## Halfway Up To the Mathematical Infinity II. Non-Descreteness, Non-Sequentiality, and Non-Locality of Post-Cantorian Transfinite Designs and the Prospectives of Quantum Both Computing and Formal Reasoning## Edward G. Belaga |

1- Institut de Recherche Mathématique Avancée, UMR 7501 ULP/CNRS, Université Louis Pasteur, 7 rue René Descartes, F-67084 Strasbourg Cedex, France

2- Laboratoire d’Imagerie et de Neurosciences Cognitives, UMR 7191 ULP/CNRS,
Université Louis Pasteur, Strasbourg, France

3- Laboratoire de Physique Théorique, UMR 7085 CNRS/ULP, Université Louis Pasteur, Strasbourg, France

4- Institut de Recherche Mathématique Avancée, Strasbourg, France

discrete, sequential, of local causality *versus* continuous, parallel, non-locality; transfinite design; universality and scalability of Quantum Computer; bulk water NMR

For at least two millennia – starting with the paradoxes of Zeno, IVth century BCE, – the practitioners of logic and mathematics were facing the *Discrete–Continuum* and closely related to it *Sequential–Parallel* dualities of formal world descriptions. And for at least two last centuries – since the appearance of the mechanistic determinism of Pierre-Simon Laplace – the philosophy and methodology of sciences has been alerted to the pertinence of the *Local–Non-Local* duality, to become eventually dominated by Laplace’s *Local Causality*, or *Locality Principle* (the local propagation of all action in the Universe) and its formal and metaphysical implications.

Then, Georg Cantor, the discoverer of the Mathematical Infinity, has extended into the Transfinite the limits of both *Descrete* and *Sequential*, with the explicitly stated purpose to definitely submit to them their *Continuum* and, respectively, *Parallel* counterparts. Then came Alonso Church, Alan Turing, and their colleagues with their computing schemata of pure *Sequential* and *Local Causality* nature [B2008], claiming that such schemata ultimately cover all imaginable computing activity (Church–Turing Thesis), – and with the Constructivist school being able to proudly proclaim that
*our point of view is to describe the mathematical operations that can be carried out by finite beings, man’s mathematics for short. In contrast, classical mathematics concerns itself with operations that can be carried out by God.*” ([B1985], p. 9)

This apparently definitive victory of *Discrete–Sequential–Local* over *Continuum–Parallel–Non-Local* came at a price; thus, Albert Einstein, the genuine discoverer of *Quantum Mechanical Non-Locality* which he was able to appreciate only as a *paradox*, has spent the last twenty years of his research activity on Local-Causation theories of everything. Ultimately, the Non-Locality – even if still referred to as *counter-intuitive* – has won over the physical research and *Quantum Computing* [NCh2000].

The present paper and its set-theoretical prequel [B2008] address and, hopefully, redress the *discrete–sequential–local* reductionist deficiencies of both Georg Cantor’s original transfinite design and closely related to it Church-Turing computing schemata. The main motivations of this work and of our closely related *Quantum Computing Project* [BGGVB2007], [BGKhRV2008] are coming from:
(*i*) an improbable synergy between counter-intuitive set-theoretical insights into the nature of the Continuum [B2008],
(*ii*) the discovery of the intrinsic ability of some, not necessary specifically designed for computing quantum systems *to act as a universal quantum computer using only its naturally available interactions*” [BKLWDiV2001], and
(*iii*) the analyses and set-theoretical interpretations of fundamental experimental prerequisites for a viable Quantum Computer, as those known under the name of *DiVincenzo Criteria*” [DiV2000].

Thus, our research on the bulk water NMR in a cylinder as a model of quantum computer, with a tube of one meter and slices of one millimeter generating eventually 1000 qubits of potentially immense error-correcting redundancy [BGGVB2007], [BGKhRV2008], finds its set-theoretical echo in the unlimited scalability of the Continuum viewed as a quantum system’ – and incidentally satisfying all other prerequisites for the most viable Quantum Computer. And *vice versa*.

- [BKLWDiV2001]
- D. Bacon, J. Kempe, D. Lidar, K. B. Whaley, D. DiVincenzo.Encoded Universality in Physical Implementations of a Quantum Computer.Proc. IQC 2001 (Intern. Conference on Experimental Implementation of Quantum Computation), p. 257-262, Sydney, Australia, arXiv: quant-ph/0102140v (2001).
- [B2008]
- E. Belaga. Halfway Up To the Mathematical Infinity I. On the Ontological and Epistemic Sustainability of Georg Cantor’s Transfinite Design. Manuscript, submitted, arXiv:0812.3207v1 (2008).
- [BGGVB2007]
- E. Belaga, D. Grucker, J. Grucker, and K. Van Schenk Brill, J. Baudon. Nuclear Spin Interferences In Bulk Water at Room Temperature. International Journal of Pure and Applied Mathematics, vol. 42, n
^{∘}2, 241-247 (2007). - [BGKhRV2008]
- E. Belaga, D. Grucker, J. Richert, T. Khalil, and K. Van Schenk Brill. Bulk Water NMR and Fermi-Dirac Statistics. Manuscript, submitted (2008).
- [B1985]
- E. Bishop. Schizophrenia in Contemporary Mathematics. In: M. Rosenblatt, ed., Errett Bishop: Reflections on Him and His Research. Contemporary Mathematics Vol. 39, American Mathematical Society, Providence, Rhode Island, pp.1-32 (1985).
- [DiV2000]
- D. DiVincenzo. The Physical Implementation of Quantum Computation. Fortschritte der Physik, Vol. 48, No. 9-11, pp. 771-783 (2000).
- [NCh2000]
- M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000).

This document was translated from L^{A}T_{E}X byH^{E}V^{E}A.