A homogeneous system for formal logic

1943 ◽  
Vol 8 (1) ◽  
pp. 1-23 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
Set Theory ◽  
General Type ◽  
Homogeneous System ◽  
Formal Logic ◽  
Strict Sense ◽  
Standard Methods ◽  
Logical Construction ◽  
Logical Type ◽  
Classical Mathematics ◽  
Axiomatic Set Theory

Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.Following the familiar simplification of Russell's theory, let us mean by a logical type in the strict sense any one of the following: (i) the totality consisting exclusively of individuals, (ii) the totality consisting exclusively of classes whose members are exclusively individuals, (iii) the totality consisting exclusively of classes whose members are exclusively classes whose members in turn are exclusively individuals, and so on. Any entity from (ii) is said to be one type higher than any entity from (i), any entity from (iii), one type higher than any entity from (ii), and so on. In systems based upon this simplified theory of types, the only significant atomic formulae involving “ε” are those asserting the membership of an entity in an entity one type higher. Thus any expression of the form “(x∈y)” is meaningless except where “y” denotes an entity of just one type higher than the type of the entity denoted by “x” It is by means of general type restrictions of this kind that the Russell and other paradoxes are avoided.

scholarly journals The Fall and Original Sin of Set Theory

Phronimon ◽  
2019 ◽  
Vol 19 ◽  
Author(s):  
Daniel Francois Strauss
Keyword(s):  
Set Theory ◽  
Original Sin ◽  
Bertrand Russell ◽  
Foundations Of Mathematics ◽  
Classical Analysis ◽  
Mathematical System ◽  
Classical Mathematics ◽  
The Difference ◽  
Infinite Totality ◽  
Axiomatic Set Theory

Hermann Weyl published a brief survey as preface to a review of The Philosophy of Bertrand Russell in 1946. In this survey he used the phrase, “The Fall and Original Sin of Set Theory.” Investigating the background of this remark will require that we pay attention to a number of issues within the foundations of mathematics. For example: Did God make the integers—as Kronecker alleged? Is mathematics set theory? Attention will also be given to axiomatic set theory and relevant ontic pre-conditions, such as the difference between number and number symbols, to number as “an aspect of objective reality” (Gödel), integers and induction (Skolem) as well as to the question if infinity—as endlessness—could be completed. In 1831 Gauss objected to viewing the infinite as something completed, which is not allowed in mathematics. It will be argued that the actual infinite is rather connected to what is present “at once,” as an infinite totality. By the year 1900 mathematicians believed that mathematics had reached absolute rigour, but unfortunately the rest of the twentieth century witnessed the opposite. The axiom of infinity ruined the expectations of logicism—mathematics cannot be reduced to logic. The intuitionism of Brouwer, Weyl and others launched a devastating attack on classical analysis, further inspired by the outcome of Gödel’s famous proof of 1931, in which he has shown that a formal mathematical system is inconsistent or incomplete. Intuitionism created a whole new mathematics, which finds no counter-part in classical mathematics. Slater remarked that within this logical paradise of Russell lurked a serpent, hidden behind the unjustified employment of the at once infinite. According to Weyl, “This is the Fall and original sin of set theory for which it is justly punished by the antinomies.” In conclusion, a few systematic distinctions are introduced.

scholarly journals Hybrid structures applied to ideals in near-rings

2021 ◽  
Author(s):  
B. Elavarasan ◽  
G. Muhiuddin ◽  
K. Porselvi ◽  
Y. B. Jun
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Real World ◽  
Rough Set Theory ◽  
Wide Spectrum ◽  
Hybrid Structure ◽  
Soft Set ◽  
Hybrid Structures ◽  
Classical Mathematics ◽  
Classical Means

AbstractHuman endeavours span a wide spectrum of activities which includes solving fascinating problems in the realms of engineering, arts, sciences, medical sciences, social sciences, economics and environment. To solve these problems, classical mathematics methods are insufficient. The real-world problems involve many uncertainties making them difficult to solve by classical means. The researchers world over have established new mathematical theories such as fuzzy set theory and rough set theory in order to model the uncertainties that appear in various fields mentioned above. In the recent days, soft set theory has been developed which offers a novel way of solving real world issues as the issue of setting the membership function does not arise. This comes handy in solving numerous problems and many advancements are being made now-a-days. Jun introduced hybrid structure utilizing the ideas of a fuzzy set and a soft set. It is to be noted that hybrid structures are a speculation of soft set and fuzzy set. In the present work, the notion of hybrid ideals of a near-ring is introduced. Significant work has been carried out to investigate a portion of their significant properties. These notions are characterized and their relations are established furthermore. For a hybrid left (resp., right) ideal, different left (resp., right) ideal structures of near-rings are constructed. Efforts have been undertaken to display the relations between the hybrid product and hybrid intersection. Finally, results based on homomorphic hybrid preimage of a hybrid left (resp., right) ideals are proved.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

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