Abstract
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.
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Partially supported by the Vice-Chancellor’s University Development Fund 23124.
Originally published in International Journal of Theoretical Physics, Vol. 44, No. 7, 2005, Due to a publishing error, authorship of the article was credited incorrectly. The corrected article is reprinted in its entirety here. The online version of the original article can be found at http://dx.doi.org/10.1007/s10773-005-7081-8
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Calude, C.S., Stay, M.A. From Heisenberg to Gödel via Chaitin. Int J Theor Phys 46, 2013–2025 (2007). https://doi.org/10.1007/s10773-006-9296-8
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DOI: https://doi.org/10.1007/s10773-006-9296-8