Foundations of real analysis and computability theory in non-Aristotelian finitary logic

This paper outlines new paradigms for real analysis and computability theory in the recently proposed non-Aristotelian finitary logic (NAFL). Constructive real analysis in NAFL (NRA) is accomplished by a translation of diagrammatic concepts from Euclidean geometry into an extension (NPAR) of the NAFL version of Peano Arithmetic (NPA). Such a translation is possible because NPA proves the existence of every infinite proper class of natural numbers that is definable in the language of NPA. Infinite sets are not permitted in NPAR and quantification over proper classes is banned; hence Cantor's diagonal argument cannot be legally formulated in NRA, and there is no `cardinality' for any collection (`super-class') of real numbers. Many of the useful aspects of classical real analysis, such as, the calculus of Newton and Leibniz, are justifiable in NRA. But the paradoxes, such as, Zeno's paradoxes of motion and the Banach-Tarski paradox, are resolved because NRA admits only closed super-classes of real numbers; in particular, open/semi-open intervals of real numbers are not permitted. The NAFL version of computability theory (NCT) rejects Turing's argument for the undecidability of the halting problem and permits hypercomputation. Important potential applications of NCT are in the areas of quantum and autonomic computing.Comment: An error corrected in the equation of Remark 9. Other comments of Version 2: 25 pages, nine references added, typos corrected. Appendix B, giving details of the formal systems, has been added. Substantially improved presentation with more details, especially in Sec. 4 on real analysis, which can be read more or less independently of other section

Logical analysis of the Bohr Complementarity Principle in Afshar's experiment under the NAFL interpretation

The NAFL (non-Aristotelian finitary logic) interpretation of quantum mechanics requires that no `physical' reality can be ascribed to the wave nature of the photon. The NAFL theory QM, formalizing quantum mechanics, treats the superposed state ($S$) of a single photon taking two or more different paths at the same time as a logical contradiction that is formally unprovable in QM. Nevertheless, in a nonclassical NAFL model for QM in which the law of noncontradiction fails, $S$ has a meaningful metamathematical interpretation that the classical path information for the photon is not available. It is argued that the existence of an interference pattern does not logically amount to a proof of the self-interference of a single photon. This fact, when coupled with the temporal nature of NAFL truth, implies the logical validity of the retroactive assertion of the path information (and the logical superfluousness of the grid) in Afshar's experiment. The Bohr complementarity principle, when properly interpreted with the time dependence of logical truth taken into account, holds in Afshar's experiment. NAFL supports, but not demands, a metalogical reality for the particle nature of the photon even when the semantics of QM requires the state $S$.Comment: 28 pages, 1 figure. Ref. [7] updated. Typo correcte