User:Boris Tsirelson/Sandbox1: Difference between revisions

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==Decidable or undecidable==
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


Theorems of a theory are, by definition, statements that follow from
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
the given axioms according to the given rules (called by different
authors inference rules, derivation rules, deduction rules,
transformation rules). Note that "a theorem" does not mean "a motivated
theorem", "an important theorem" etc., not even "an already discovered
theorem". All theorems are just an indiscriminate stream of logical
consequences of the axioms. It is impossible to list all theorems,
since they are infinitely many. However, an endless algorithmic
process can generate theorems, only theorems, and all theorems in the
sense that every theorem will be generated, sooner or later. (In more
technical words: the set of all theorems is recursively enumerable.)
 
An open question (in a mathematical theory) is a statement neither
proved nor disproved. It is possible (in principle, not necessarily in
practice) to run the theorem-generating algorithm waiting for one of
two events: either the given statement appears to be a theorem, or its
negation does; in both cases the (formerly) open question is
decided. However,
(a) there is no guarantee that it will be decided in the first
10<sup>6</sup> steps of the algorithm, nor in the first
10<sup>1000</sup> steps, nor in any time estimated beforehand;
(b) worse, there is no guarantee that it will be decided at all.
 
A statement is called independent (in other words, undecidable) in the
given theory, if it is not a theorem, but also its negation is not a
theorem.

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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