Ordinal logics

2018 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Effective Means ◽  
Formal System ◽  
Formal Systems ◽  
Alan Turing ◽  
Kurt Gödel ◽  
Boot Strap ◽  
Completeness Result ◽  
Systematic Development ◽  
Intended Use ◽  
Formal Statement

By an ordinal logic is meant any uniform effective means of associating a logic (that is, an effectively generated formal system) with each effective ordinal representation. This notion was first introduced and studied by Alan Turing in 1939 as a means to overcome the incompleteness of sufficiently strong consistent formal systems, established by Kurt Gödel in 1931. The first ordinal logic to consider, in view of Gödel’s results, would be that obtained by iterating into the constructive transfinite the process of adjoining to each system the formal statement expressing its consistency. For that ordinal logic, Turing obtained a completeness result for the class of true statements of the form that all natural numbers have a given effectively decidable property. However, he also showed that any ordinal logic (such as this) which is strictly increasing with increasing ordinal representation cannot have the property of invariance: in general, different representations of the same ordinal will have different sets of theorems attached to them. This makes the choice of representation a crucial one, and without a clear rationale as to how that is to be made, the notion of ordinal logic becomes problematic for its intended use. Research on ordinal logics lapsed until the late 1950s, when it was taken up again for more systematic development. Besides leading to improvements of Turing’s results in various respects (both positive and negative), the newer research turned to restrictions of ordinal logics by an autonomy (or ‘boot-strap’) condition which limits the choice of ordinal representations admitted, by requiring their recognition as such in advance.

Modality and folk psychology

2019 ◽  
Vol 28 (1) ◽  
pp. 19-27
Author(s):  
Ja. O. Petik
Keyword(s):  
Modal Logic ◽  
Brain Activity ◽  
Formal System ◽  
Philosophy Of Logic ◽  
Psychological States ◽  
Formal Systems ◽  
Modern Psychology ◽  
Philosophical Interpretation ◽  
Important Direction

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic. The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.

Degrees of formal systems

1958 ◽  
Vol 23 (4) ◽  
pp. 389-392 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Sufficient Conditions ◽  
Formal System ◽  
Independent Interest ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Axiomatizable Theory

In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of Gödel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable.In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A. (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.

scholarly journals NEW GENERATION OF ADAPTABLE MOBILE MEANS OF ORGANISATIONAL REPAIR

2018 ◽  
pp. 32-34 ◽  
Author(s):  
V. L. Krivolapov ◽  
A. F. Strakhov
Keyword(s):  
Effective Means ◽  
Key Factors ◽  
Time Period ◽  
Foreign Countries ◽  
Repair Systems ◽  
New Generation ◽  
Technological Opportunities ◽  
Technical Readiness ◽  
Further Development ◽  
Intended Use

One of the key factors of ensuring the required level of the AD WME technical readiness for the intended use is minimisation of the time period necessary for restoration of the AD WME systems after their failures. Over the years, JSC Principle Production and Technical Company Granit has per-formed a cycle of R&D activities aimed at creating effective means of the AD WME maintenance and organisational repair at their deployment sites as well as at developing innovative maintenance and organisational repair technologies. Following the results of these activities, a set of adaptable unified mobile repair and diagnostic equipment (RDE)was created and then exported in order to ensure the maintenance and organisational repair of the AD WME pieces delivered within military technical cooperation with foreign countries. Further development of this focus area resulted in creation of a set of unified mobile organisational repair systems. This article considers main functional and technical as well as technological opportunities of the unified mobile organisational repair systems. Their prototypes are previously developed RDE for export.

scholarly journals Observações sobre a meta final do modo de fazer filosofia de Ludwig Wittgenstein

2019 ◽  
Vol 22 (3) ◽  
pp. 411-438
Author(s):  
Gustavo Augusto Fonseca
Keyword(s):  
Ludwig Wittgenstein ◽  
Basic Concept ◽  
Foundations Of Mathematics ◽  
Newtonian Physics ◽  
Alan Turing ◽  
Kurt Gödel ◽  
Perfect Model ◽  
Proposal A ◽  
Heinrich Hertz

In The Principles of Mechanics, physicist Heinrich Hertz argues that instead of replying to the question “what is force?” like physicists and philosophers had been doing unsuccessfully, Newtonian physics should be reformulated without considering “force” a basic concept. Decades after Hertz’s book, Ludwig Wittgenstein considered the physicist’s proposal a perfect model for how philosophical problems should be solved, to the point that he made it the foundation of his way of doing philosophy. This article addresses Wittgenstein’s way of doing philosophy, while it also proposes the reason why he failed in solving the philosophical problems — as did Hertz in his project on reformulating Newtonian physics without considering the concept “force”. And to illustrate Wittgenstein’s failure, it examines his disputes with mathematicians Kurt Gödel and Alan Turing on the foundations of mathematics.

Prologue

2017 ◽  
Author(s):  
Jan von Plato
Keyword(s):  
Information Society ◽  
Formal Languages ◽  
Foundations Of Mathematics ◽  
Direct Path ◽  
John Von Neumann ◽  
Von Neumann ◽  
Alan Turing ◽  
Kurt Gödel

It may sound paradoxical, but if around 1930 Kurt Gödel had not thought very deeply about the foundations of mathematics, there would be no information society in the form in which we have it today. Gödel’s solitary work was the single most important factor in the development of precise theories of formal languages, ones that through the coding he invented could be handled by a machine. Likewise, his work led to precise notions of algorithmic computability from which a direct path led to the first theoretical ideas of a computer, in the work of Alan Turing in 1936 and John von Neumann some years later....

On Absolutely Unsolvable Problems

2019 ◽  
Vol 4 (3) ◽  
Author(s):  
Liesbeth De Mol
Keyword(s):  
Research Program ◽  
Alan Turing ◽  
Kurt Gödel ◽  
Unsolvable Problems

It was not Alan Turing or Kurt Gödel, but Emil Post, who reflected most deeply on issues of absolutely unsolvable problems. He proposed an open-ended research program that was abandoned upon his death.

HEALTH TECHNOLOGY ASSESSMENT IN THE UNITED KINGDOM

2000 ◽  
Vol 16 (2) ◽  
pp. 591-625 ◽  
Author(s):  
Steven H. Woolf ◽  
Chris Henshall
Keyword(s):  
United Kingdom ◽  
Health Service ◽  
Health Technology Assessment ◽  
Technology Assessment ◽  
Private Insurance ◽  
Effective Means ◽  
Formal System ◽  
Health Technology ◽  
Health Coverage ◽  
The United Kingdom

The National Health Service (NHS) provides universal health coverage for all British citizens. Most services are free of charge, although modest copayments are sometimes applied. About 11% of the population also has private insurance. General practitioners, generally the first point of contact for accessing the system, are independent contractors who serve as gatekeepers for specialist and hospital services and enjoy substantial clinical autonomy. Hospitals are public and are regionalized, but the 1990 reforms made them self-governing trusts that contract with local purchasers (health authorities and general practitioner fundholders). Reforms beginning in 1990 moved the NHS away from a centralized administrative structure to more pluralistic arrangements in which competition, as well as management, influences how services develop. Health technology and health technology assessment (HTA) have gained increasing attention in the NHS during this period, as part of a wider NHS Research and Development (R&D) Strategy. The strategy promotes a knowledge-based health service with a strong research infrastructure and the capacity to critically review its own needs. HTA is the largest and most developed of the programs within the strategy. It has a formal system for setting assessment priorities involving widespread consultation within the NHS, and a National Co-ordinating Centre for Health Technology Assessment. The stategy supports related centers such as the U.K. Cochrane Centre and the NHS Centre for Reviews and Dissemination. A hallmark of the HTA program is strong public participation. The United Kingdom has made a major commitment to HTA and to seeking effective means of reviewing and disseminating evidence.

Benchmarks for reasoning with syntax trees containing binders and contexts of assumptions

2017 ◽  
Vol 28 (9) ◽  
pp. 1507-1540 ◽  
Author(s):  
AMY FELTY ◽  
ALBERTO MOMIGLIANO ◽  
BRIGITTE PIENTKA
Keyword(s):  
Formal System ◽  
Abstract Syntax ◽  
Qualitative Comparison ◽  
Separate Paper ◽  
Logical Frameworks ◽  
Design And Implementation ◽  
Formal Systems ◽  
Reasoning Systems ◽  
Fundamental Research ◽  
Research Questions

A variety of logical frameworks supports the use of higher order abstract syntax in representing formal systems. Although these systems seem superficially the same, they differ in a variety of ways, for example, how they handle acontextof assumptions and which theorems about a given formal system can be concisely expressed and proved. Our contributions in this paper are two-fold: (1) We develop a common infrastructure and language for describing benchmarks for systems supporting reasoning with binders, and (2) we present several concrete benchmarks, which highlight a variety of different aspects of reasoning within a context of assumptions. Our work provides the background for the qualitative comparison of different systems that we have completed in a separate paper. It also allows us to outline future fundamental research questions regarding the design and implementation of meta-reasoning systems.

scholarly journals PDE boundary conditions that eliminate quantum weirdness: a mathematical game inspired by Kurt Gödel and Alan Turing

2021 ◽  
Vol 20 ◽  
pp. 211-239
Author(s):  
Jeffrey Boyd
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Single Point ◽  
Alan Turing ◽  
Elementary Wave ◽  
Kurt Gödel ◽  
Mathematical Game ◽  
Directional Wave ◽  
Elementary Waves ◽  
Quantum Mathematics

Although boundary condition problems in quantum mathematics (QM) are well known, no one ever used boundary conditions technology to abolish quantum weirdness. We employ boundary conditions to build a mathematical game that is fun to learn, and by using it you will discover that quantum weirdness evaporates and vanishes. Our clever game is so designed that you can solve the boundary condition problems for a single point if-and-only-if you also solve the “weirdness” problem for all of quantum mathematics. Our approach differs radically from Dirichlet, Neumann, Robin, or Wolfram Alpha. We define domain Ω in one-dimension, on which a partial differential equation (PDE) is defined. Point α on ∂Ω is the location of a boundary condition game that involves an off-center bi-directional wave solution called Æ, an “elementary wave.” Study of this unusual, complex wave is called the Theory of Elementary Waves (TEW). We are inspired by Kurt Gödel and Alan Turing who built mathematical games that demonstrated that axiomatization of all mathematics was impossible. In our machine quantum weirdness vanishes if understood from the perspective of a single point α, because that pinpoint teaches us that nature is organized differently than we expect.

Degrees of unsolvability associated with classes of formalized theories

1957 ◽  
Vol 22 (2) ◽  
pp. 161-175 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Formal System ◽  
Point Of View ◽  
Wide Sense ◽  
Recursively Enumerable Set ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
The Difference ◽  
Degrees Of Unsolvability ◽  
Order Predicate Calculus

In his well-known paper [11], Post founded a general theory of recursively enumerable sets, which had its metamathematical source in questions about the decision problem for deducibility in formal systems. However, in centering attention on the notion of degree of unsolvability, Post set a course for his theory which has rarely returned to this source. Among exceptions to this tendency we may mention, as being closest to the problems considered here, the work of Kleene in [8] pp. 298–316, of Myhill in [10], and of Uspenskij in [15]. It is the purpose of this paper to make some further contributions towards bridging this gap.From a certain point of view, it may be argued that there is no real separation between metamathematics and the theory of recursively enumerable sets. For, if the notion of formal system is construed in a sufficiently wide sense, by taking as ‘axioms’ certain effectively found members of a set of ‘formal objects’ and as ‘proofs’ certain effectively found sequences of these objects, then the set of ‘provable statements’ of such a system may be identified, via Gödel's numbering technique, with a recursively enumerable set; and conversely, each recursively enumerable set is identified in this manner with some formal system (cf. [8] pp. 299–300 and 306). However, the pertinence of Post's theory is no longer clear when we turn to systems formalized within the more conventional framework of the first-order predicate calculus. It is just this restriction which serves to clarify the difference in spirit of the two disciplines.

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