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Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Cited by 8 (7 self)
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
Relativizing Relativity
, 2000
"... this article; nor should they be blamed for any misconception and fallacy of the author ..."
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Cited by 4 (4 self)
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this article; nor should they be blamed for any misconception and fallacy of the author
Selfreference and Logic
 Phi News
, 2002
"... Tarski's schema T plays a central role in each of these formalizations. a In particular, we show that each of the classical paradoxes of selfreference can be reduced to lIf the sentence is true, what it states must be the case. But it states that it itself is not true. Thus, if it is true, it ..."
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Cited by 2 (0 self)
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Tarski's schema T plays a central role in each of these formalizations. a In particular, we show that each of the classical paradoxes of selfreference can be reduced to lIf the sentence is true, what it states must be the case. But it states that it itself is not true. Thus, if it is true, it is not true. On the contrary assumption, if the sentence is not true, then what it states must not be the case and, thus, it is true. Therefore, the sentence is true iff it is not true. 2 Often cases of selfreference will fit into more than one of these categories. aTarski's schema T is the set of all firstorder logical equivalences T(rg TM)  g where g is any sentence and rg is a term denoting g. schema T. This leads us to a discussion of schema T, the problems it gives rise to, and how to circumvent these problems. The first part of the essay does not require any training in mathematical logic. Part I: SelfReference We start out by taking a closer look at paradoxes related to selfr
Quantum recursion theory
, 2009
"... Incompleteness and undecidability theorems have to be revised in view of quantum information and computation theory. qrt.tex 1 As has already been pointed out in Gödel’s centennial paper on the incompleteness af arithmetic [1], the classical undecidability theorems of formal logic [2] and the theory ..."
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Incompleteness and undecidability theorems have to be revised in view of quantum information and computation theory. qrt.tex 1 As has already been pointed out in Gödel’s centennial paper on the incompleteness af arithmetic [1], the classical undecidability theorems of formal logic [2] and the theory of computable functions [4, 5] are based on semantical pardoxes such as the liar [6] or Richard’s paradox. The method of diagonalization, which was first applied by Cantor for a proof of the undenumerability of real numbers [7], has been applied by Turing for a proof of the recursive undecidability of the halting problem [8]. The halting problem is the problem of whether or not an arbitrary algorithm terminates or produces a particular output and terminates. Assume that the halting problem is decidable. Turing [8] proved that this assumption yields a contradiction. To construct the contradiction, consider
Quantum algorithmic information theory
, 2008
"... The agenda of quantum algorithmic information theory, ordered ‘topdown, ’ is the quantum halting amplitude, followed by the quantum algorithmic information content, which in turn requires the theory of quantum computation. The fundamental atoms processed by quantum computation are the quantum bits ..."
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The agenda of quantum algorithmic information theory, ordered ‘topdown, ’ is the quantum halting amplitude, followed by the quantum algorithmic information content, which in turn requires the theory of quantum computation. The fundamental atoms processed by quantum computation are the quantum bits which are dealt with in quantum information theory. The theory of quantum computation will be based upon a model of universal quantum computer whose elementary unit is a twoport interferometer capable of arbitrary U(2) transformations. Basic to all these considerations is quantum theory, in particular Hilbert space quantum mechanics.
THE ONE, THE MANY, AND THE QUANTUM
, 2000
"... The problem of understanding quantum mechanics is in large measure the problem of finding appropriate ways of thinking about the spatial and temporal aspects of the physical world. The standard, substantival, settheoretic conception of space is inconsistent with quantum mechanics, and so is the doc ..."
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The problem of understanding quantum mechanics is in large measure the problem of finding appropriate ways of thinking about the spatial and temporal aspects of the physical world. The standard, substantival, settheoretic conception of space is inconsistent with quantum mechanics, and so is the doctrine of local realism, the principle of local causality, and the mathematical physicist’s golden calf, determinism. The said problem is made intractable by our obtruding onto the physical world a theoretical framework that is more detailed than the physical world. This framework portraits space and time as infinitely and intrinsically differentiated, whereas the physical world is only finitely differentiated spacewise and timewise, namely to the extent that spatiotemporal relations and distinctions are warranted by facts. This has the following consequences: (i) The contingent properties of the physical world, including the times at which they are possessed, are indefinite and extrinsic. (ii) We cannot think of reality as being built “from the bottom up”, out of locally instantiated physical properties. Instead we must conceive of the physical world as being built “from the top down”: By entering into a multitude of spatial relations with itself, “existence itself ” takes on both the aspect of a spatially differentiated world and the aspect of a multiplicity of formless relata, the fundamental particles. At the root of our interpretational difficulties is the “cookie cutter paradigm”, according to which the world’s synchronic multiplicity is founded on the introduction of surfaces that carve up space in the manner of threedimensional cookie cutters. The neurophysiological underpinnings of this insidious notion are discussed. 1 1
Notes prepared by
, 2001
"... In Chapter II of LMCS we looked at the propositional proof system PC and resolution theorem proving. A number of ideas have been developed regarding just how one sets up the basic structure of a proof system. We will look at some of the main ones here. 1.1 Definition of a propositional logic DEFINIT ..."
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In Chapter II of LMCS we looked at the propositional proof system PC and resolution theorem proving. A number of ideas have been developed regarding just how one sets up the basic structure of a proof system. We will look at some of the main ones here. 1.1 Definition of a propositional logic DEFINITION 1 A propositional logic 1 consists of • S, a set of connectives • C, the associated algebra of connectives • X, a set of propositional variables • C, a set of propositional constants • T(X), the set of propositional formulas over X • Axioms • Rules of inference. DEFINITION 2 Given a set of propositional formulas Σ and a propositional formula ϕ we say
Hierarchical Relationships "isa": Distinguishing Belonging, Inclusion and Part/of Relationships.
"... In thesauri, conceptual structures or semantic networks, relationships are too often vague. For instance, in terminology, the relationships between concepts are often reduced to the distinction established by standard (ISO 704, 1987) and (ISO 1087, 1990) between hierarchical relationships (genusspe ..."
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In thesauri, conceptual structures or semantic networks, relationships are too often vague. For instance, in terminology, the relationships between concepts are often reduced to the distinction established by standard (ISO 704, 1987) and (ISO 1087, 1990) between hierarchical relationships (genusspecies relationships and part/whole relationships) and nonhierarchical relationships (“time, space, causal relationships, etc.”). The semantics of relationships are vague because the principal users of these relationships are industrial actors (translators of technical handbooks, terminologists, dataprocessing specialists, etc.). Nevertheless, the consistency of the models built must always be guaranteed... One possible approach to this problem consists in organizing the relationships in a typology based on logical properties. For instance, we typically use only the general relation “Isa”. It is too vague. We assume that general relation “Isa ” is characterized by asymmetry. This asymmetry is specified in: (1) the belonging of one individualizable entity to a distributive class, (2) Inclusion among distributive classes and (3) relation part of (or “composition”). 1.
Set Theory
"... Set Theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. In particular, he proved that there are different infinite cardinalities: the quantity of ..."
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Set Theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. In particular, he proved that there are different infinite cardinalities: the quantity of natural numbers is strictly smaller than the quantity of real numbers. Cantor formalized and studied the notions of ordinal and cardinal numbers. Set theory considers a universe of sets which is ordered by the membership or element relation ∈. All other mathematical objects are coded into this universe and studied within this framework. In this way, set theory is one of the foundations of mathematics. All of the information that will be covered by the exams can be found in this text, as well as most of the exercises that will be discussed in the tutorials. The grading scheme is as follows. • One final exam, worth 65%. • Two midterms, each worth 15%, for a total of 30%. • Three homework problems (explained below), each worth 1%, for a total of 3%. • Presentation of one problem in a tutorial, worth 2%. Each week, when exercises for the tutorials are handed out, some of them will be starred (∗). Each student must submit a solution to one starred problem assigned before the first midterm, one assigned between the first and second midterms and one assigned after the second midterm. These solutions must be carefully written up and submitted to J. Franklin by the tutorial for which they have been assigned. The other problems will not be graded. If you wish to know whether your solution is correct, you are welcome to submit these assignments to J. Franklin as well.