Models Of Set Theory Research Articles

Test groups for Whitehead groups

We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns out to be equivalent to the question of when the tensor product of two Whitehead groups is Whitehead. We investigate what happens in different models of set theory.

PFA(S)[S]: More Mutually Consistent Topological Consequences of PFA and V = L

Abstract Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable L-spaces or compact S-spaces. The model is one of the form PFA(S)[S], where S is a coherent Souslin tree.

Cofinality and measurability of the first three uncountable cardinals

This paper discusses models of set theory without the Axiom of Choice. We investigate all possible patterns of the cofinality function and the distribution of measurability on the first three uncountable cardinals. The result relies heavily on a strengthening of an unpublished result of Kechris: we prove (under A D \mathsf {AD} ) that there is a cardinal κ \kappa such that the triple ( κ , κ + , κ + + ) (\kappa ,\kappa ^+,\kappa ^{++}) satisfies the strong polarized partition property.

Nonstandard tools of nonsmooth analysis

This is an overview of the basic tools of nonsmooth analysis which are grounded on nonstandard models of set theory. By way of illustration we give a criterion for an infinitesimally optimal path of a general discrete dynamic system.

Accessing the switchboard via set forcing

AbstractWe force a property of cardinals first proved relatively consistent by Sargsyan, that of being supercompact but not \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{HOD}$\end{document}‐supercompact, starting from a model of set theory which does not satisfy \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${GCH}$\end{document} and that contains supercompact cardinals.

Aspects of predicative algebraic set theory III: sheaves

This is the third instalment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a ‘predicative category with small maps’ which axiomatizes the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are sets (in some axiomatic set theory). The main result of the present paper is that such predicative categories with small maps are stable under internal sheaves. We discuss the sheaf models of constructive set theory this leads to, as well as ideas for future work.

Realizability models refuting Ishiharaʼs boundedness principle

Ishiharaʼs boundedness principleBD-N was introduced in Ishihara (1992) [5] and has turned out to be most useful for constructive analysis, see e.g. Ishihara (2001) [6]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from NN to N is sequentially continuous and in which the axiom of choice from NN to N holds. Since the latter is known to be inconsistent with the statement that all functions from NN to N are continuous these models refute BD-N.

Constructive toposes with countable sums as models of constructive set theory

We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive set theory. We show that any constructive topos with countable coproducts provides a model of a standard constructive set theory, CZFExp (that is, the variant of Aczel’s Constructive Zermelo–Fraenkel set theory CZF obtained by weakening Subset Collection to the Exponentiation axiom). The model is constructed as a category of classes, using ideas derived from Joyal and Moerdijk’s programme of algebraic set theory. A curiosity is that our model always validates the axiom V=Vω1 (in an appropriate formulation). It follows that the full Separation schema is always refuted.

An Extension of a Permutative Model of Set Theory

In this paper we present a new axiomatic model of set theory called the Extended Fraenkel Mostowski model. It is dened by replacing an axiom of the Fraenkel- Mostowski model with a consequence of it; the other axioms of the Fraenkel-Mostowski model are left unchanged in the new Extended Fraenkel-Mostowski model. We use the theory of groups to dene the sets both in the Extended Fraenkel- Mostowski and in the Fraenkel-Mostowski approach. Several algebraic properties of the sets in the Fraenkel-Mostowski model remain also valid in the Extended Fraenkel- Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel- Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel- Mostowski model. Mathematics Subject Classication

The Usual Model Construction for NFU Preserves Information

The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo–Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper is that the restriction of the membership relation of the original model of set theory with automorphism to the domain of the Boffa model is first-order definable in the language of NFU. In particular, all information about the extensions in the original model of the urelements of the model of NFU is definable in terms of NFU. A corollary (answering a question of Thomas Forster) is that the urelements in a Boffa model are not homogeneous.

Paradox and Potential Infinity

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

Some problems in automata theory which depend on the models of set theory

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a B\"uchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter B\"uchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A)^-$ has cardinal $2^{\aleph_0}$. (3). There is a model $V_3$ of ZFC in which $L(A)^-$ has cardinal $\aleph_1$ with $\aleph_0<\aleph_1<2^{\aleph_0}$. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape B\"uchi automaton $B$. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter omega-languages and for complements of infinitary rational relations accepted by 2-tape B\"uchi automata. We infer from the proof of the above results that basic decision problems about 1-counter omega-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter omega-language (respectively, infinitary rational relation) is countable is in $\Sigma_3^1 \setminus (\Pi_2^1 \cup \Sigma_2^1)$. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

Generalizing realizability and Heyting models for constructive set theory

This article presents a common generalization of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalization of the Boolean models for classical set theory which are a variant of forcing, while realizability is a decidedly constructive method that has first been developed for number theory by Kleene and was later very fruitfully adapted to constructive set theory. In order to achieve the generalization, a new kind of structure (applicative topologies) is introduced, which contains both elements of formal topology and applicative structures. This approach not only deepens the understanding of class models and leads to more efficiency in proofs about these kinds of models, but also makes it possible to prove new results about the two special cases that were not known before and to construct new models.

Granular Computing Models for Knowledge Uncertainty

PDF HTML阅读 XML下载 导出引用 引用提醒 知识不确定性问题的粒计算模型 DOI: 10.3724/SP.J.1001.2011.03954 作者: 作者单位: 作者简介: 通讯作者: 中图分类号: 基金项目: 国家自然科学基金(61073146); 重庆市自然科学基金(2008BA2017); 重庆市杰出青年科学基金(2008BA2041) Granular Computing Models for Knowledge Uncertainty Author: Affiliation: Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:知识不仅是构成人类认知能力的重要基石,也是智能科学研究的基础问题之一.随着智能科学技术研究的发展,知识的不确定性研究受到人们的普遍关注.知识的不确定性来源于知识本身的不确定性以及受外界(客观世界)影响而导致的不确定性.从粒计算模型的角度分析了模糊集理论模型、粗糙集理论模型、商空间理论模型以及其他扩展粒计算模型中知识的不确定性问题,并对知识不确定性问题的研究工作进行了讨论和总结,对有待研究的重要问题进行了展望. Abstract:Knowledge is not only an important cornerstone that constructs the cognitive ability in human beings, but is also one of the basic issues in intelligence science. With the development of intelligence science and technology, the study of knowledge uncertainty is attracting more and more attention. Knowledge uncertainty includes the inherent uncertainty of knowledge itself and the effect of external influence on knowledge. In the view of granular computing (GrC), the knowledge uncertainties of the fuzzy set theory model, the rough set theory model, the quotient space theory model, and some other related granular computing models are studied in this paper. The state-of-the-art and key issues of knowledge uncertainty are discussed. 参考文献 相似文献 引证文献

An application of testing fuzzy hypotheses: Soil study on the bioavailability of cadmium

Fuzzy set theory models situations in which the uncertainty is due to the non-precise (fuzzy) environment. One such case is testing the hypotheses problem where hypotheses are fuzzy rather than crisp and the data are crisp. Pais and Benton (1997) [1] present a suitable amount of cadmium absorption which is more appropriate for modeling by a fuzzy subset. In this paper, on real-world agricultural data generated by Ivani (2007) [2], the suitableness of the mean absorption cadmium in the plant from the polluted soil is tested at the given significance level. Due to uncertainty in the suitable amount of cadmium absorption, in order to test this amount, we use fuzzy hypotheses testing instead of classical hypotheses testing. The fuzzy p-value approach is used for this test to conclude whether or not the mean absorption of cadmium coincides with the proposed amounts by Pais and Benton [1]. As expected, in fuzzy hypotheses testing, the degree of acceptance or rejection of the null fuzzy hypothesis is computed for each treatment of pollution. The data are from two plants: radish and cress, which are experimented on in soil polluted with CdNO3 salt. The results showed that using classical hypotheses testing may lead to contradictory decisions, and the proposed fuzzy hypotheses testing is a rational substitute for classical hypotheses testing when the environment is in a vague status.

The cardinal inequality α2 < 2α

In ZFC set theory (i.e., Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) any two cardinal numbers are comparable. However, this may not be valid in ZF (i.e., Zermelo-Fraenkel set theory modulo AC). In this paper, we study the strength of the inequality α2 < 2α (in ZF, for every infinite cardinal number α, 2α ≰ α2; see [13]), where α is either the cardinality of special sets (see Definition 1 below) which are expressed as disjoint unions of finite, pairwise equipotent, sets lacking choice functions, or the cardinality of sets in specific permutation models of ZF0 set theory (i.e., ZF without the Axiom of Regularity).

PREDICTING FINANCIAL DISTRESS OF CHINESE LISTED COMPANIES USING ROUGH SET THEORY AND SUPPORT VECTOR MACHINE

Effectively predicting corporate financial distress is an important and challenging issue for companies. The research aims at predicting financial distress using the integrated model of rough set theory (RST) and support vector machine (SVM), in order to find a better early warning method and enhance the prediction accuracy. After several comparative experiments with the dataset of Chinese listed companies, rough set theory is proved to be an effective approach for reducing redundant information. Our results indicate that the SVM performs better than the BPNN when they are used for corporate financial distress prediction.

Fuzzy Set Theory and Grey System Model for Forecasting the Trend of IC Product

Fuzzy Set Theory and Grey System Model for Forecasting the Trend of IC Product

The representation and processing of uncertain problems

Uncertainty is everywhere, and there are many researches on uncertain problems. Soft Computing combined intelligent paradigms as Probabilistic Reasoning, Fuzzy Logic to deal with pervasive imprecision and uncertainty of the real-world problems. In this paper, four main Soft Computing methods, namely, Probability Theory, Fuzzy Set Theory, Rough Set Theory and Cloud Model are introduced briefly, and their representation and measure of uncertainty are discussed respectively.

Substandard models of finite set theory

AbstractA survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set‐theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)