine all co-k-restricted min-wise independent groups.

Interval covers of a linearly ordered set

1996 ◽  
pp. 1-13
Author(s):  
R. Aharoni ◽  
A. Hajnal ◽  
E. C. Milner
Keyword(s):  
Ordered Set ◽  
Linearly Ordered Set

On a theorem of Fleischer

1986 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
G. Mehta
Keyword(s):  
Ordered Set ◽  
Separation Theorem ◽  
Order Topology ◽  
Topological Spaces ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractFleischer proved that a linearly ordered set that is separable in its order topology and has countably many jumps is order-isomorphic to a subset of the real numbers. The object of this paper is to extend Fleischer's result and to prove it in a different way. The proof of the theorem is based on Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

An open mapping theorem for o-minimal structures

2001 ◽  
Vol 66 (4) ◽  
pp. 1817-1820 ◽  
Author(s):  
Joseph Johns
Keyword(s):  
Elementary Proof ◽  
Ordered Set ◽  
Cartesian Product ◽  
Closed Field ◽  
Product Topology ◽  
Open Mapping Theorem ◽  
Open Set ◽  
Minimal Structure ◽  
Background Material ◽  
Linearly Ordered Set

We fix an arbitrary o-minimal structure (R, ω, …), where (R, <) is a dense linearly ordered set without end points. In this paper “definable” means “definable with parameters from R”, We equip R with the interval topology and Rn with the induced product topology. The main result of this paper is the following.Theorem. Let V ⊆ Rnbe a definable open set and suppose that f: V → Rnis a continuous injective definable map. Then f is open, that is, f(U) is open whenever U is an open subset of V.Woerheide [6] proved the above theorem for o-minimal expansions of a real closed field using ideas of homology. The case of an arbitrary o-minimal structure remained an open problem, see [4] and [1]. In this paper we will give an elementary proof of the general case.Basic definitions and notation. A box B ⊆ Rn is a Cartesian product of n definable open intervals: B = (a1, b1) × … × (an, bn) for some ai, bi, ∈ R ∪ {−∞, +∞}, with ai < bi, Given A ⊆ Rn, cl(A) denotes the closure of A, int(A) denotes the interior of A, bd(A) ≔ cl(A) − int(A) denotes the boundary of A, and ∂A ≔ cl(A) − A denotes the frontier of A, Finally, we let π: Rn → Rn− denote the projection map onto the first n − 1 coordinates.Background material. Without mention we will use notions and facts discussed in [5] and [3]. We will also make use of the following result, which appears in [2].

The Fuzzy Description Logic ALCFH with Hedge Algebras as Concept Modifiers

2003 ◽  
Vol 7 (3) ◽  
pp. 294-305 ◽  
Author(s):  
Steffen Hölldobler ◽  
◽  
Hans-Peter Störr ◽  
Tran Dinh Khang ◽  
Keyword(s):  
Knowledge Base ◽  
Decision Procedure ◽  
Description Logic ◽  
Ordered Set ◽  
Fuzzy Description ◽  
Linearly Ordered Set

In this paper we present the fuzzy description logic ALCFH introduced, where primitive concepts are modified by means of hedges taken from hedge algebras. ALCFH is strictly more expressive than Fuzzy-ALC defined in [11]. We show that given a linearly ordered set of hedges primitive concepts can be modified to any desired degree by prefixing them with appropriate chains of hedges. Furthermore, we define a decision procedure for the unsatisfiability problem in ALCFH, and discuss knowledge base expansion when using terminologies, truth bounds, expressivity as well as complexity issues. We extend [8] by allowing modifiers on non-primitive concepts and extending the satisfiability procedure to handle concept definitions.

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