The Lost Melody Phenomenon

A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set $\{x\}$ is computable while $x$ itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for $ITRM$s. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for $ITRM$s. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that $0^{\sharp}$ exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting $\alpha$-register machines and give some information on the question for which of these machines there are lost melodies

A Heuristic Approach to the Quantum Measurement Problem: How to Distinguish Particle Detectors from Ordinary Objects

Elementary particle detectors fall broadly into only two classes: phase-transformation devices, such as the bubble chamber, and charge-transfer devices like the Geiger-Mueller tube. Quantum measurements are seen to involve transitions from a long-lived metastable state (e. g., superheated liquid or a gas of atoms between charged capacitor plates) to a thermodinamically stable condition. A detector is then a specially prepared object undergoing a metastable-to-stable transformation that is significantly enhanced by the presence of the measured particle, which behaves, in some sense, as the seed of a process of heterogeneous nucleation. Based on this understanding of the operation of a conventional detector, and using results of orthogonality-catastrophe theory, we argue that, in the thermodynamic limit, the pre-measurement Hamiltonian is not the same as that describing the detector during or after the interaction with a particle and, thus, that superpositions of pointer states (Schroedinger cats) are unphysical because their time evolution is ill defined. Examples of particle-induced changes in the Hamiltonian are also given for ordinary systems whose macroscopic parameters are susceptible to radiation damage, but are not modified by the interaction with a single particle.Comment: 14 pages, 2 figure

Towards a Church-Turing-Thesis for Infinitary Computations

We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines ($IAM$s) of this which will turn out to be equivalent in strength to the Ordinal Turing Machines defined by P. Koepke

A Note on OTM-Realizability and Constructive Set Theories

We define an ordinalized version of Kleene's realizability interpretation of intuitionistic logic by replacing Turing machines with Koepke's ordinal Turing machines (OTMs), thus obtaining a notion of realizability applying to arbitrary statements in the language of set theory. We observe that every instance of the axioms of intuitionistic first-order logic are OTM-realizable and consider the question which axioms of Friedman's Intuitionistic Set Theory (IZF) and Aczel's Constructive Set Theory (CZF) are OTM-realizable. This is an introductory note, and proofs are mostly only sketched or omitted altogether. It will soon be replaced by a more elaborate version

Concerns of Pastoral Ministry With a Biblical Perspective From the Gospel of Mark

The area of study for this paper was pastoral ministry. The research focused on three areas of concern: the recruitment of persons for pastoral ministry; the attrition of pastors from the parish ministry: and, the training of ministers