Some proofs of independence in axiomatic set theory

1956 ◽  
Vol 21 (3) ◽  
pp. 291-303 ◽  
Author(s):  
Elliott Mendelson
Keyword(s):  
Set Theory ◽  
The Other ◽  
Formal Systems ◽  
System A ◽  
Similar Proof ◽  
Independence Results ◽  
The One ◽  
Axiomatic Set Theory ◽  
Gödel’S Theorem

1. Gödel's theorem that sufficiently strong formal systems cannot prove their own consistency and Tarski's method for constructing truth-definitions can be combined to give several independence results in axiomatic set theory. In substance, the following theorems can be obtained: (a) The existence of inaccessible ordinals is not provable from the axioms of set theory, if these axioms are consistent, (b) The axiom of infinity is independent of the other axioms, if these other axioms are consistent, (c) The axiom of replacement is independent of the other axioms, if these other axioms are consistent. In all cases, V = L will be included as an axiom.The result (a) concerning inaccessible ordinals already has been proved in Shepherdson [10] and Mostowski [6], but their proofs are somewhat different from the one given here. According to Mostowski [6], Kuratowski essentially had a proof of (a) in 1924. Propositions (b) and (c) have been proved, for axiomatic set theory without the axiom V = L, by Bernays [1] pp. 65–69. The method of proof used in this paper is due to Firestone and Rosser [2].An outline of a similar proof along these lines is given in Rosser [7] pp. 60–62.As our system G of set theory, we choose Gödel's system A, B, C, as given in [4], except for the following changes.

The General Idea Behind Gödel’s Proof

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
The Other ◽  
General Idea ◽  
Original Proof ◽  
Extensive Class ◽  
The One ◽  
Axiomatic Set Theory ◽  
Gödel’S Theorem ◽  
Mathematical Systems

In the next several chapters we will be studying incompleteness proofs for various axiomatizations of arithmetic. Gödel, 1931, carried out his original proof for axiomatic set theory, but the method is equally applicable to axiomatic number theory. The incompleteness of axiomatic number theory is actually a stronger result since it easily yields the incompleteness of axiomatic set theory. Gödel begins his memorable paper with the following startling words. . . . “The development of mathematics in the direction of greater precision has led to large areas of it being formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems to date are, on the one hand, the Principia Mathematica of Whitehead and Russell and, on the other hand, the Zermelo-Fraenkel system of axiomatic set theory. Both systems are so extensive that all methods of proof used in mathematics today can be formalized in them—i.e. can be reduced to a few axioms and rules of inference. It would seem reasonable, therefore, to surmise that these axioms and rules of inference are sufficient to decide all mathematical questions which can be formulated in the system concerned. In what follows it will be shown that this is not the case, but rather that, in both of the cited systems, there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis of the axioms.” . . . Gödel then goes on to explain that the situation does not depend on the special nature of the two systems under consideration but holds for an extensive class of mathematical systems. Just what is this “extensive class” of mathematical systems? Various interpretations of this phrase have been given, and Gödel’s theorem has accordingly been generalized in several ways. We will consider many such generalizations in the course of this volume. Curiously enough, one of the generalizations that is most direct and most easily accessible to the general reader is also the one that appears to be the least well known.

information. How do produced quantities influence the costs per unit? How can costs, calculated at different times, be compared? What is the best way to distribute the overheads? etc.. .. After the setting up of the accounting system, a long process of maturation began. This is evident, on the one hand, from the discussions of the Board of Directors and, on the other hand from the differences between the two sets of accounts approved by the Board of Directors in 1832 and 1872. The structure of the Com­ pany evolved considerably between 1832 and 1880: two mergers occurred, the first one in 1858 with Saint-Quirin, a glass manufac­ turer, and the second one in 1872 with Perret-Olivier, whose fields of activity were mining and chemistry. After the second merger, the sales figures for chemistry outstripped the sales of glass and mirrors and during this time the Company had grown to include 16 branches in France and Germany. DISCUSSIONS ON INDUSTRIAL ACCOUNTING All the questions dealing with the setting up of a management accounting system were discussed by the Boards of Directors. In most cases, the solutions were only practical ones. There never seemed any intent or desire by the Company to make any theory or any generalization of those practical solutions. Direct and indirect costs. The distinction between direct and indirect cost was made first in 1829 with regards to labor charges.9 Salaries, of which a comprehensive list is given above, will be separated into two groups: 1) Those concerning directly and specially with the manufacturing process. 2) Those concerning administration. At the end of the year, the former will be divided and included in the suitable items of expenses; then the latter will be included in the overheads. However, direct labor is likely to have included only the wages of workers having a permanent job, and excluded those of the day laborer, which are by their very nature fluctuating. In the soda factory, the majority of workers were day laborers, thus making it difficult to estimate precisely the ratio between direct and indirect labor charges. Production level and cost per unit. In the previously quoted chief accountant’s report concerning the financial year 1827-1828,

2014 ◽  
pp. 259-259
Keyword(s):  
Board Of Directors ◽  
Boards Of Directors ◽  
Indirect Costs ◽  
The Other ◽  
Production Level ◽  
Accounting System ◽  
Direct And Indirect Costs ◽  
System A ◽  
The Board Of Directors ◽  
The One

The Polarizability of Molecular Hydrogen H2

1936 ◽  
Vol 32 (2) ◽  
pp. 260-264 ◽  
Author(s):  
C. E. Easthope
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Molecular Hydrogen ◽  
Applied Field ◽  
Minimum Energy ◽  
Wave Functions ◽  
The Other ◽  
System A ◽  
The One ◽  
Two Parameter

1. The problem of calculating the polarizability of molecular hydrogen has recently been considered by a number of investigators. Steensholt and Hirschfelder use the variational method developed by Hylleras and Hassé. For ψ0, the wave function of the unperturbed molecule when no external field is present, they take either the Rosent or the Wang wave function, while the wave functions of the perturbed molecule were considered in both the one-parameter form, ψ0 [1+A(q1 + q2)] and the two-parameter form, ψ0 [1+A(q1 + q2) + B(r1q1 + r2q2)], where A and B are parameters to be varied so as to give the system a minimum energy, q1 and q2 are the coordinates of the electrons 1 and 2 in the direction of the applied field as measured from the centre of the molecule, and r1 and r2 are their respective distances from the same point. Mrowka, on the other hand, employs a method based on the usual perturbation theory. Their numerical results are given in the following table.

scholarly journals A Review on Outfit Fashion Recommendation System

2021 ◽  
pp. 220-222
Author(s):  
Bhagyshree Pravin Bhure ◽  
Pratiksha Tulshiram Bansod ◽  
Monali Shivram Amgaokar ◽  
Savita Pralhad Lodiwale ◽  
Anjali Pravin Orkey ◽  
...  
Keyword(s):  
Network Topology ◽  
Recommendation System ◽  
Web Applications ◽  
Living Standards ◽  
The Other ◽  
Convolutional Network ◽  
System A ◽  
Training Technique ◽  
The One ◽  
Time And Energy

With the quick rise in living standards, people's shopping passion grew, and their desire for clothing grew as well. A growing number of people are interested in fashion these days. However, when confronted with a large number of garments, consumers are forced to try them on multiple times, which takes time and energy. As a result of the suggested Fashion Recommendation System, a variety of online fashion businesses and web applications allow buyers to view collages of stylish items that look nice together. Clients and sellers benefit from such recommendations. On the one hand, customers can make smarter shopping decisions and discover new articles of clothes that complement one other. Complex outfit recommendations, on the other hand, assist vendors in selling more products, which has an impact on their business. FashionNet is made up of two parts: a feature network for extracting features and a matching network for calculating compatibility. A deep convolutional network is used to achieve the former. For the latter, a multi-layer completely connected network topology is used. For FashionNet, you must create and compare three different architectures. To achieve individualised recommendations, a two-stage training technique was created.

What is Neologicism?

2006 ◽  
Vol 12 (1) ◽  
pp. 60-99 ◽  
Author(s):  
Bernard Linsky ◽  
Edward N. Zalta
Keyword(s):  
20Th Century ◽  
Set Theory ◽  
The Other ◽  
Early 20Th Century ◽  
Foundations Of Mathematics ◽  
The One

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Frege's [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

scholarly journals Lines and Semi-Countably Differentiable Primes

2021 ◽  
Vol 70 (2) ◽  
pp. 90-98
Author(s):  
Abigaël Alkema
Keyword(s):  
Set Theory ◽  
The Other ◽  
Real Case ◽  
Central Problem ◽  
Linear Graph ◽  
Other Hand ◽  
Non Linear ◽  
Recent Developments ◽  
Axiomatic Set Theory

Let l(u)⊃ |G|. A central problem in higher non-linear graph theoryis the construction of projective numbers. We show that Recent developments in axiomatic set theory [6] have raised the questionof whetherEis not dominated byl. On the other hand, the work in [6, 24] did not consider the hyper-real case.

The Artistic and Religious Canon in the Christian and Muslim Traditions

2021 ◽  
Vol 18 (4) ◽  
pp. 390-397
Author(s):  
Alfia K. Shayakhmetova
Keyword(s):  
Comparative Analysis ◽  
Religious Tradition ◽  
The Other ◽  
Religious Art ◽  
Narrow Sense ◽  
Artistic Style ◽  
System A ◽  
Religious Cult ◽  
Religious Systems ◽  
The One

The article presents a comparative analysis of the musical component of the artistic and religious canon in the orthodox direction of Christianity (Orthodoxy) and Islam. The author considers the concept of canon in a broad sense as a special type of holistic artistic-style system. In a narrow sense, it is considered as an artistic method with its own specific musical and ritual code. The musical beginning is an integral component of a religious cult and, consequently, of the liturgical canon in the Muslim and Christian traditions. Studying music as an artistic component of a particular religious tradition is one of the most popular trends in modern musicology.Religious art is canonical regardless of the ideological differences between religious systems. A canon as an integral art system is characterized by a number of patterns that manifest themselves at all levels of its structure, thus acting as a norm of tradition and, at the same time, as a way of preserving and transmitting this norm, and this transmission is of a variable type. In the article, the term “canon” is understood in the context of the culturological concepts of canon revealed in the works of V.V. Bychkov, A.F. Losev, Yu.M. Lotman, Yu.N. Plakhov, P.A. Florensky. The canon is understood as an artistic method, on the one hand, and as a special artistic and stylistic system (a set of rules that exist virtually), on the other.The article clarifies the theoretical ideas about the canon as a carrier of the norm of tradition in relation to the field of art.

Set theory, different systems of

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Set Theory ◽  
The Other ◽  
Von Neumann ◽  
Class Theory ◽  
Iterative Conception ◽  
The One ◽  
Limitation Of Size ◽  
As If

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory. Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals). Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception. Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.

scholarly journals La mente mecánica

2018 ◽  
Vol 1 (1) ◽  
Author(s):  
Juan Miguel Suay Belenguer
Keyword(s):  
Quantum Mechanics ◽  
Philosophy Of Mind ◽  
Human Mind ◽  
The Other ◽  
Turing Machines ◽  
Other Hand ◽  
The One ◽  
Gödel’S Theorem

Resumen: La mente humana es capaz de razonar, de manera similar que lo haría un ordenador, sobre cuestiones que son formuladas algorítmicamente, pero tam­bién es capaz de realizar otras funciones que algunos autores consideran que son imposibles de simular por una máquina. Los diferentes respuestas a cómo fun­ciona la mente han sido abordadas por la filosofía de la mente, la lógica, la psico­logía y la neurología, incluso hoy en día por la mecánica cuántica. En este trabajo intentaré realizar un compendio de las algunas teorías que han apoyado por un lado la posibilidad, y por otro la imposibilidad, de una mente mecánica.Palabras clave: filosofía de la mente, intencionalidad, dualismo, maquinas de Tu­ring, Teorema de Gödel. Abstract: Human mind is capable of reasoning, as much as a computer would do, on issues algorithmically formulated, but it is also able to play other roles which are regarded by some authors impossible for a machine to mimic. The different answers to how human mind works have been addressed by philosophy of mind, logics, psychology and neurology, and nowadays even by quantum mechanics. In this paper I will intend to present an overall review of the several theories that have supported on the one hand the feasibility, and on the other hand the impos­sibility, of a mechanical mind.Key words: Philosophy of mind, Intentionality, dualism, Turing machines, Gödel’s theorem. Recibido: 07/09/2011. Aprobado: 10/12/2012. 

LINGUA CHARACTERICA AND CALCULUS RATIOCINATOR: THE LEIBNIZIAN BACKGROUND OF THE FREGE-SCHRÖDER POLEMIC

2020 ◽  
pp. 1-36
Author(s):  
JOAN BERTRAN-SAN MILLÁN
Keyword(s):  
Formal System ◽  
Mathematical Concept ◽  
Boolean Logic ◽  
The Other ◽  
Logical Theory ◽  
Specific Content ◽  
System A ◽  
Scientific Disciplines ◽  
Ideal Language ◽  
The One

Abstract After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme.

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