The -spectrum of a linear order

2001 ◽  
Vol 66 (2) ◽  
pp. 470-486 ◽  
Author(s):  
Russell Miller
Keyword(s):  
Linear Order ◽  
Turing Degree ◽  
Ordinary Type

AbstractSlaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable degree, but not 0. Since our argument requires the technique of permitting below a set, we include a detailed explantion of the mechanics and intuition behind this type of permitting.

scholarly journals Amenable equivalence relations and Turing degrees

1991 ◽  
Vol 56 (1) ◽  
pp. 182-194 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Linear Order ◽  
Order Type ◽  
Positive Answer ◽  
Turing Degree ◽  
Partial Answer ◽  
Equivalence Relations ◽  
Turing Degrees

In [12] Slaman and Steel posed the following problem:Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order preservingly in <d for a cone of d's?They had already obtained a partial answer to this question by showing that there is no such d ↦ <d with <d of order type ζ = ω* + ω on a cone. Already the possibility that <d has order type ζ · ζ was left open.We use here, ideas and methods associated with the concept of amenability (of groups, actions, equivalence relations, etc.) to prove some general results from which one can obtain a positive answer to the above problem.

scholarly journals Spectra of structures and relations

2007 ◽  
Vol 72 (1) ◽  
pp. 324-348 ◽  
Author(s):  
Valentina S. Harizanov ◽  
Russell G. Miller
Keyword(s):  
Random Graph ◽  
Linear Order ◽  
Turing Degree ◽  
Linear Orders ◽  
Degree Spectrum

AbstractWe consider embeddings of structures which preserve spectra: if g : ℳ → with computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on ). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on . Finally, we consider the extent to which all spectra of unary relations on the structure ℒ may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c′ ≥ τ 0″.

Autostability spectra for decidable structures

2016 ◽  
Vol 28 (3) ◽  
pp. 392-411 ◽  
Author(s):  
NIKOLAY BAZHENOV
Keyword(s):  
Linear Order ◽  
Turing Degree ◽  
Turing Degrees ◽  
Decidable Structure ◽  
Autostability Spectrum

We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure$\mathcal{S}$, the SC-autostability spectrum of$\mathcal{S}$is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of$\mathcal{S}$. The degree of SC-autostability for$\mathcal{S}$is the least degree in the spectrum (if such a degree exists).We prove that for a computable successor ordinal α, every Turing degree c.e. in and above0(α)is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above0(2β+1)is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative ton-constructivizations.

scholarly journals η-representation of sets and degrees

2008 ◽  
Vol 73 (4) ◽  
pp. 1097-1121 ◽  
Author(s):  
Kenneth Harris
Keyword(s):  
Linear Order ◽  
Monotonic Function ◽  
Turing Degree

AbstractWe show that a set has an η-representation in a linear order if and only if it is the range of a 0′-computable limitwise monotonic function. We also construct a Δ3 Turing degree for which no set in that degree has a strong η-representation, answering a question posed by Downey.

The (Im)purity Levels of Communal Meals within the Qumran Movement

2016 ◽  
Vol 7 (1) ◽  
pp. 102-122
Author(s):  
Cecilia Wassén
Keyword(s):  
Careful Study ◽  
Ordinary Type ◽  
The Common ◽  
The Temple

Scholars usually take for granted that the sectarian members of the Qumran movement ate their common meals in full purity at a level that is often compared to that of the priests serving in the temple. This assumption rests on the interpretation of hatohorah, “the purity,” as pertaining to common meals. But a careful study of a range of texts, including the important Tohorot A, leads to a more nuanced picture. Accordingly, it is important to distinguish between the common, everyday meals of the movement and the special meals. Whereas a mild level of impurity of the participants was accepted at the ordinary type of communal meals, special meals required purity. Even at these pure meals, there were variations concerning the required level of purity depending on the occasion.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

scholarly journals Progenitors and Hydrodynamics of Type II and lb Supernovae

1996 ◽  
Vol 145 ◽  
pp. 137-147
Author(s):  
S. E. Woosley ◽  
T. A. Weaver ◽  
R. G. Eastman
Keyword(s):  
Black Holes ◽  
Light Curve ◽  
Mass Loss ◽  
Massive Stars ◽  
The Other ◽  
Shock Interaction ◽  
Type Ii ◽  
Residual Hydrogen ◽  
Ordinary Type ◽  
The One

We review critical physics affecting the observational characteristics of those supernovae that occur in massive stars. Particular emphasis is given to 1) how mass loss, either to a binary companion or by a radiatively driven wind, affects the type and light curve of the supernova, and 2) the interaction of the outgoing supernova shock with regions of increasing pr3 in the stellar mantle. One conclusion is that Type II-L supernovae may occur in mass exchanging binaries very similar to the one that produced SN 1993J, but with slightly larger initial separations and residual hydrogen envelopes (∼1 Mʘ and radius ∼ several AU). The shock interaction, on the other hand, has important implications for the formation of black holes in explosions that are, near peak light, observationally indistinguishable from ordinary Type II-p and lb supernovae.

Two results on borel orders

1989 ◽  
Vol 54 (3) ◽  
pp. 865-874 ◽  
Author(s):  
Alain Louveau
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Lexicographic Ordering ◽  
Countable Ordinal

AbstractWe prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

Wtt-degrees and T-degrees of r.e. sets

1983 ◽  
Vol 48 (4) ◽  
pp. 921-930 ◽  
Author(s):  
Michael Stob
Keyword(s):  
Minimal Pair ◽  
Turing Degree ◽  
Turing Reducibility

AbstractWe use some simple facts about the wtt-degrees of r.e. sets together with a construction to answer some questions concerning the join and meet operators in the r.e. degrees. The construction is that of an r.e. Turing degree a with just one wtt-degree in a such that a is the join of a minimal pair of r.e. degrees. We hope to illustrate the usefulness of studying the stronger reducibility orderings of r.e. sets for providing information about Turing reducibility.

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