A critical review of randomness criteria shows that no-go theorems severely restrict the validity of actual "proofs" of undecidability. It is suggested to test microphysical undecidability by physical processes with low extrinsic complexity, such as polarized laser light. The publication and distribution of a sequence of pointer readings generated by such methods is proposed. Unlike any pseudorandom sequence generated by finite deterministic automata, the postulate of microscopic randomness implies that this sequence can be safely applied for all purposes requireing stochasticity and high complexity.

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.

Abstract. Is the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP-complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical algorithms on the one hand, and physical computers on the other, finds a natural explanation if the universe is assumed to be algorithmic; that is, that physical reality is the product of discrete subphysical information processing equivalent to the actions of a probabilistic Turing machine. This assumption can be reconciled with the observed exponentiality of quantum systems at microscopic scales, and the consequent possibility of implementing Shor's quantum polynomial time algorithm at that scale, provided the degree of superposition is intrinsically, finitely upper-bounded. If this bound is associated with the quantum-classical divide (the Heisenberg cut), a natural resolution to the quantum measurement problem arises. From this viewpoint, macroscopic classicality is an evidence that the universe is in BPP, and both questions raised above receive affirmative answers. A recently proposed computational model of quantum measurement, which relates the Heisenberg cut to the discreteness of Hilbert space, is briefly discussed. A connection to quantum gravity is noted. Our results are compatible with the philosophy that mathematical truths are independent of the laws of physics.

We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51]. That means that there does not exist any reasonable (consistent) finite formal system from which all mathematical truth is derivable. And there exists a "huge" number [11] of mathematical assertions (e.g., the continuum hypothesis, the axiom of choice) which are independent of any particular formal system. That is, they as well as their negations are compatible with the formal system. Can such formal incompleteness be translated into physics or the natural sciences in general? Is there some question about the nature of things which is provable unknowable for rational thought? Is it conceivable that the natural phenomena, even if they occur deterministically, do not allow their complete d...

Physical systems can be characterized by several types of complexity measures which indicate the computational resources employed.

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The logic–linguistic structure of quantum physics is analysed. The role of formal systems and interpretations in the representation of nature is investigated. The problems of decidability, completeness, and consistency can affect quantum physics in different ways. Bohr’s complementarity is of great interest, because it is a contradictory proposition. We shall see that the flowing of time prevents the birth of contradictions in nature, because it makes a cut between two different, but complementary aspects of the reality. PACS 03.65.Bz: Foundations, theory of measurement, miscellaneous theories. PACS 02.10.By: Logic and foundations. 1 1

The most peculiar, specifically quantum, features of quantum mechanics — quantum nonlocality, indeterminism, interference of probabilities, quantization, wave function collapse during measurement — are explained on a logical-geometrical basis. It is shown that truths of logical statements about numerical values of quantum observables are quantum observables themselves and are represented in quantum mechanics by density matrices of pure states. Structurally, quantum mechanics is a result of applying non-Abelian symmetries to truth operators and their eigenvectors — wave 1 functions. Wave functions contain information about conditional truths of all possible logical statements about physical observables and their correlations in a given physical system. These correlations are logical, hence nonlocal, and exist when the system is not observed. We analyze the physical conditions and logical and decision-making operations involved in the phenomena of wave function collapse and unpredictability of the results of measurements. Consistent explanations of the Stern-Gerlach and EPR-Bohm experiments are presented.

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

Consistent use of paradoxes in deriving constraints on the dynamics of physical systems and of no-go-theorems