Glimpses beyond the Two Millennia Old Bondage of the Archimedean Axiom: Part 4

No-Cloning in Reduced Power Algebras (by Elemer E. Rosinger): Abstract: The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity. Based on that fact, here it is shown that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely, one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C.

Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers (by Elemer E. Rosinger): Abstract: This, under the present form, is a replacement that is a two part paper in which the new second part was brought together with my recently posted arxiv paper, upon the suggestion of the arxiv moderators.

Quantum Foundations: Is Probability Ontological? (by Elemer E. Rosinger): Abstract: It is argued that the Copenhagen Interpretation of Quantum Mechanics, founded ontologically on the concept of probability, may be questionable in view of the fact that within Probability Theory itself the ontological status of the concept of probability has always been, and is still under discussion.

Leaving the Aristotelean Realm: Some Comments Inspired by the Articles of Elemer E. Rosinger (by Matti Pitkänen): Abstract: In the following I represent some comments on the articles of Elemer Rosinger as a physicist from the point of view of Topological Geometrodynamics. The construction of ultrapower fields (loosely surreals) is associated with physics using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Non-standard numbers are compared with the numbers generated by infinite primes and it is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Ʌ= N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S1 represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized -possibly infinite and infinitesimal- algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals.