On a set theory of bernays

1967 ◽  
Vol 32 (3) ◽  
pp. 319-321 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Free Variable ◽  
Reflection Principle ◽  
Formal Definition ◽  
Von Neumann ◽  
Starting Point ◽  
Definition Of ◽  
Image Position

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

Existence of classes and value specification of variables

1950 ◽  
Vol 15 (2) ◽  
pp. 103-112 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Definition Of ◽  
Image Position

In mathematics, when we want to introduce classes which fulfill certain conditions, we usually prove beforehand that classes fulfilling such conditions do exist, and that such classes are uniquely determined by the conditions. The statements which state such unicity and existence of classes are in mathematical logic consequences of the principles of extensionality and class existence. In order to illustrate how these principles enable us to introduce classes into systems of mathematical logic, let us consider the manner in which Gödel introduces classes in his book on set theory.For instance, before introducing the definition of the non-ordered pair of two classesGödel puts down as its justification the following two axioms:By A4, for every two classesyandzthere exists at least one non-ordered pairwof them; and by A3,wis uniquely determined in A4.

Forcing for the impredicative theory of classes

1972 ◽  
Vol 37 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Rolando Chuaqui
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Inaccessible Cardinal ◽  
Consistency Proof ◽  
Suitable Form ◽  
Definition Of ◽  
Class Construction ◽  
The Continuum

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

Inner Models and Large Cardinals

1995 ◽  
Vol 1 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Ronald Jensen
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Ordinal Number ◽  
Infinite Cardinal ◽  
Recursive Definition ◽  
Von Neumann ◽  
Ordinal Numbers ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

Foundation versus induction in Kripke-Platek set theory

1998 ◽  
Vol 63 (4) ◽  
pp. 1399-1403
Author(s):  
Domenico Zambella
Keyword(s):  
Set Theory ◽  
Transitive Closure ◽  
Free Variable ◽  
Minimal Set ◽  
Open Formula ◽  
Open Induction ◽  
Image Position

We denote by KP_ the fragment of set-theory containing the axioms of extensionality, pairing, union and foundation as well as the schemas of ∆0-comprehension and ∆0-collection, that is: Kripke-Platek set-theory (KP) with the axiom of foundation in place of the ∈-induction schema. The theory KP is obtained by adding to KP_ the schema of ∈-inductionUsing ∈-induction it is possible to prove the existence of the transi tive closure without appealing to the axiom of infinity (see, e.g., [1]). Vice versa, when a theory proves the existence of the transitive closure, some induction is immediately ensured (by foundation and comprehension). This is not true in general: e.g., the whole of Zermelo-Fraenkel set-theory without the axiom of infinity does not prove ∈-induction (in fact, it does not prove the existence of the transitive closure; see, e.g., [3]). Open-induction is the schema of ∈-induction restricted to open formulas. We prove the following theorem.KP_ proves open-induction.We reason in a fixed but arbitrary model of KP_ whom we refer to as the model. The language is extended with a name for every set in the model. We call this constants parameters. Let φ(x) be a satisfiable open-formula possibly depending on parameters and with no free variable but x. We show that φ(x) is satisfied by an ∈-minimal set, that is, a set a such that φ(a) and (∀x ∈ a) ¬φ(x). We assume that no ordinal satisfies φ(x), otherwise the existence of a ∈-minimal set follows from foundation and comprehension.

On the axiom of extensionality – Part I

1956 ◽  
Vol 21 (1) ◽  
pp. 36-48 ◽  
Author(s):  
R. O. Gandy
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Formal System ◽  
Free Variable ◽  
Simple Theory ◽  
Predicate Calculus ◽  
Simple System ◽  
System A ◽  
Image Position

In part I of this paper it is shown that if the simple theory of types (with an axiom of infinity) is consistent, then so is the system obtained by adjoining axioms of extensionality; in part II a similar metatheorem for Gödel-Bernays set theory will be proved. The first of these results is of particular interest because type theory without the axioms of extensionality is fundamentally rather a simple system, and it should, I believe, be possible to prove that it is consistent.Let us consider — in some unspecified formal system — a typical expression of the axiom of extensionality; for example:where A(h) is a formula, and A(f), A(g) are the results of substituting in it the predicate variagles f, g for the free variable h. Evidently, if the system considered contains the predicate calculus, and if h occurs in A(h) only in parts of the form h(t) where t is a term which lies within the range of the quantifier (x), then 1.1 will be provable. But this will not be so in general; indeed, by introducing into the system an intensional predicate of predicates we can make 1.1 false. For example, Myhill introduces a constant S, where ‘Sϕψχω’ means that (the expression) ϕ is the result of substituting ψ for χ in ω.

Significant Digits and the Significand

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Benford’S Law ◽  
Formal Definition ◽  
Floating Point Arithmetic ◽  
Natural Domain ◽  
Starting Point ◽  
Formal Definitions ◽  
Benford's Law ◽  
Probability Spaces ◽  
Definition Of ◽  
Point Arithmetic

Benford's law is a statement about the statistical distribution of significant (decimal) digits or, equivalently, about significands, namely fraction parts in floating-point arithmetic. Thus, a natural starting point for any study of Benford's law is the formal definition of significant digits and the significand function. This chapter contains formal definitions, examples, and graphs of significant digits and the significand (mantissa) function, and the probability spaces needed to formulate Benford's law precisely, including the crucial natural domain of “events,” the so-called significand σ‎-algebra.

Wildness implies undecidability for lattices over group rings

1997 ◽  
Vol 62 (4) ◽  
pp. 1429-1447 ◽  
Author(s):  
Carlo Toffalori
Keyword(s):  
Finite Group ◽  
Group Rings ◽  
Starting Point ◽  
Torsionfree Module ◽  
Group A ◽  
The One ◽  
Definition Of ◽  
Image Position ◽  
A Finite Group ◽  
The Right

Let G be a finite group. A Z [G]-lattice is a finitely generated Z-torsionfree module over the group ring Z [G]. There is a general conjecture concerning classes of modules over sufficiently recursive rings, and linking wildness and undecidability. Given a finite group G, Z [G] is sufficiently recursive, and our aim here is just to investigate this conjecture for Z [G]-lattices. In this setting, the conjecture says thatif and only ifIn particular, we wish to deal here with the direction from the left to the right, so the one assuring that wildness implies undecidability. Of course, before beginning the analysis of this problem, one should agree upon a sharp definition of wildness for lattices. But, for our purposes, one might alternatively accept as a starting point a general classification of wild Z [G]-lattices when G is a finite p-group for some prime p, based on the isomorphism type of G. This is due to several authors and can be found, for instance, in [3]. It says that, when p is a prime and G is a finite p-group, thenif and only if.More precisely, the representation type of Z [G]-lattices is finite when G is cyclic of order ≤ p2, tame domestic when G is the Klein group [1], tame non-domestic when G is cyclic of order 8 [11].So our claim might be stated as follows.

Inner models for set theory—Part II

1952 ◽  
Vol 17 (4) ◽  
pp. 225-237 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Complete Model ◽  
Universal Class ◽  
Propositional Function ◽  
Class V ◽  
Completeness Condition ◽  
Inner Models ◽  
Condition C ◽  
Definition Of ◽  
Image Position

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I (page 186). The condition (c) of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition (c) and the completeness conditions 1.41, 1.42 in terms of Vm, we may state the definition in the form:3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM(U)’ (‘Udetermines a super-complete model’) thus:whereψ(U) is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ(U) contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form (after having replaced all defined symbols which occur by the corresponding formulae in primitive notation) and then replacing ‘(Am)ϕ(Am) bywhich is in turn replaced by, and similarly replacing ‘(xm)ϕ(xm)’ by ‘(xm)ϕ(xm)’ by ‘(X)(X ϵ U ▪ ⊃ ▪ ϕ(X)), andThusψ(U) is obtained in primitive notation.

Holism, Chinese Medicine and Systems Ideologies: Rewriting the Past to Imagine the Future

2018 ◽  
Author(s):  
Volker Scheid
Keyword(s):  
Eighteenth Century ◽  
Chinese Medicine ◽  
Medical Humanities ◽  
Science And Technology Studies ◽  
Human Beings ◽  
The Past ◽  
Starting Point ◽  
The Us ◽  
Shared Objective ◽  
Definition Of

This chapter explores the articulations that have emerged over the last half century between various types of holism, Chinese medicine and systems biology. Given the discipline’s historical attachments to a definition of ‘medicine’ that rather narrowly refers to biomedicine as developed in Europe and the US from the eighteenth century onwards, the medical humanities are not the most obvious starting point for such an inquiry. At the same time, they do offer one advantage over neighbouring disciplines like medical history, anthropology or science and technology studies for someone like myself, a clinician as well as a historian and anthropologist: their strong commitment to the objective of facilitating better medical practice. This promise furthermore links to the wider project of critique, which, in Max Horkheimer’s definition of the term, aims at change and emancipation in order ‘to liberate human beings from the circumstances that enslave them’. If we take the critical medical humanities as explicitly affirming this shared objective and responsibility, extending the discipline’s traditional gaze is not a burden but becomes, in fact, an obligation.

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