Set Theory

A Combinatorial Method for a Context Comparison

by Jiří Milička

Published in Issues in Quantitative Linguistics 2 2011

When comparing the use of two word types within one text, we can do it by comparing the contexts in which they occur.... more When comparing the use of two word types within one text, we can do it by comparing the contexts in which they occur. We pick all the tokens that occur e.g. immediatelly to the right of the word A and immediatelly to the right of the word B, thus getting two multiple subsets of text. This paper offers a method for comparing such subsets (and its use is not limited only to the field of linguistics). The method is based on comparing the cardinality of the intersection of the two multiple subsets and a model which characterizes the average cardinality of all possible subsets of a given length from the given text. The model is derived algebraically.

A Theory of Hyperfinite Sets

by Petr Andreev

Co-authored with E.I. Gordon

We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS— which is based on the idea of the existence... more We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS— which is based on the idea of the existence of proper subclasses of large finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS, and present some applications.

"Advice to the Humanities scholar: Do some math!"

by William J. Urban

presentation delivered April 8, 2011

Variables, generality and existence

by henry laycock

DRAFT CONFERENCE VERSION - ALMOST IDENTICAL WITH PUBLISHED VERSION

In that semantic tradition of which Frege and Russell are among the most distinguished members, the project of... more In that semantic tradition of which Frege and Russell are among the most distinguished members, the project of formalizing natural-language sentences is not simply a matter of developing smooth and effective techniques for the representation of reasoning. Over and above the representation of valid inference as valid, and invalid inference as invalid, there is a further objective. Logic in this tradition is what Frege himself famously calls a concept-script, the import of the notion being chiefly that in natural languages, as Frege emphasizes, ‘the connection of words corresponds only partially to the structure of concepts’, thereby compelling the logician to ‘conduct an ongoing struggle against language and grammar, insofar as they fail to give clear expression to the logical’. In the more recent past, a kindred overall approach is forcefully expounded in the work of Quine, who writes, albeit with a positivistic slant, that

the simplification and clarification of logical theory to which a canonical logical notation contributes is not only algorithmic, it is also conceptual ... each elimination of obscure constructions or notions that we manage to achieve, by paraphrase into more lucid elements, is a clarification of the conceptual scheme of science.

The approach is one with which I find myself in general sympathy; indeed the contrast between clear and less-than-clear ‘expressions of the logical’ is vital to the thesis of this work. Though it has not always received the understanding and respect which it deserves, the ideal of a logically transparent language represents, in my estimation, no merely interesting episode in the history of ideas. It embodies, rather, a permanently valid insight, an enduringly valuable ideal for any analytical conception of philosophy.

Words Without Objects (BOOK)

by henry laycock

Clarendon Press, Oxford (this is not the complete book, unfortunately)

CLICK ON THE 'DOWNLOAD' - NOT THE 'QUICK VIEW' [ERROR!]

The book seeks to resolve the so-called ‘problem of mass nouns’ — a problem which cannot be resolved on the basis of a... more The book seeks to resolve the so-called ‘problem of mass nouns’ — a problem which cannot be resolved on the basis of a conventional system of logic. It is not, for instance, possible to explicate assertions of the existence of air, oil, or water through the use of quantifiers and variables which take objectual values. The difficulty is attributable to the semantically distinctive status of non-count nouns — nouns which, although not plural, are nonetheless akin to plural nouns in being semantically non-singular. Such are the semantics of a non-singular noun, that there can be no such single thing or object as the thing of which the noun is true. However, standard approaches to understanding non-singular nouns tend to be reductive, construing them as singular expressions — expressions which, in the case of non-count nouns, are true of ‘parcels’ or ‘quantities’ of stuff, and in the case of plural nouns, are true of ‘plural entities’ or ‘sets’. It is argued that both approaches are equally misguided, that there are no distinctive objects in the extensions of non-singular nouns. With plural nouns, their extensions are identical with those of the corresponding singular expressions. With non-count nouns, because they are not plural, there can be no corresponding singular expressions. In consequence, there are no objects in the extensions of non-count nouns at all. In short, there are no such things as instances of stuff: the world of space and time contains not merely large numbers of discrete concrete things or individuals of diverse kinds, but also large amounts of sheer undifferentiated concrete stuff. Metaphysically, non-singular reference in general is an arbitrary modality of reference, ungrounded in the realities to which it is non-ideally or intransparently correlated.

Any sum of parts which are water is water

by henry laycock

HUMANA.MENTE
International Journal of Philosophical Studies founded in Florence in 2007. Official journal of the Italian Philosophical Society

Issue 19 - December 2011
COMPOSITION, COUNTERFACTUALS AND CAUSATION
The idea behind this issue is to offer a representation of the most recent theories and position which are emerging in the debate and take David Lewis as their main theoretical source, critical target, or point of departure

ABSTRACT. Mereological entities often seem to violate ‘ordinary’ ideas of what a concrete object can be like, behaving... more ABSTRACT. Mereological entities often seem to violate ‘ordinary’ ideas of what a concrete object can be like, behaving more like sets than like Aristotelian substances. However, the mereological notions of ‘part’, ‘composition’, and ‘sum’ or ‘fusion’ appear to find concrete realisation in the actual semantics of mass nouns. Quine notes that ‘any sum of parts which are water is water’; and the wine from a single barrel can be bottled and distributed around the globe without affecting its identity. Is there here, as some have claimed, a ‘natural’ or ‘innocent’ form of mereology? The claim rests on the assumption that what a mass noun such as ‘wine’ denotes – the wine from a single barrel , for example – is indeed a unit of a special type, the sum or fusion of its many ‘parts’. The assumption is, however, open to question on semantic grounds.

Surreal Canonical Linear Orders

by Tyler Neylon

Georg Cantor showed that any countable linear order (aka a total order) can be embedded in the rationals.  In... more Georg Cantor showed that any countable linear order (aka a total order) can be embedded in the rationals.  In this paper we present a natural extension of this idea, showing that, for any cardinality, a certain subset of the surreals can be the receiver of an order embedding for any linear order of that size.  In particular we define a linear order S_d such that, for any linear order X of cardinality at most \aleph_d, X can be embedded in S_d.  Under the generalized continuum hypothesis, S_d also has cardinality \aleph_d.

Philosophical foundations of set theories of Georg Cantor and Petr Vopěnka.

by Kirill Gabrusenko

Published in 'Tomsk State University Bulletin', № 339 October 2010
Russian title "Философские основания теорий множеств Георга Кантора и Петра Вопенка"

Considering infinite as actually infinite was denied in philosophy and science from the antiquity till 1870s. It... more Considering infinite as actually infinite was denied in philosophy and science from the antiquity till 1870s. It sprang up as a result of Aristotelian analysis of Classical Greek notion “απειρον”, the sense of which was “immensity” or “limitlessness”, and was formalized in the formula “Infinitum actu non datur”. Georg Cantor introduced the concept of set in the middle of the 19th century, which represented assemblies of things as an individual object, particularly infinite assemblies. Cantor’s set theory assumes that all its objects are formed at all; i.e., actually given. A Slovak mathematician, Petr Vopenka, proposed a crucially new sense of “infinity” – natural infinity – that is the result of considering of a sufficiently large set by a finite observer. This considering is characterised by increasing of unsharpness closer to horizon – the border, which constraints the view intended afield or inward if it has no clear obstruction. Unsharpness is not a disadvantage and allows to abstract wholeness from specialties, or to consider specialties, rejecting their integrality. Sharpness is a subcase of unsharpness. Horizon is not fixed and can move while observer is approaching. G. Cantor and P. Vopenka considered infinity from different points of view. So we need to answer a question: what is the cause of this difference? We can discover parts of ontological basis of G. Cantor’s ideas in his works and authentically reconstruct it. He affirmed that mathematical objects have two types of reality – intrasubject (immanent) and transsubject (transient). Their connection is apodictical, and the immanent reality is primary so mathematics has to take into account only this type of reality. The foregoing allows us to determine the philosophical position of G. Cantor as a position of classical Platonism. In his works, P. Vopenka mentions about phenomenological rebuilding of the set theory and mathematics in whole. The main method is in the displacement of the investigators’ point of view to the point of view of a finite observer, as contrary to God’s point of view in Cantor’s set theory. There is very frequent usage of words “phenomenon” and “horizon” in contexts and senses specific to phenomenology in Vopenka’s texts. So we can conclude that philosophical position of P. Vopenka is very close to the Husserlian phenomenology. We come to the conclusion that the philosophical basis of Cantor’s set theory is Platonism, whereas Vopenka’s alternative set theory is founded on the Husserlian phenomenology; and this determines differences in the sense of infinity of the noted authors.

(2012c) Towards a Geometrical Syntax: a Formalization of Radical Minimalism

by Diego Gabriel Krivochen

Schroder-Bernstein Theorem

by Siddharth Vishwanath

The logic of sheaves, sheaf forcing and the independence of the Continuum Hypothesis

by J. Benavides

An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on... more An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on this logic. Using these tools, it is presented an alternative proof of the independence of the Continuum Hypothesis; which simplifies and unifies the classical boolean and intuitionistic approaches, avoiding the difficulties linked to the categorical machinery of the topoi based approach.

Object

by henry laycock

I have converted my own copy of this 2010 piece, which appears in the Stanford Encyclopedia of Philosophy, from WP into PDF for this document.
A LIVE-LINKS COMPLETE BIBLIOGRAPHY FOLLOWS
http://philpapers.org/sep/object/

ABSTRACT. The Frege / Russell account of the object-concept is here called into question. The most general category or... more ABSTRACT. The Frege / Russell account of the object-concept is here called into question. The most general category or concept of an object is a formal one -- a logico-semantic category which is not (as is commonly supposed) exhaustive of what may be thought or said to be. Bona fide objects, whether abstract or concrete, must be countable - 'no entity without identity' (and hence without distinctness). But stuff or matter is not countable and cannot be understood in terms of objects. The issue is significant, if only because the predicate calculus rests upon the object-concept: non-count nouns have no place within the notation

Proper forcing remastered

by Giorgio Venturi

Co-authored with Professor Boban Velickovic. Presented at the Appalachian set theory workshop on October 15th, 2011, at University of Illinois at Chicago, by Professor Velickovic. Avalaible online on the webpage of the workshop: http://www.math.cmu.edu/~eschimme/Appalachian/Index.html

We present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss... more We present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: a variation of the Friedman-Mitchell poset for adding a club with finite conditions, the consistency of the existence of an \omega_2 increasing chain in (\omega_1^{\omega_1}, <_{fin} ), originally proved by Koszmider, and the existence of a thin very tall superatomic Boolean algebra, originally proved by Baumgartner-Shelah. We expect that the present method will have many more applications.

Hilbert, completeness and geometry

by Giorgio Venturi

This paper is an extended verion of a talk given in Mialano on 21st June 2011, for the workshop "Due giornate di studio sulla filosofia della matematica", organized by the SELP, and sponsored by the AILA.

This paper aims to show how the mathematical content of Hilbert’s Axiom of Completeness consists in an attempt to... more This paper aims to show how the mathematical content of Hilbert’s Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how Hilbert’s conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert’s foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert’s position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert’s for the completeness of set theor

Leaving Cantor’s Paradise through Paul Cohen’s Golden Door

by J. Benavides

All different formulations of Quantum Mechanics show that Quantum Reality do not obey some laws of classical logic. So... more All different formulations of Quantum Mechanics show that Quantum Reality do not obey some laws of classical logic. So far, this has been seen more as a curiosity or as a deficiency in our models. However, Quantum Computation results show that these non-classical logic features are not just a deficiency in our models, but the true essence of Quantum Reality. Once we accept this, we should start using non classical logics, not just to measure, but also to understand Quantum Reality. The first consequence of this, it is that widespread notions like those of Planck Scale limit, the Measurement problem and the Discrete vs Continuum duality, become non-sense concepts in a non-classical logic conception of reality. Until now it seems we have unconsciously ignored that properties of the structures we study depend on the principles of logic we employ in studying them. Fortunately, when we use this non classical logics to understand reality, their semantics open new shocking perspectives to relate General Relativity and Quantum Mechanics.

Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics.

by J. Benavides

Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of Quantum Variable Sets is constructed, which... more Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of Quantum Variable Sets is constructed, which generalizes and simplifies the analogous construction developed by Takeuti on boolean valued models of set theory. Over this model, two alternative proofs of Takeuti’s correspondence, between self adjoint operators and the real numbers of the model, are given. This approach results to be more constructive, showing a direct relation with the Gelfand representation theorem, and revealing also the importance of these results with respect to the interpretation of Quantum Mechanics in close connection with the Deutsch-Everett multiversal interpretation of quantum theory. Finally, it is shown how in this context the notion of genericity and the corresponding generic model theorem can help to explain the emergence of classicality in Quantum Mechanics also in close connection with the Deutsch-Everett perspective.

Phenomenological Sociography and Time Travel

by Peter Theodoropoulos

Related to questions of Time, structure and aspectual dimensions of relational time. Related theories that I am working on to discuss this are syntactogenerics and syntactogenesis, Syntacto generics and Semanto Generics, Syntactogenesis and Semantogenesis relations, and the nature of time as emminative, immanent and transcendent meaning both virtual and ontological. Please offer your input if you so wish.

What is time? How is time non linear? What are the relational or assemblages of subjective structure, how can we... more What is time? How is time non linear? What are the relational or assemblages of subjective structure, how can we create a taxonomy of types of relational time? By what features does time seem linear, and what is the nature of conditioning, or ritual space that instantiates regularities IN the mind? How is this relation both physical and immaterial? IS there such thing as the immaterial? Can time be said to be Immaterial and materializable? Is time instantiative or phasic, normative and bound to rules or flexible? How do we understand distinctions in subjective and objective time, how can we quantify and compare and contrast variables of each? Is there such thing as a phase shift in which ontological bias may be met with nomological or noumenal objectivity, if so what are the expectations, means of discernment and systems available to explore critique, modify and experiment with time relations? What, finally, are some means to codify and structure simulations of these relations utilizing electromagnetic fields that may interface with the Neural correlates of Consciousness study to explore results? What should our ethical considerations with regards to this sort of study be? How was your day?

The concept of axiom in Hilbert's thought

by Giorgio Venturi

New version of this paper (2012)

We discuss the concept of axiom in Hilbert's thought, finding two different concepts: the first one from the period of... more We discuss the concept of axiom in Hilbert's thought, finding two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the developement of Hilbert's proof theory. Both conception are linked to two different notion of intuition. Finally we show that, although Godel's theorems ends Hilbert's program, the first concept of axiom is still tenable and indeed very similar to how axioms are considered in modern set theory.

Context-Awareness: Control Over Disclosure and Privacy In a Social Environment

by Varun Singh

Modern mobile phones in the near future will adapt theirapplications, services, and look-and-feel based on theirusage... more Modern mobile phones in the near future will adapt theirapplications, services, and look-and-feel based on theirusage and environment. This ability of the devices toadapt based on user’s context provides new challengesin areas of context sharing, context management, and inthe associated area of privacy. Inadequate relationshipmodels make control over disclosure impossible, whichmanifests itself in either sharing too much information- compromising ones privacy, or sharing too little - con-straining the use of context in a shared environment. Inthis paper, we define a new pseudo-hierarchical tag basedrelationship model to overcome the constraints of controlover disclosure. Furthermore, we discuss some of the pit-falls of sharing context on user’s privacy.

Formalistics of Generalization

by Victor Porton