Non-standard models for formal logics

1950 ◽  
Vol 15 (2) ◽  
pp. 113-129 ◽  
Author(s):  
J. Barkley Rosser ◽  
Hao Wang
Keyword(s):  
Number Theory ◽  
Standard Model ◽  
Formal Logic ◽  
Precise Definition ◽  
Ordinal Numbers ◽  
Equality Relation ◽  
Standard Models ◽  
Definition Of ◽  
Positive Integers

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex (for instance, if it is adequate for number theory), then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either:(a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model.(b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦.(c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦.

scholarly journals Statistical Methodology for the Definition of Standard Model for Energy Analysis of Residential Buildings in Korea

Energies ◽  
2020 ◽  
Vol 13 (21) ◽  
pp. 5796
Author(s):  
Hye-Ryeong Nam ◽  
Seo-Hoon Kim ◽  
Seol-Yee Han ◽  
Sung-Jin Lee ◽  
Won-Hwa Hong ◽  
...  
Keyword(s):  
Standard Model ◽  
Energy Analysis ◽  
Residential Buildings ◽  
Statistical Methodology ◽  
National Scale ◽  
The Standard Model ◽  
Perform Cluster Analysis ◽  
Standard Models ◽  
Components Analysis ◽  
Definition Of

This study was conducted to propose an optimal methodology for deriving a standard model from existing residential buildings. To strategically improve existing residential buildings, it is necessary to identify standard models that can be used as quantitative standards. In this study, a total of six methods were established for different algorithms in the dimensionality reduction and clustering stage of the data preprocessing stage. In addition, a total of 22,342 households’ data were analyzed, and a total of 26 variables were used to perform cluster analysis. The process of method 6 (data pre-processing, principal components analysis, clustering [K-medoids], verification) was proposed as a way to derive the standard model from the existing Korean housing. The method proposed in this study is capable of deriving a number of standard models considering all variables (n) in a single analysis. The representative building derived in this study contains a lot of building data, so it can be effectively used for planning and research related to buildings on a regional and national scale. In addition, this process can be applied to various buildings to derive representative buildings.

Formal development of ordinal number theory

1955 ◽  
Vol 20 (2) ◽  
pp. 95-104
Author(s):  
Steven Orey
Keyword(s):  
Number Theory ◽  
Ordinal Number ◽  
Ordered Sets ◽  
Stringent Condition ◽  
Easy Consequence ◽  
Basic Properties ◽  
Ordinal Numbers ◽  
Order Types ◽  
Proper Subclass ◽  
Definition Of

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

A Lower Bound for the Scholz-Brauer Problem

1969 ◽  
Vol 21 ◽  
pp. 675-683 ◽  
Author(s):  
Kenneth B. Stolarsky
Keyword(s):  
Lower Bound ◽  
Precise Definition ◽  
Addition Chains ◽  
Definition Of ◽  
Image Position ◽  
Positive Integers

In (6)Scholz asked if the inequality1.1held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating1.2It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.

Initial Polymerization Sites Indicated During Mitochondrial Oxidation of Diaminobenzidine after Toluene Treatment

1977 ◽  
Vol 35 ◽  
pp. 408-409
Author(s):  
W. A. Shannon ◽  
M. A. Matlib
Keyword(s):  
Membrane Permeability ◽  
Cytochrome Oxidase ◽  
Rat Liver ◽  
Precise Definition ◽  
Cytochemical Localization ◽  
Oxidative Reaction ◽  
Mitochondrial Oxidation ◽  
Definition Of ◽  
Exogenous Substrates

Numerous studies have dealt with the cytochemical localization of cytochrome oxidase via cytochrome c. More recent studies have dealt with indicating initial foci of this reaction by altering incubation pH (1) or postosmication procedure (2,3). The following study is an attempt to locate such foci by altering membrane permeability. It is thought that such alterations within the limits of maintaining morphological integrity of the membranes will ease the entry of exogenous substrates resulting in a much quicker oxidation and subsequently a more precise definition of the oxidative reaction.The diaminobenzidine (DAB) method of Seligman et al. (4) was used. Minced pieces of rat liver were incubated for 1 hr following toluene treatment (5,6). Experimental variations consisted of incubating fixed or unfixed tissues treated with toluene and unfixed tissues treated with toluene and subsequently fixed.

Authentic Culinary Tourism Experiences: The Perspectives of Locals

2020 ◽  
Author(s):  
Susan C. Graham
Keyword(s):  
Prince Edward Island ◽  
Precise Definition ◽  
Scholarly Work ◽  
Culinary Tourism ◽  
Work Related ◽  
Napa Valley ◽  
Definition Of

Culinary experiences have long been an important aspect of tourism. For many destinations, culinary offerings have become ubiquitous with the place – pasta in Italy, wine in the Loire- or Napa Valley, or curry in India. As tourists increasingly seek out authentic touristic experiences, including culinary experiences, the question arises regarding what constitutes an authentic culinary experience in a place. While authentic and authenticity are terms widely used in the tourism literature, a precise definition of what those terms mean and a method for identifying that which is authentic remains elusive. Research regarding authenticity in tourism suggests that locals occupy a ‘place of privilege’ with respect to determining the authenticity of a touristic experience because of their connection to and context in relation to the place. This paper examines the perspectives of Prince Edward Island (PEI) residents with respect to what constitutes an authentic culinary touristic experience in which visitors to Canada’s smallest province can partake and that provide those visitors with a glimpse of what life in PEI is or was really like, and provides a voice for an underrepresented group in the authenticity discourse. Results show that authentic culinary experiences transcend food, and encompass people, places, and experiences in ways that enrich touristic endeavours, and that locals understand and interpret authenticity in ways that both conform to and differ from existing scholarly work related to tourism authenticity, and span objective, existential, and constructive authenticity.

Two Theories of Effective Government

2017 ◽  
Author(s):  
Johannes Lindvall
Keyword(s):  
Political Science ◽  
Political Institutions ◽  
Public Policies ◽  
Methodological Approach ◽  
Decision Makers ◽  
Precise Definition ◽  
Policy Changes ◽  
Political Decision ◽  
Definition Of ◽  
The Relationship

This chapter introduces the problem of “reform capacity” (the ability of political decision-makers to adopt and implement policy changes that benefit society as a whole, by adjusting public policies to changing economic, social, and political circumstances). The chapter also reviews the long-standing discussion in political science about the relationship between political institutions and effective government. Furthermore, the chapter explains why the possibility of compensation matters greatly for the politics of reform; provides a precise definition of the concept of reform capacity; describes the book's general approach to this problem; and discusses the ethics of compensating losers from reform; and presents the book's methodological approach.

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

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