caa a22 4500 463197003 CHVBK 20180406164925.0 cr unu---uuuuu 170326e20070801xx s 000 0 eng 10.1007/s10773-006-9296-8 doi (NATIONALLICENCE)springer-10.1007/s10773-006-9296-8 From Heisenberg to Gödel via Chaitin [Elektronische Daten] [Cristian Calude, Michael Stay] In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a "formal uncertainty principle” which implies Chaitin's information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics. Springer Science+Business Media, LLC, 2007 algorithmic randomness nationallicence uncertainty nationallicence incompleteness nationallicence Calude Cristian Department of Computer Science, The University of Auckland, Auckland, New Zealand aut Stay Michael Department of Computer Science, The University of Auckland, Auckland, New Zealand aut International Journal of Theoretical Physics Springer US; http://www.springer-ny.com 46/8(2007-08-01), 2013-2025 0020-7748 46:8<2013 2007 46 10773 https://doi.org/10.1007/s10773-006-9296-8 text/html Onlinezugriff via DOI 1 other jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10773-006-9296-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Calude Cristian Department of Computer Science, The University of Auckland, Auckland, New Zealand aut NATIONALLICENCE 700 1- Stay Michael Department of Computer Science, The University of Auckland, Auckland, New Zealand aut NATIONALLICENCE 773 0- International Journal of Theoretical Physics Springer US; http://www.springer-ny.com 46/8(2007-08-01), 2013-2025 0020-7748 46:8<2013 2007 46 10773 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer