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 three-dimensional objects, (iii) in the theory of effective computability (Church-Turhrg 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.

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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 self-reference 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 self-reference will fit into more than one of these categories. aTarski's schema T is the set of all first-order logical equivalences T(r-g 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: Self-Reference We start out by taking a closer look at paradoxes related to self-r

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. This text contains all information relevant for the exams. Furthermore, the exercises in this text are those which will be demonstrated in the tutorials. Each sheet of exercises contains some important ones marked with a star and some other ones. You have to hand in an exercise marked with a star in Weeks 3 to 6, Weeks 7 to 9 and Weeks 10 to 12; each of them gives one mark. Furthermore, you can hand in any further exercises, but they are only checked for correctness. There will be two mid term exams and a final exam; the mid term exams count 15 marks each and the final exam counts 67 marks.

The agenda of quantum algorithmic information theory, ordered ‘top-down, ’ 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 two-port interferometer capable of arbitrary U(2) transformations. Basic to all these considerations is quantum theory, in particular Hilbert space quantum mechanics. 1 Information is physical, so is computation qait.tex The reasoning in constructive mathematics [17, 18, 19] and recursion theory, at least insofar as their applicability to worldly things is concerned, makes implicit assumptions about the operationalizability of the entities of discourse. It is this postulated correspondence between practical and theoretical objects, subsumed by the Church-Turing thesis, which confers power to the formal methods. Therefore, any finding in physics concerns the formal sciences; at least insofar as

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, set-theoretic 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 three-dimensional cookie cutters. The neurophysiological underpinnings of this insidious notion are discussed. 1 1

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