Requirement systems

1995 ◽  
Vol 60 (1) ◽  
pp. 222-245 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Recent Result ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The One ◽  
Scott Set

Methods for carrying out transfinitely nested priority constructions have been developed by Harrington [7] and by Ash [2, 1, 3, 4]. Ash's method has different versions, with later ones becoming simpler. Lemmp and Lerman [11] have also developed a method, for finitely nested constructions. Ash formulated abstractly the object of a nested priority construction, and he proved a metatheorem for what he called “α-systems”, listing conditions which guarantee the success of the construction. Harrington's method of “workers”, at least in its original, informal state, seems more flexible than Ash's α-systems.In [10, 9], there are finite and transfinite versions of a metatheorem for workers. The statements are complicated, and these metatheorems have not proved to be very useful. The present paper gives a new transfinite metatheorem. The statement is considerably simpler than the one in [9], although not so simple as that in [3]. The new metatheorem grew, in part, out of an effort to find a new proof of Ash's metatheorem. The new metatheorem yields the one in [3], and it seems more flexible. A different generalization of Ash's metatheorem will be given in [5].Ash's metatheorem is easier to use than the one in the present paper, and the result in [3] is certainly the one to use wherever it applies. Here we give one application of the new metatheorem which does not seem to follow from the result in [3]. This is a theorem on models “representing” a given Scott set, which implies one half of a recent result of Solovay [18], on Turing degrees of models of particular completions of Peano arithmetic (PA).

On the Rates of Convergence of Chlodovsky–Durrmeyer Operators and their Bézier Variant

2009 ◽  
Vol 16 (4) ◽  
pp. 693-704
Author(s):  
Harun Karsli ◽  
Paulina Pych-Taberska
Keyword(s):  
Recent Result ◽  
Pointwise Convergence ◽  
Rates Of Convergence ◽  
Durrmeyer Operators ◽  
The One ◽  
Special Case ◽  
Appl Math ◽  
Bézier Variant

Abstract We consider the Bézier variant of Chlodovsky–Durrmeyer operators 𝐷𝑛,α for functions 𝑓 measurable and locally bounded on the interval [0,∞). By using the Chanturia modulus of variation we estimate the rate of pointwise convergence of (𝐷𝑛,α 𝑓) (𝑥) at those 𝑥 > 0 at which the one-sided limits 𝑓(𝑥+), 𝑓(𝑥–) exist. In the special case α = 1 the recent result of [Ibikli, Karsli, J. Inequal. Pure Appl. Math. 6: 12, 2005] concerning the Chlodovsky–Durrmeyer operators 𝐷𝑛 is essentially improved and extended to more general classes of functions.

An introduction to γ-recursion theory (or what to do in KP – Foundation)

1990 ◽  
Vol 55 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The Subject ◽  
Admissible Ordinals ◽  
Limited Induction ◽  
Well Foundedness

The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

Some highly saturated models of Peano arithmetic

2002 ◽  
Vol 67 (4) ◽  
pp. 1265-1273
Author(s):  
James H. Schmerl
Keyword(s):  
Regular Cardinal ◽  
Peano Arithmetic ◽  
Saturated Model ◽  
Elementary Extension ◽  
Saturated Models ◽  
The One ◽  
Recursively Saturated

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is a λ-saturated model of PA such that ∣M∣ > λ, then has an elementary extension of the same cardinality which is also λ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf() = λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

scholarly journals Degrees of unsolvability of continuous functions

2004 ◽  
Vol 69 (2) ◽  
pp. 555-584 ◽  
Author(s):  
Joseph S. Miller
Keyword(s):  
Continuous Functions ◽  
Turing Degrees ◽  
Total Degree ◽  
Classical Analysis ◽  
Degree Relative ◽  
Enumeration Degrees ◽  
Standard Notion ◽  
Scott Set ◽  
Degrees Of Unsolvability ◽  
Satisfactory Theory

Abstract.We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ [0,1] computes a non-computable subset of ℕ there is a non-total degree between Turing degrees a <Tb iff b is a PA degree relative to a; ⊆ 2ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f ∈ [0,1] of non-total degree; and there are computably incomparable f, g ∈ [0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

Two further combinatorial theorems equivalent to the 1-consistency of peano arithmetic

1983 ◽  
Vol 48 (4) ◽  
pp. 1090-1104 ◽  
Author(s):  
Peter Clote ◽  
Kenneth Mcaloon
Keyword(s):  
Peano Arithmetic ◽  
Infinite Subset ◽  
Linear Ordering ◽  
Tree Property ◽  
Preceding Theorem ◽  
Linear Orderings ◽  
Finite Set ◽  
The One ◽  
König’S Lemma

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.

scholarly journals WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS

2018 ◽  
Vol 83 (1) ◽  
pp. 392-394
Author(s):  
LINDA BROWN WESTRICK
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Turing Degrees ◽  
Image Position ◽  
Scott Set

AbstractLet ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$, where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$.

A minimal pair of Π10 classes

1971 ◽  
Vol 36 (1) ◽  
pp. 66-78 ◽  
Author(s):  
Carl G. Jockusch ◽  
Robert I. Soare
Keyword(s):  
Peano Arithmetic ◽  
Minimal Degree ◽  
Minimal Pair ◽  
Turing Degrees ◽  
Complete Extension ◽  
Recursively Enumerable

A pair of sets (A0, A1) forms a minimal pair if A0 and A1 are nonrecursive, and if whenever a set B is recursive both in A0 and in A1 then B is recursive. C. E. M. Yates [8] and independently A. H. Lachlan [4] proved the existence of a minima] pair of recursively enumerable (r.e.) sets thereby establishing a conjecture of G. E. Sacks [6]. We simplify Lachlan's construction, and then generalize this result by constructing two disjoint pairs of r.e. sets (A0, B0) and (A1B1) such that if C0 separates (A0, A1 and C1 separates (B0, B1), then C0 and C1 form a minimal pair. (We say that C separates (A0, A1) if A0 ⊂ C and C ∩ = ∅.) The question arose in our study of (Turing) degrees of members of certain classes, where we proved the weaker result [2, Theorem 4.1] that the above pairs may be chosen so that C0 and C2 are merely Turing incomparable. (There we used a variation of the weaker result to improve a result of Scott and Tennenbaum that no complete extension of Peano arithmetic has minimal degree.)

Generalizations of the one-dimensional version of the Kruskal-Friedman theorems

1989 ◽  
Vol 54 (1) ◽  
pp. 100-121 ◽  
Author(s):  
L. Gordeev
Keyword(s):  
Monotone Function ◽  
Formal Theory ◽  
Peano Arithmetic ◽  
One Dimensional ◽  
First Order ◽  
Countable Ordinal ◽  
Finite Sequences ◽  
Kruskal’S Theorem ◽  
Dimensional Version ◽  
The One

The paper [Schütte + Simpson] deals with the following one-dimensional case of Friedman's extension (see in [Simpson 1]) of Kruskal's theorem ([Kruskal]). Given a natural number n, let Sn+1 be the set of all finite sequences of natural numbers <n + 1. If s1 = (a0,…,ak) ∈Sn+1 and s2 = (b0,…,bm) ∈Sn + 1, then a strictly monotone function f: {0,…, k} → {0,…, m} is called an embedding of s1 into s2 if the following two assertions are satisfied:1) ai, = bf(i), for all i < k;2) if f(i) < j < f(i + 1) then bj > bf(i+1), for all i < k, j < m.Then for every infinite sequence s1, s2,…,sk,… of elements of Sn + 1 there exist indices i < j and an embedding of si into Sj. That is, Sn+1 forms a well-quasi-ordering (wqo) with respect to embeddability. For each n, this statement W(Sn+1) is provable in the standard second order conservative extension of Peano arithmetic. On the other hand, the proof-theoretic strength of the statements W(Sn+1) grows so fast that this formal theory cannot prove the limit statement ∀nW(Sn+1). The appropriate first order -versions of these combinatory statements preserve their proof-theoretic strength, so that actually one can speak in terms of provability in Peano arithmetic. These are the main conclusions from [Schütte + Simpson].We wish to extend this into the transfinite. That is, we take an arbitrary countable ordinal τ > 0 instead of n + 1 and try to obtain an analogous “strong” combinatory statement about finite sequences of ordinals < τ.

Completions of PA: Models and enumerations of representable sets

1998 ◽  
Vol 63 (3) ◽  
pp. 1063-1082 ◽  
Author(s):  
Alex M. McAllister
Keyword(s):  
Peano Arithmetic ◽  
The Family ◽  
Scott Set

AbstractWe generalize a result on True Arithmetic (ℐA) by Lachlan and Soare to certain other completions of Peano Arithmetic (PA). If ℐ is a completion of PA, then Rep(ℐ) denotes the family of sets X ⊆ ω for which there exists a formula φ(x) such that for all n ∈ ω, if n ∈ X, then ℐ ⊢ φ(S(n) (0)) and if n ∉ X, then ℐ ⊢ ┐φ(S(n)(O)). We show that if S, J ⊆ P(ω) such that S is a Scott set, J is a jump ideal, S ⊂ J and for all X ∈ J, there exists C ∈ S such that C is a “coding” set for the family of subtrees of 2<ω computable in X, and if ℐ is a completion of PA Such that Rep(ℐ) = S, then there exists a model A of ℐ such that J is the Scott set of A and no enumeration of Rep(ℐ) is computable in A. The model A of ℐ is obtained via a new notion of forcing.Before proving our main result, we demonstrate the existence of uncountably many different pairs (S, J) satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2<ω. In particular, C C ⊆ ω is a coding set for the family of subtrees of 2<ω computable in X if and only if for all trees T ⊆ 2<ω computable in X, if χc is a path through T, then there exists σ ∈ T such that σ ⊂ χc and every extension of σ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2<ω. We show they are identical.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

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