The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

Evaluation of Hessenberg determinants via generating function approach

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4945-4962 ◽  
Author(s):  
Emrah Kılıç ◽  
Talha Arıkan
Keyword(s):  
Generating Function ◽  
Arbitrary Constant ◽  
Function Method ◽  
Higher Order ◽  
Function Approach ◽  
New Method ◽  
Factor Matrix ◽  
New Approach ◽  
Constant Coefficients ◽  
The Subject

In this paper, we will present various results on computing of wide classes of Hessenberg matrices whose entries are the terms of any sequence. We present many new results on the subject as well as our results will cover and generalize earlier many results by using generating function method. Moreover, we will present a new approach on computing Hessenberg determinants, whose entries are general higher order linear recursions with arbitrary constant coefficients, based on finding an adjacency-factor matrix. We will give some interesting showcases to show how to use our new method.

Higher-order differential equations and higher-order lagrangian mechanics

1986 ◽  
Vol 99 (3) ◽  
pp. 565-587 ◽  
Author(s):  
M. Crampin ◽  
W. Sarlet ◽  
F. Cantrijn
Keyword(s):  
Order Differential Equation ◽  
Higher Order ◽  
Point Of View ◽  
Lagrangian Mechanics ◽  
First Order ◽  
The Subject ◽  
Tangent Bundles ◽  
Order Tangent ◽  
Higher Order Differential Equations ◽  
Number Of Authors

The study of higher-order mechanics, by various geometrical methods, in the framework of the theory of higher-order tangent bundles or jet spaces, has been undertaken by a number of authors recently: for example, Tulczyjew [16, 17], Rodrigues [14, 15] de León [8], Krupka and Musilova [11, and references therein]. In this article we wish to complement these studies by approaching the subject from a new point of view, one which we developed for second-order differential equation fields and first-order Lagrangian mechanics in [19]. In particular, our aim is to show that many of the results we obtained there may be extended to the higher-order case.

Higher order reflection principles

1989 ◽  
Vol 54 (2) ◽  
pp. 474-489 ◽  
Author(s):  
M. Victoria Marshall R.
Keyword(s):  
Measurable Cardinal ◽  
Reflection Principle ◽  
Higher Order ◽  
Weakly Compact ◽  
Class Theory ◽  
First Order ◽  
Transitive Set ◽  
Reflection Principles ◽  
Image Position ◽  
Order Reflection

In [1] and [2] there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See [3]. These axioms are formulated in first order logic with ∈:(A1)Extensionality.(A2)Class specification. Ifϕis a formula andAis not free inϕ, thenNote that “xis a set“ can be written as “∃u(x∈u)”.(A3)Subsets.Note also that “B⊆A” can be written as “∀x(x∈B→x∈A)”.(A4)Reflection principle. Ifϕ(x)is a formula, thenwhere “uis a transitive set” is the formula “∃v(u∈v) ∧ ∀x∀y(x∈y∧y∈u→x∈u)” andϕPuis the formulaϕrelativized to subsets ofu.(A5)Foundation.(A6)Choice for sets.We denote byB1the theory with axioms (A1) to (A6).The existence of weakly compact and-indescribable cardinals for everynis established inB1by the method of defining all metamathematical concepts forB1in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see [1]. There is a proof of the consistency ofB1assuming the existence of a measurable cardinal; see [4] and [5]. In [6] several set and class theories with reflection principles are developed. In them, the existence of inaccessible cardinals and some kinds of indescribable cardinals can be proved; and also there is a generalization of indescribability for higher-order languages using only class parameters.The purpose of this work is to develop higher order reflection principles, including higher order parameters, in order to obtain other large cardinals.

Eliminate of the Nonlinearity in Heterodyne Interferometer

2008 ◽  
Vol 381-382 ◽  
pp. 255-258
Author(s):  
Wen Mei Hou ◽  
L.Y. Shen ◽  
X. Ren
Keyword(s):  
Higher Order ◽  
Compensation System ◽  
Phase Compensation ◽  
Order Error ◽  
New Approach ◽  
Heterodyne Interferometer ◽  
First Order ◽  
Simple Phase ◽  
Set Up

A new approach is found to eliminate the nonlinearity of heterodyne interferometer. It uses a simple phase compensation system by rotating a polarizer which is set up before the detector in the interferometer. This method can be used in most science researches and the industry precision nanometer measurement, and it is valid for all nonlinearity errors whether it is a first-order or a higher-order error, or caused by any reasons.

Logic, sheaves, and factorization systems

1993 ◽  
Vol 58 (3) ◽  
pp. 872-893
Author(s):  
G. P. Monro
Keyword(s):  
Full Subcategory ◽  
Functional Relation ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
Completeness Proof ◽  
Factorization System ◽  
First Order ◽  
Functional Relations

In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.

Norm inequalities involving derivatives

1978 ◽  
Vol 82 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
W. D. Evans ◽  
A. Zettl
Keyword(s):  
General Class ◽  
New Method ◽  
Elementary Operator ◽  
Best Constant ◽  
New Approach ◽  
Theoretic Proof ◽  
General Class Of Functions ◽  
Image Position ◽  
Class Of Functions

SynopisisA new method for studying inequalities of the type ‖y(r)‖2<ε‖Sk−ry(k)‖2 + K(ε)‖S−ry‖2 and ‖y′‖2≦ Kp(S)‖Sy″‖ ‖S−1y‖ is presented here. With this new approach we obtain new and far reaching extensions of previously known inequalities of this sort as well as simpler proofs of the known cases. In addition we obtain an inequality of type ‖Sy′‖<ε‖(Sy′)′‖ + K(ε)‖y‖ for a general class of functions S. Also we give an elementary operator-theoretic proof of Everitt's characterization of the best constant as well as all cases of equality for

Transfinite recursive progressions of axiomatic theories

1962 ◽  
Vol 27 (3) ◽  
pp. 259-316 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Number Theory ◽  
Functional Calculus ◽  
Higher Order ◽  
Order Number ◽  
First Order ◽  
Definable Set ◽  
Recursively Enumerable

The theories considered here are based on the classical functional calculus (possibly of higher order) together with a set A of non-logical axioms; they are also assumed to contain classical first-order number theory. In foundational investigations it is customary to further restrict attention to the case that A is recursive, or at least recursively enumerable (an equivalent restriction, by [1]). For such axiomatic theories we have the well-known incompleteness phenomena discovered by Godei [6]. Quite far removed from such theories are those based on non-constructive sets of axioms, for example the set of all true sentences of first-order number theory. According to Tarski's theorem, there is not even an arithmetically definable set of axioms A which will give the same result (cf. [18] for exposition).

A New Approach for Determining Epilayer Strain Relaxation and Composition Through High Resolution X-Ray Diffraction

1995 ◽  
Vol 379 ◽  
Author(s):  
K. M. Matney ◽  
M.S. Goorsky
Keyword(s):  
High Resolution ◽  
Small Angle ◽  
Experimental Error ◽  
Strain Relaxation ◽  
New Method ◽  
X Ray Diffraction ◽  
New Approach ◽  
X Ray ◽  
First Order ◽  
Ray Geometry

ABSTRACTWe developed a new method of determining epilayer relaxation (along one direction) and composition using a symmetric and any single asymmetric high resolution x-ray diffraction scan. The previous use of small angle approximations can be very detrimental to calculated results and should be avoided. This new method does not employ small angle approximations or first order Taylor approximations, producing accurate results. The effect of x-ray geometry (glancing incident versus glancing exit) on the analysis of epilayer composition and strain is also reviewed. It is also shown that the glancing exit geometry is generally less susceptible to experimental error.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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