INITIAL SEGMENTS OF THE ENUMERATION DEGREES

2016 ◽  
Vol 81 (1) ◽  
pp. 316-325 ◽  
Author(s):  
HRISTO GANCHEV ◽  
ANDREA SORBI
Keyword(s):  
Local Structure ◽  
Initial Segment ◽  
Nonzero Element ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Image Position

AbstractUsing properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

DEEP CLASSES

2016 ◽  
Vol 22 (2) ◽  
pp. 249-286 ◽  
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Mutual Information ◽  
Initial Segment ◽  
Computability Theory ◽  
Binary Sequences ◽  
Algorithmic Randomness ◽  
Positive Probability ◽  
Probabilistic Algorithm ◽  
Image Position ◽  
The Relationship ◽  
Mass Problems

AbstractA set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.

Type-raising operations on cardinal and ordinal numbers in Quine's “New foundations”

1973 ◽  
Vol 38 (1) ◽  
pp. 59-68 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Set Theory ◽  
Initial Segment ◽  
Initial Section ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Basic Facts ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Increasing Function ◽  
Standard Operations

In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, T(n) on cardinal numbers and U(α) on ordinal numbers, which are defined by the equationsfor each set x and well ordering R. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas T(x) = y and U(x) = y are stratified when y is assigned a type one higher than x. The importance of T and U stems from the following facts: (i) each of T and U is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand U commute with most of the standard operations on cardinal and ordinal numbers.These basic facts are discussed in §1. In §2 we prove in NF that the exponential function 2n is not 1-1. Indeed, there exist cardinal numbers m and n which satisfyIn §3 we prove the following technical result, which has many important applications. Suppose f is an increasing function from an initial segment S of the set NO of ordinal numbers into NO and that f commutes with U.

Describing ordinals using functionals of transfinite type

1972 ◽  
Vol 37 (1) ◽  
pp. 35-47 ◽  
Author(s):  
Peter Aczel
Keyword(s):  
Initial Segment ◽  
Normal Functions ◽  
Definition Of ◽  
Image Position

Bachmann, in [2] shows how certain ordinals <Ω(Ω = Ω1 where Ωξ is the (1 + ξ)th infinite initial ordinal) may be described from below using suitable descriptions of ordinals <Ω2. The aim of this paper is to consider another approach to describing ordinal <Ω and compare it with the Bachmann method. Our approach will use functionals of transfinite type based on Ω.The Bachmann method consists in denning a hierarchy of normal functions ϕδ: Ω → Ω (i.e. continuous and strictly increasing) for δ ≤ η0 < Ω2, starting with ϕ0(λ) = ω1 + λ. The definition of depends on a suitable description of the ordinals ≤ η0. This is obtained by defining a hierarchy 〈Fδ ∣ δ ≤ Ω2〉 of normal functions Fδ: Ω2 → Ω2 analogously to the definition of the initial segment 〈ϕδ ∣ δ ≤ Ω〉 of . The ordinal η0 is .Note. Our description of Bachmann's hierarchies will differ slightly from those in Bachmann's paper. Let and denote the hierarchies in [2]. Then as Bachmann's normal functions are not defined at 0 we let for λ, δ < Ω2. Bachmann defines for 0 < λ < Ω2 but it seems more natural to omit this so that we let . The situation is analogous for and leads to the following definitions:where n < ω and ξ is a limit number of cofinality Ω, and

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

scholarly journals RELATIVE DEFINABILITY OF n-GENERICS

2018 ◽  
Vol 83 (04) ◽  
pp. 1345-1362
Author(s):  
WEI WANG
Keyword(s):  
Positive Integer ◽  
Initial Segment ◽  
Cohen Forcing ◽  
Image Position ◽  
Generic Set

AbstractA set $G \subseteq \omega$ is n-generic for a positive integer n if and only if every ${\rm{\Sigma }}_n^0$ formula of G is decided by a finite initial segment of G in the sense of Cohen forcing. It is shown here that every n-generic set G is properly ${\rm{\Sigma }}_n^0$ in some G-recursive X. As a corollary, we also prove that for every $n > 1$ and every n-generic set G there exists a G-recursive X which is generalized ${\rm{lo}}{{\rm{w}}_n}$ but not generalized ${\rm{lo}}{{\rm{w}}_{n - 1}}$. Thus we confirm two conjectures of Jockusch [4].

UNIVERSAL ROSSER PREDICATES

2017 ◽  
Vol 82 (1) ◽  
pp. 292-302 ◽  
Author(s):  
MAKOTO KIKUCHI ◽  
TAISHI KURAHASHI
Keyword(s):  
Standard Model ◽  
Initial Segment ◽  
Nonstandard Model ◽  
Complete Extension ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Incompleteness Theorems ◽  
Image Position ◽  
Gödel’S Incompleteness Theorems

AbstractGödel introduced the original provability predicate in the proofs of Gödel’s incompleteness theorems, and Rosser defined a new one. They are equivalent in the standard model ${\mathbb N}$ of arithmetic or any nonstandard model of ${\rm PA} + {\rm Con_{PA}} $, but the behavior of Rosser’s provability predicate is different from the original one in nonstandard models of ${\rm PA} + \neg {\rm Con_{PA}} $. In this paper, we investigate several properties of the derivability conditions for Rosser provability predicates, and prove the existence of a Rosser provability predicate with which we can define any consistent complete extension of ${\rm PA}$ in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. We call it a universal Rosser predicate. It follows from the theorem that the true arithmetic ${\rm TA}$ can be defined as the set of theorems of ${\rm PA}$ in terms of a universal Rosser predicate in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. By using this theorem, we also give a new proof of a theorem that there is a nonstandard model M of ${\rm PA} + \neg {\rm Con_{PA}} $ such that if N is an initial segment of M which is a model of ${\rm PA} + {\rm Con_{PA}} $ then every theorem of ${\rm PA}$ in N is a theorem of $\rm PA$ in ${\mathbb N}$. In addition, we prove that there is a Rosser provability predicate such that the set of theorems of $\rm PA$ in terms of the Rosser provability predicate is inconsistent in any nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $.

On maximal theories

1991 ◽  
Vol 56 (3) ◽  
pp. 885-890 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Open Problem ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Finite Collection ◽  
Extension Problem ◽  
Recursive Theory ◽  
Image Position ◽  
Induction Scheme

Let S be a recursive theory. Let a theory T* consisting of Σ1 sentences be called maximal (with respect to S) if T* is maximal consistent with S, i.e. there is no Σ1 sentence consistent with T* + S which is not in T*.A maximal theory with respect to IΔ0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem.Let us recall that IΔ0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA− together with the induction scheme restricted to bounded formulas.The main open problem concerning the end-extendability of models of IΔ0 is the following:(*) Does every model of IΔ0 + BΣ1 have a proper end-extension to a model of IΔ0?Here BΣ1 is the following collection scheme:where φ runs over bounded formulas and may contain parameters.It is well known(see [KP]) that if I is a proper initial segment of a model M of IΔ0, then I satisfies IΔ0 + BΣ1.For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of IΔ0 + BΣ1 which has no proper end-extension to a model of IΔ0 under the assumption IΔ0 ⊢¬Δ0H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T* where S = IΔ0.Moreover, T*, which is the set Σ1(M) of all Σ1 sentences true in M, is not codable in M.

Scanning probes for new energy materials: Probing local structure and function

MRS Bulletin ◽  
2012 ◽  
Vol 37 (7) ◽  
pp. 633-637 ◽  
Author(s):  
Nina Balke ◽  
Dawn Bonnell ◽  
David S. Ginger ◽  
Martijn Kemerink
Keyword(s):  
Local Structure ◽  
Structure And Function ◽  
Energy Materials ◽  
Scanning Probes ◽  
New Energy ◽  
Image Position ◽  
And Function

Abstract

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