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?”

Simple r. e. degree structures

1987 ◽  
Vol 52 (1) ◽  
pp. 208-213
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Recursion Theory ◽  
Turing Degrees ◽  
Local Version ◽  
Admissible Set ◽  
Jump Operator ◽  
Admissible Sets ◽  
Closed Sets ◽  
Locally Countable ◽  
To Come ◽  
Admissible Ordinals

Much of recursion theory centers on the structures of different kinds of degrees. Classically there are the Turing degrees and r. e. Turing degrees. More recently, people have studied α-degrees for α an ordinal, and degrees over E-closed sets and admissible sets. In most contexts, deg(0) is the bottom degree and there is a jump operator' such that d' is the largest degree r. e. in d and d' > d. Both the degrees and the r. e. degrees usually have a rich structure, including a relativization to the cone above a given degree.A natural exception to this pattern was discovered by S. Friedman [F], who showed that for certain admissible ordinals β the β-degrees ≥ 0′ are well-ordered, with successor provided by the jump.For r. e. degrees, natural counterexamples are harder to come by. This is because the constructions are priority arguments, which require only mild restrictions on the ground model. For instance, if an admissible set has a well-behaved pair of recursive well-orderings then the priority construction of an intermediate r. e. degree (i.e., 0 < d < 0′) goes through [S]. It is of interest to see just what priority proofs need by building (necessarily pathological) admissible sets with few r. e. degrees.Harrington [C] provides an admissible set with two r. e. degrees, via forcing. A limitation of his example is that it needs ω1 (more accurately, a local version thereof) as a parameter. In this paper, we find locally countable admissible sets, some with three r. e. degrees and some with four.

Omitting types: application to recursion theory

1972 ◽  
Vol 37 (1) ◽  
pp. 81-89 ◽  
Author(s):  
Thomas J. Grilliot
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Significant Contribution ◽  
Hard Core ◽  
Completeness Theorem ◽  
Recursion Theory ◽  
Omitting Types ◽  
Admissible Ordinals ◽  
Models Of Set Theory

Omitting-types theorems have been useful in model theory to construct models with special characteristics. For instance, one method of proving the ω-completeness theorem of Henkin [10] and Orey [20] involves constructing a model that omits the type {x ≠ 0, x ≠ 1, x ≠ 2,···} (i.e., {x ≠ 0, x ≠ 1, x ≠ 2,···} is not satisfiable in the model). Our purpose in this paper is to illustrate uses of omitting-types theorems in recursion theory. The Gandy-Kreisel-Tait Theorem [7] is the most well-known example. This theorem characterizes the class of hyperarithmetical sets as the intersection of all ω-models of analysis (the so-called hard core of analysis). The usual way to prove that a nonhyperarithmetical set does not belong to the hard core is to construct an ω-model of analysis that omits the type representing the set (Application 1). We will find basis results for and s — sets that are stronger than results previously known (Applications 2 and 3). The question of how far the natural hierarchy of hyperjumps extends was first settled by a forcing argument (Sacks) and subsequently by a compactness argument (Kripke, Richter). Another problem solved by a forcing argument (Sacks) and then by a compactness argument (Friedman-Jensen) was the characterization of the countable admissible ordinals as the relativized ω1's. Using omitting-types technique, we will supply a third kind of proof of these results (Applications 4 and 5). S. Simpson made a significant contribution in simplifying the proof of the latter result, with the interesting side effect that Friedman's result on ordinals in models of set theory is immediate (Application 6). One approach to abstract recursiveness and hyperarithmeticity on a countable set is to tenuously identify the set with the natural numbers. This approach is equivalent to other approaches to abstract recursion (Application 7). This last result may also be proved by a forcing method.

Standard foundations for nonstandard analysis

1992 ◽  
Vol 57 (2) ◽  
pp. 741-748 ◽  
Author(s):  
David Ballard ◽  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Natural Place ◽  
Nonstandard Set Theory ◽  
Standard Set ◽  
The Subject ◽  
Internal Sets ◽  
Well Foundedness

In the thirty years since its invention by Abraham Robinson, nonstandard analysis has become a useful tool for research in many areas of mathematics. It seems fair to say, however, that the search for practically satisfactory foundations for the subject is not yet completed. New proposals, intended to remedy various shortcomings of older approaches, continue to be put forward. The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features. We do this by working in Zermelo-Fraenkel set theory with non-well-founded sets. It has always been clear that the axiom of regularity may fail for external sets. The previous approaches either avoid non-well-foundedness by considering only that fragment of nonstandard set theory that is well-founded (over individuals; enlargements of Robinson and Zakon [17]) or reluctantly live with it (various axiomatic nonstandard set theories). Ballard and Davidon [2] were the first to propose constructive use for non-well-foundedness in the foundations of nonstandard analysis. In the present paper we adopt a very strong anti-foundation axiom. In the resulting more or less “usual” set theory, the (to the “standard” mathematician) unfamiliar concepts of standard, external and internal sets can be defined and their requisite properties proved (rather than postulated, as is the case in axiomatic nonstandard set theories).

Σn sets which are Δn-incomparable (uniformly)

1974 ◽  
Vol 39 (2) ◽  
pp. 295-304 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
The Other ◽  
Direct Generalization ◽  
Natural Notion ◽  
A Priority ◽  
Definition Of ◽  
Admissible Ordinals ◽  
Priority Argument

In this paper we will present an application of generalized recursion theory to (noncombinatorial) set theory. More precisely we will combine a priority argument in α-recursion theory with a forcing construction to prove a theorem about the interdefinability of certain subsets of admissible ordinals.Our investigation was prompted by G. Sacks and S. Simpson asking [6] if it is obvious that there are, for each Σn-admissible α, Σn (over Lα) subsets of α which are Δn-incomparable. If one understands “B is Δn in C” to mean that there are Σn/Lα reduction procedures which put out B and when one feeds in C, then the answer is an unqualified “yes.” In this sense “Δn in” is a direct generalization of “α-recursive in” (replace Σ1 by Σn in the definition) and so amenable to the methods of [7, §§3, 5]. Indeed one simply chooses a complete Σn−1 set A and mimics the construction of [6] as modified in [7, §5] to produce two α-A-r.e. sets B and C neither of which is α-A-recursive in the other. By the remarks on translation [7, §3] this will immediately give the desired result for this definition of “Δn in.”There is, however, the more obvious and natural notion of “Δn in” to be considered: B is Δn in C iff there are Σn and Πn formulas of ⟨Lα, C⟩ which define B.

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.

Friedberg Numbering in Fragments of Peano Arithmetic and α-Recursion Theory

2013 ◽  
Vol 78 (4) ◽  
pp. 1135-1163 ◽  
Author(s):  
Wei Li
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Fragments Of Peano Arithmetic

AbstractIn this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchy Lα, where α is Σ1 admissible. We prove that(1) Over P− + BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and(2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.

Complex Spacetimes and the Newman-Janis Trick

2021 ◽  
Author(s):  
◽  
Del Rajan
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Mathematical Theory ◽  
Kerr Black Hole ◽  
Complex Variables ◽  
Complex Manifolds ◽  
Schwarzschild Black Hole ◽  
The Subject ◽  
Short Method

<p>In this thesis, we explore the subject of complex spacetimes, in which the mathematical theory of complex manifolds gets modified for application to General Relativity. We will also explore the mysterious Newman-Janis trick, which is an elementary and quite short method to obtain the Kerr black hole from the Schwarzschild black hole through the use of complex variables. This exposition will cover variations of the Newman-Janis trick, partial explanations, as well as original contributions.</p>

XVII. On the aberrations of compound lenses and object-glasses

1821 ◽  
Vol 111 ◽  
pp. 222-267 ◽  
Keyword(s):  
Mathematical Theory ◽  
The Subject ◽  
Experimental Researches ◽  
Analytical Skill

It has not unfrequently of late been made a subject of re­proach to mathematicians who have occupied themselves with the theory of the refracting telescope, that the practical be­nefit derived from their speculations has been by no means commensurate to the expenditure of analytical skill and labour they have called for, and that from all the abstruse researches of Clairaut, Euler, D'Alembert, and other celebrated geometers, nothing hitherto has resulted beyond a mass of complicated formulæ, which, though confessedly exact in theory, have never yet been made the basis of con­struction for a single good instrument, and remain therefore totally inapplicable, or at least unapplied, in practice. The simplest considerations, indeed, suffice for the correction of that part of the aberration which arises from the different refrangibility of the differently coloured rays; and accord­ingly, this part of the mathematical theory of refracting telescopes was soon brought to perfection, and has received no important accession since the original invention of the achromatic object-glass. Indeed the theoretical considera­tions advanced on this part of the subject by Euler and D'Alembert have even had a tendency to retard its advancement, by appearing to establish relations among the relative refractive powers of media on rays of different colours which later experimental researches have exploded.

scholarly journals The World Is Not a Theorem

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1467
Author(s):  
Stuart Kauffman ◽  
Andrea Roli
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Generative Process ◽  
Behavioral Adaptations ◽  
The World

The evolution of the biosphere unfolds as a luxuriant generative process of new living forms and functions. Organisms adapt to their environment, exploit novel opportunities that are created in this continuous blooming dynamics. Affordances play a fundamental role in the evolution of the biosphere, for organisms can exploit them for new morphological and behavioral adaptations achieved by heritable variations and selection. This way, the opportunities offered by affordances are then actualized as ever novel adaptations. In this paper, we maintain that affordances elude a formalization that relies on set theory: we argue that it is not possible to apply set theory to affordances; therefore, we cannot devise a set-based mathematical theory to deduce the diachronic evolution of the biosphere.

scholarly journals NOVEL “ORANG-ORANG PROYEK”: SEJARAH ORDE BARU

TELAGA BAHASA ◽  
2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ramis Rauf
Keyword(s):  
Set Theory ◽  
Positive Relationship ◽  
New Order ◽  
The Novel ◽  
Literary Works ◽  
Alain Badiou ◽  
History Of ◽  
The Subject

This study wants to reveal the truth procedures in Ahmad Tohari's novel Orang-Orang Proyek, as a part of an event and a factor in the presence of a new subject. This research would answer the problem: how was the subjectification of Ahmad Tohari in Orang-Orang Proyek novel as truth procedures? This study used the set theory by Alain Badiou. The set theory explained that within a set there were members of "Existing" or Being and events as "Plural" members.  The results proved that the subjectivity between Tohari and New Order events produced literary works: Orang-Orang Proyek. This happened because there was a positive relationship between the author and the event as well as on the naming of the event. Not only as of the subject but also do a fidelity to what he believed to be a truth. The truth procedures or the void—originating from the New Order event—was in the history of the making of a bridge in a village in Java island, Indonesia during the New Order period that filled with corruption, collusion, and nepotism. Tohari then embodied it in his novel. By the presences of the novel, we could know the category of Tohari's presentation as a new subject such as faithful, reactive, and obscure.

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