A note on the hyperarithmetical hierarchy

1970 ◽  
Vol 35 (3) ◽  
pp. 429-430 ◽  
Author(s):  
H. B. Enderton ◽  
Hilary Putnam
Keyword(s):  
Hyperarithmetical Hierarchy ◽  
Limit Ordinal

The hyperarithmetical hierarchy assigns a degree of unsolvability hγ to each constructive ordinal γ. This assignment has the properties that h0 is the recursive degree and hγ+1 is the jump h′γ of hγ. For a limit ordinal λ < ω1 it is not so easy to define hγ. The original definitions used systems of notations for ordinals, see Spector [6]. There are also later notation-free definitions due to Shoenfield (unpublished) and to Hensel and Putnam [4].

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

scholarly journals A Generalization of Final Rank of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1118-1122 ◽  
Author(s):  
Doyle O. Cutler ◽  
Paul F. Dubois
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Basic Subgroup ◽  
Primary Abelian Group ◽  
Limit Ordinal ◽  
Final Rank ◽  
Definition Of ◽  
Group Recall

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

scholarly journals On Crossed Products of a Sfield

1963 ◽  
Vol 22 ◽  
pp. 65-71 ◽  
Author(s):  
Masatoshi Ikeda
Keyword(s):  
Integral Domain ◽  
Free Abelian Group ◽  
Group Extension ◽  
Crossed Products ◽  
Torsion Free Abelian Group ◽  
Outer Automorphisms ◽  
Identity Automorphism ◽  
Limit Ordinal ◽  
Group B ◽  
Torsion Free

In the previous paper [3] the author has shown a possibility to construct a series of sfields by taking sfields of quotients of split crossed products of a sfield. In this paper the same problem is treated, and, by considering general crossed products, a generalization of the previous result is given: Let K be a sfield and G be the join of a well-ordered ascending chain of groups Gα of outer automorphisms of K such that a) G1 is the identity automorphism group, b) Gα is a group extension of Gα-1 by a torsion-free abelian group for each non-limit ordinal α, and c) for each limit ordinal α. Then an arbitrary crossed product of K with G is an integral domain with a sfield of quotients Q and the commutor ring of K in Q coincides with the centre of K.

Intrinsic bounds on complexity and definability at limit levels

2009 ◽  
Vol 74 (3) ◽  
pp. 1047-1060 ◽  
Author(s):  
John Chisholm ◽  
Ekaterina B. Fokina ◽  
Sergey S. Goncharov ◽  
Valentina S. Harizanov ◽  
Julia F. Knight ◽  
...  
Keyword(s):  
Computable Structure ◽  
Additional Relation ◽  
Scott Family ◽  
Limit Ordinal

AbstractWe show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

MASS PROBLEMS AND HYPERARITHMETICITY

2007 ◽  
Vol 07 (02) ◽  
pp. 125-143 ◽  
Author(s):  
JOSHUA A. COLE ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Equivalence Class ◽  
Cantor Space ◽  
Mass Problem ◽  
Ordinal Numbers ◽  
Hyperarithmetical Hierarchy ◽  
Index Sets ◽  
Weakly Reducible ◽  
Mass Problems

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the Cantor space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0(α), the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X (X ∈ P and BLR (X) ⊆ BLR (Y))} where BLR (X) is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete.

On Cantor-Bendixson spectra containing (1,1). II

1985 ◽  
Vol 50 (3) ◽  
pp. 611-618 ◽  
Author(s):  
Annalisa Marcja ◽  
Carlo Toffalori
Keyword(s):  
Boolean Algebras ◽  
Stable Theory ◽  
Definable Subset ◽  
Limit Ordinal ◽  
Image Position ◽  
Countable Models

Let T be a (countable, complete, quantifier eliminable) ω-stable theory; an analysis of T, and consequently a classification of ω-stable theories, can be done by looking at the Boolean algebras B(M) of definable subsets of its countable models M (as usual, we often confuse a definable subset of M with the class of formulas defining it). If M ⊨ T, ∣M∣ = ℵ0, then, for every LM-formula ϕ(v) and for every ordinal α, we define a relation(CB = Cantor-Bendixson, of course) by induction on α:CB-rank ϕ(v) ≥ 0 if ϕ(M) ≠ ∅CB-rank ϕ(v) ≥ λ for λ a limit ordinal, if CB-rank ϕ(v) ≥ for all v < λ;CB-rank ϕ(v)≥ α + 1 if, for all n ∈ ω,(*) there are LM-formulas ϕ0(v), …, ϕn − 1(v) such thatIt is well known that the ω-stability of T implies that, for every consistent LM-formula ϕ(v), there is exactly one ordinal α < ω1 such that CB-rank ϕ(v) ≥ α and CB-rank ϕ(v)≱α + 1. Therefore we define:CB-rank ϕ(v) = αCB-degree ϕ(v) = d if d is the maximal n ∈ ω satisfying (*); andCB-type ϕ(v) = (α, d).

scholarly journals The isomorphism problem for computable Abelian p-groups of bounded length

2005 ◽  
Vol 70 (1) ◽  
pp. 331-345 ◽  
Author(s):  
Wesley Calvert
Keyword(s):  
Recent Work ◽  
Structure Theorem ◽  
Isomorphism Problem ◽  
Linear Orders ◽  
Computable Structures ◽  
Hyperarithmetical Hierarchy ◽  
Working Out

AbstractTheories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples.We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian p-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.

scholarly journals The hyper-archimedean kernel sequence of a lattice-ordered group

1974 ◽  
Vol 10 (3) ◽  
pp. 337-349 ◽  
Author(s):  
Jorge Martinez
Keyword(s):  
Large Class ◽  
Stone Space ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Free Products ◽  
Limit Ordinal

The hyper-archimedean kernel Ar(G) of a lattice-ordered group (hence forth l–group) is the largest hyper-archimedean convex l–subgroup of the l–group G. One defines Arσ (G), for an ordinal σ as if a is a limit ordinal, and as the unique l–ideal with the property that Arσ(G)/Ar.σ–1(G) = Ar(G/Arσ–1(G)), otherwise. The resulting "Loewy"-like sequence of characteristic l–ideals, Ar(G) ⊆ Ar2(G) ⊆ … ⊆ Arσ (G) ⊆ …, is called the hyper-archimedean kernel sequence. The first result of this note says that each Arσ(G) ⊆ Ar(G)”.Most of the paper concentrates on archimedean l–groups; in particular, the hyper-archimedean kernels are identified for: D(X), where X is a Stone space, a large class of free products of abelian l–groups, and certain l–subrings of a product of real groups.It is shown that even for archimedean l–groups the hyper-archimedean kernel sequence may proceed past Ar(G).

Building iteration trees

1991 ◽  
Vol 56 (4) ◽  
pp. 1369-1384 ◽  
Author(s):  
Alessandro Andretta
Keyword(s):  
Limit Ordinal ◽  
Woodin Cardinal ◽  
Iteration Trees

AbstractIt is shown, assuming the existence of a Woodin cardinal δ, that every tree ordering on some limit ordinal λ < δ with a cofinal branch is the tree ordering of some iteration tree on V.

scholarly journals ELEMENTARY AMENABLE SUBGROUPS OF R. THOMPSON'S GROUP F

2005 ◽  
Vol 15 (04) ◽  
pp. 619-642 ◽  
Author(s):  
MATTHEW G. BRIN
Keyword(s):  
Piecewise Linear ◽  
Amenable Group ◽  
Solvable Groups ◽  
Amenable Groups ◽  
Elementary Class ◽  
Limit Ordinal ◽  
Thompson’S Group ◽  
Thompson's Group F

We study the subgroups of R. J. Thompson's group F and PLo(I), the group of orientation preserving, piecewise linear self homeomorphisms of [0, 1]. We exhibit, for each non-limit ordinal α ≤ ω2 + 1, an elementary amenable group of elementary class α (under Chou's stratification of elementary amenable groups) that is a subgroup of F and thus of PLo(I). We also give examples that negatively answer a question of Sapir about non-solvable groups in F and PLo(I).

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