An ordinal analysis of stability

This paper is the first in a series of three which culminates in an ordinal analysis of ¹₂-comprehension. On the set-theoretic side ¹₂-comprehension corresponds to Kripke-Platek set theory, KP, plus ₁-separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all >, is -stable, i.e. L is a ₁-elementary substructure of L. The objective of this paper is to give an ordinal analysis of a scenario of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of ¹₂-comprehension is greatly facilitated by explicating certain simpler cases first.This paper introduces an ordinal representation system based on -indescribable cardinals which is then employed for determining an upper bound for the proof–theoretic strength of the theory KPi+ is +-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.The results in this paper were obtained in 1995 when the author was a Heisenberg Fellow of the German Science Foundation, Deutsche Forschungsgemeinschaft.     

Proof theory and ordinal analysis

In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis.Dedicated to K. Schütte on the occasion of his 80th birthdayWork partly supported by a grant of the Volkswagenstiftung     

S-PLUS code for ordinal correspondence analysis

Summary  Correspondence analysis is a popular graphical tool used to analyse contingency tables. In the past, it has commonly been performed by applying a singular value decomposition to a transformation of the data in the contingency table. A recent advance in its theory is to perform a bivariate moment decomposition instead. This approach is especially useful for the detection of linear and non-linear associations between ordinal variables; a feature not readily available using singular value decomposition. This paper outlines S-PLUS code that will perform correspondence analysis using bivariate moment decomposition. It also includes a simple plotting function that will enable the graphical interpretation of the different levels of association.     

An ordinal analysis for theories of self-referential truth

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of ${\Sigma^1_1}The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of S¹₁{\Sigma^1_1} dependent choice.     

列联资料的有向聚类分析及其应用

本文针对单向有序列联资料 ,提出了有序因素的秩效应概念。在秩效应排序的基础上 ,构造了平均秩效应原则 ,并对因素各水平进行有向聚类分析。利用该方法对大学生隐性教育调查资料进行了剖析     

Adequateness and interpretability of objective functions in ordinal data analysis

Objective functions that are applied in ordinal data analysis must be adequate, i.e. carefully adapted to the structure of the observed data. In addition, any analysis of data that is based upon objective functions must lead to interpretable results. After a general characterization of adequate objective functions in ordinal data analysis, therefore, the particular problems of constructing adequate and interpretable dissimilarity coefficients and correlation coefficients in ordinal data analysis, stress measures (stress functions) in non-metric scaling and generalized stress measures or correlation coefficients in any theory of rank estimation will be discussed.     

Iterating ordinal definability

The paper presents a uniform way of obtaining by forcing descending sequences of the iterations of HOD and describes their structure. Among other things the following results are proved:Theorem. Any model M of ZFC can be obtained by the transfinite iteration of HOD in some generic extension N of M, i.e. M=(HOD^On)^N.Theorem (ZFC). For any ordinal α there is a complete Boolean algebra B with a decreasing sequence of derivatives of length α.Moreover some other results about ordinal definability are reproved using the introduced method. This method is a refinement of the classical method of coding introduced by McAloon in [11].     

A statistical model for the analysis of ordinal level dependent variables

This paper develops a model, with assumptions similar to those of the linear model, for use when the observed dependent variable is ordinal. This model is an extension of the dichotomous probit model, and assumes that the ordinal nature of the observed dependent variable is due to methodological limitations in collecting the data, which force the researcher to lump together and identify various portions of an (otherwise) interval level variable. The model assumes a linear eflect of each independent variable as well as a series of break points between categories for the dependent variable. Maximum likelihood estimators are found for these parameters, along with their asymptotic sampling distributions, and an analogue of R ² (the coefficient of determination in regression analysis) is defined to measure goodness of fit. The use of the model is illustrated with an analysis of Congressional voting on the 1965 Medicare Bill.     

Aggregation of ordinal information

Our interest is with the fusion of information which has an ordinal structure. Information fusion in this environment requires the availability of ordinal aggregation operations. Basic ordinal operations are first introduced. Next we investigate conjunctive and disjunction aggregations of ordinal information. The idea of a pseudo-log in the ordinal environment is presented. We discuss the introduction of a zero like point on an ordinal scale along with the related ideas of bipolarity (positive and negative values) and uni-norm aggregation operators. We introduce mean like aggregation operators as well weighted averages on a ordinal scale. The problem of selecting between ordinal models is considered.     

Coordinatization of ordinal structures

Dependencies between attributes in ordinal data contexts are algebraically described. Suitable conditions are analysed which allow coordinatizations of ordinal contexts (ordinal structures) by ordered n-quasigroups, ordered abelian groups, and ordered fields. The presented development offers a new approach to conjoint measurement.     

Algorithmic decision rule using ordinal criteria importance coefficients with a first ordinal metric scale

An exact computational method is proposed for the preferability comparison of various solution variants in multicriteria problems with importance-ordered criteria using a common scale along which the growth of preferences slows down.     

Classification trees for ordinal variables

We introduce new criteria to obtain classification trees for ordinal response variables. At this aim, Breiman et al. (Classification and regression trees. Wadsworth, Belmont, 1984), extended their twoing criterion to the ordinal case. Following CART procedure, we extend the well known Gini–Simpson criterion to the ordinal case. Referring to the exclusivity preference property (introduced by Taylor and Silverman in Stat Comput 3:147–161, 1993, for the nominal case), suitably modified for the ordinal case, a second criterion is introduced. The hereby proposed methods are compared with the ordered twoing criterion via simulations.     

Rectangular products with ordinal factors

Let A and B be subspaces of an ordinal. It is proved that the product A×B is countably paracompact if and only if it is rectangular. Before this main result, we discuss several covering properties of products with one ordinal factor. In particular, for every paracompact space X, it is proved that the product X×A is paracompact if so is A.     

Shortest paths with ordinal weights

We investigate the single-source-single-destination “shortest” path problem in directed, acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated path vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated path vectors from source to sink.     

Locating a waste treatment facility by using stochastic multicriteria acceptability analysis with ordinal criteria

In this paper we describe a real-life application of an ordinal multicriteria method in the context of choosing a location for a waste treatment facility near Lappeenranta in South-Eastern Finland. The associated environmental impact assessment (EIA) procedure is briefly described. The application was characterized by two interesting properties: no preference information was available, and only ordinal measurements for the criteria were available. The large amount of data obtained was then analyzed using the SMAA-O method – Stochastic Multicriteria Acceptability Analysis with Ordinal criteria designed for this problem setting. SMAA-O converts ordinal information into cardinal data by simulating all possible mappings between ordinal and cardinal scales that preserve the given rankings. As with the basic SMAA-method, the decision makers' (DMs) unknown or partly known preferences are at the same time simulated by choosing weights randomly from appropriate distributions. The main results of the analysis are acceptability indices for alternatives describing the variety of preferences that could make each alternative the best choice. Based on these and additional considerations, the DMs made the final choice for the location of the plant.