Models Of Set Theory Research Articles

Thoughts on Klumpenhouwer Networks and Mathematical Models

[1] In the past seventeen or so years since the publication of pioneering work by David Lewin (Lewin 1990) and Henry Klumpenhouwer (Klumpenhouwer 1991), the theory of Klumpenhouwer networks (hereafter referred to as K-nets) has established itself as an important topic in music theory, but it seems that it is still in its youth. Michael Buchler's recent critical examination of K-net theory (Buchler 2007) is indicative of both its standing in the discipline of music theory and the need for ongoing evaluation of its premises. In this essay I respond to one of the four central issues in Buchler's thoughtful article: K-nets and their association with dual transformation. The juxtaposition of K-nets and dual transformation leads to a broader engagement with some fundamentals and phenomenological and metaphorical considerations of mathematical models in music theory. Remodeling K-nets as dual transformations, as Buchler advocates (Buchler 2007, 7), indeed elucidates transformational voice leading, but dual transformations also dispense with the network organization of K-nets, and thus evince fundamentally different structures than K-net isographies.K-nets, set theory, and graph theory[2] K-nets are typically employed to model sets of pitch classes through internal transformational pairings of pitch classes by the canonical operations of transposition (Tn) and inversion (In).(1) While a relatively large number of K-net representations can be constructed for any set, the number of well-formed K-net representations that can be constructed is considerably smaller.(2) Selecting one K-net representation from the (well-formed) alternatives is an interpretive act; when employed in analysis, the selection is based on musical context, usually in consideration of the relationship of one K-net to its successor(s) or predecessor(s). We turn now to theoretical aspects of K-nets.[3] K-nets call upon the mathematical models of set theory and graph theory, models of different conceptual origins, as Jeffrey Johnson describes:Graph-theoretical constructs are not dependent on Cartesian coordinates, but only on the abstract relationship between dots and connecting lines. Graph theory had its origin in solutions to puzzles and games rather than more intense desires to describe motion, quantify measurement, and, bring order to scientific inquiry that impelled other branches of mathematics (Johnson 1997, 12).A graph models relations between pairs of elements, and consists of two sets: a set of vertices and a set of edges. The vertex set in a K-net consists of pitch classes, while the edge set consists of ordered pairs of vertices, each representing the transformational relation between a pair of pitch classes. K-nets are directed graphs (also called digraphs), because each edge has a specific direction or orientation.(3) (The double-headed arrow representing the inversion operation on pitch classes, identifying the operation as an involution, stands for edges with dual orientations between a single pair of vertices.) Strictly speaking, a graph is comprised simply of dots or points without content (the vertices) and connecting lines (the edges); it is an abstract construction independent of the elements that motivate and occupy the vertices. A network is a graph whose vertices have assigned values.(4) A network is thus, in a sense, an applied graph.[4] K-net theory adopts the terms node for vertex and arrow for edge, following David Lewin's definition of node/arrow system,(5) which corresponds to digraph, as described above. The distinction between graph and network likewise follows Lewin's distinction between transformation graph and transformation network (Lewin 1987), and is nicely summarized by Julian Hook: "The graph is...the more abstract of the two structures; a network adds information to a graph, bringing it to life in a real musical setting via the node labels" (Hook 2007, 7). …

Data analysis based on discernibility and indiscernibility

Rough set theory models similarities and differences of objects based on the notions of indiscernibility and discernibility. With respect to any subset of attributes, one can define two pairs of dual relations: the strong indiscernibility and weak discernibility relations, and the weak indiscernibility and strong discernibility relations. The similarities of objects are examined by the indiscernibility relations, and the differences by the discernibility relations, respectively. Alternatively, one can construct an indiscernibility matrix to represent the family of strong indiscernibility or weak discernibility relations. One also can construct a discernibility matrix to represent the family of strong discernibility or weak indiscernibility relations. The consideration of the matrix-counterpart of relations, and the relation-counterpart of matrices, brings more insights into rough set theory. Based on indiscernibility and discernibility, three different types of reducts can be constructed, keeping the indiscernibility, discernibility, and indiscernibility-and-discernibility relations, respectively. Although the indiscernibility reducts have been intensively studied in the literature, the other two types of reducts are relatively new and require more attention. The existing methods for constructing the indiscernibility reducts also can be applied to construct the other two types of reducts. An empirical experiment for letter recognition is reported for demonstrating the usefulness of the discussed relations and reducts.

Transfer principle in quantum set theory

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

Formula-inaccessible cardinals and a characterization of all natural models of Zermelo-Fraenkel set theory

E. Zermelo (1930) and J. C. Sheperdson (1952) proved that a cumulative set is a standard model of von Neumann-Bernays-Gödel set theory if and only if for some inaccessible cardinal number . The problem of a canonical form for all natural models of ZF theory turned out to be more complicated. Since the notion of a model of ZF theory cannot be defined by a finite set of formulae, we introduce a new notion of (strongly) formula-inaccessible cardinal number using a schema of formulae and its relativization on the set , and prove a formula-analogue of the Zermelo-Sheperdson theorem giving a canonical form for all natural models of ZF theory.

Feinstrukturtheorie und Innere Modelle

The workshop Fine Structure Theory and Inner Models , organised by Ronald Jensen (Berlin), Menachem Magidor (Jerusalem) and Ralf Schindler (Münster) was held April 30th - May 6th, 2006. It was attended by most of the leading researchers in the area. Fine structure theory was initiated by the first organizer, R. Jensen, in the 70ies. It has been exploited ever since for producing a series of spectacular results in set theory. One such is Jensen's Covering Lemma for Gödel's constructible hierarchy, L , which says, put informally, that the universe V of all sets either resembles L to a large extent or else is very different from L . Later on, various people (most of which were participants of this workshop) proved versions of the Covering Lemma for larger inner models. The main goal of “Fine structure theory and inner model theory” is to construct fine structural inner models of set theory, i.e., definable transitive proper class-sized models of the standard axiom system {\sf ZFC} of set theory, which reflect the large cardinal structure of the universe, but at the same time admit a fine structure that makes it possible to analyze them in great detail and prove various combinatorial properties in them. Other applications of such inner models are consistency strength investigations, and they can be used as a tool for proving implications which don't mention inner models at all, but for which no “direct” proof is in sight. A large cardinal concept is one such that \mathsf{ZFC} cannot prove that there is an incarnation of it. Our area is the key tool for uncovering the large cardinal structure which is implicit in many (not only set theoretic) hypotheses. In fact, often a given statement which doesn't mention large cardinals at all and a statement about the existence of models with large cardinals turn out to be two sides of the same coin. Breathtaking results by Martin, Steel, Woodin, and others in the 80ies and 90ies have shown that the large cardinal concept of a Woodin cardinal is a crucial one here. The main issues of this area are the following. The conference had 16 participants. 13 talks were given, and they covered both pure and applied parts of inner model theory. Because this was a gathering of true specialists, there was no need for overview-style talks and we could concentrate on issues which are at the focus of current research. The talks came with intriguing results, but also with promising new perspectives for upcoming research. We had very lively discussions. It was a fruitful workshop, and many of the ideas which were exchanged are sure to be further elaborated in the near future.

Addition and multiplication of sets

AbstractOrdinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non‐empty elements of any model of set theory can be partitioned into infinitely many submodels, each isomorphic to the original model. Also any model of set theory has an isomorphic extension in which the empty set of the original model is non‐empty. Among other results, the relations between the arithmetical operations and the transitive closure and the adductive hierarchy are elucidated. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Using a model of cellular automata and classification methods for prediction of time series with memory

The paper deals with the problem of prediction of time series with memory for which classical prediction methods are frequently inadequate. A method is proposed that is based on a model of cellular automata, classification methods, and fuzzy set theory. The accuracy of models based on this method is estimated.

Ordinal computations

The notion of ordinal computability is defined by generalising standard Turing computability on tapes of length . In this paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo–Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.

CAUTIOUS RELIABILITY ANALYSIS OF MULTI-STATE AND CONTINUUM-STATE SYSTEMS BASED ON THE IMPRECISE DIRICHLET MODEL

Cautious reliability estimates of multi-state and continuum-state systems are studied in the paper under condition that initial data about reliability of components are given in the form of interval-valued observations, measurements or expert judgments. The interval-valued information is processed by means of a set of the imprecise Dirichlet model which can be regarded as a set of Dirichlet distributions. The developed model of reliability provides cautious reliability measures when the number of observations or measurements is rather small. It can be viewed as an extension of models based on random set theory and robust statistical models. A numerical example illustrates the proposed model and an algorithm for computing the system reliability.

On Evans's Vague Object from Set Theoretic Viewpoint

Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects.

Skolem and pessimism about proof in mathematics

Attitudes towards formalization and proof have gone through large swings during the last 150 years. We sketch the development from Frege's first formalization, to the debates over intuitionism and other schools, through Hilbert's program and the decisive blow of the Gödel Incompleteness Theorem. A critical role is played by the Skolem-Lowenheim Theorem, which showed that no first-order axiom system can characterize a unique infinite model. Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescient paper, he even sketches the possibility of interesting new models for set theory itself, something later realized by the method of forcing. This is in contrast to Hilbert's belief that mathematics could resolve all its questions. We discuss the role of new axioms for set theory, questions in set theory itself, and their relevance for number theory. We then look in detail at what the methods of the predicate calculus, i.e. mathematical reasoning, really entail. The conclusion is that there is no reasonable basis for Hilbert's assumption. The vast majority of questions even in elementary number theory, of reasonable complexity, are beyond the reach of any such reasoning. Of course this cannot be proved and we present only plausibility arguments. The great success of mathematics comes from considering 'natural problems', those which are related to previous work and offer a good chance of being solved. The great glories of human reasoning, beginning with the Greek discovery of geometry, are in no way diminished by this pessimistic view. We end by wishing good health to present-day mathematics and the mathematics of many centuries to come.

Putnam and Constructibility

I discuss and try to evaluate the argument about constructible sets made by Putnam in ‘ ”Models and Reality”, and some of the counterarguments directed against it in the literature. I shall conclude that Putnam’s argument, while correct in substance, nevertheless has no direct bearing on the philosophical question of unintended models of set theory.

CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max , which satisfy [Formula: see text] and prove: Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X]2→2with[Formula: see text], [Formula: see text] (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of (2ω)2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which (1) ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; (2) ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3).

Adaptive clustering algorithm for recycling cell formation: An Application of fuzzy ART neural networks

The recycling cell formation problem means that disposal products are classified into recycling part families using group technology in their end-of-life phase. Disposal products have the uncertainties of product status by usage influences during product use phase, and recycling cells are formed design, process and usage attributes. In order to deal with the uncertainties, fuzzy set theory and fuzzy logic-based neural network model are applied to recycling cell formation problem for disposal products. Fuzzy C-mean algorithm and a heuristic approach based on fuzzy ART neural network is suggested. Especially, the modified Fuzzy ART neural network is shown that it has a good clustering results and gives an extension for systematically generating alternative solutions in the recycling cell formation problem. Disposal refrigerators are shown as examples.

Zermelo and Set Theory

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

Compact spaces and the pseudoradial property, II

There is a model of set theory in which all compact spaces of weight at most ω 2 are pseudoradial. We show that 2 ω 2 can be pseudoradial. It is easily seen that if 2 ω 2 is pseudoradial, then all compact spaces of weight at most ω 2 are also pseudoradial.

Models of set theory with definable ordinals

A DO model (here also referred to a Paris model) is a model Open image in new window of set theory all of whose ordinals are first order definable in Open image in new window. Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the following:

Leibnizian models of set theory

Abstract.A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF. T has a Leibnizian model if and only if T proves LM. Here we prove:Theorem A. Every complete theory T extending ZF + LM has nonisomorphic countable Leibnizian models.Theorem B. If κ is a prescribed definable infinite cardinal ofa complete theory T extending ZF + V = OD, then there are nonisomorphic Leibnizian models of T of power ℵ1such thatis ℵ1-like.Theorem C. Every complete theory T extendingZF + V = ODhas nonisomorphic ℵ1-like Leibnizian models.

Computational adequacy for recursive types in models of intuitionistic set theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.

Formula balancing and continuously valuated models

Uniform spaces can be Cauchy-completed; and if the base space was a first-order structure, this structure can be naturally extended to the completion. While common in algebra, this construction has been more recently used to produce new models of special set theories. We investigate here a natural way to twist the semantics of any structure according to a uniformity on its universe. We use it to relate the (classical first-order) theories of structures and dense substructures and apply it to the case of Cauchy-completions.