User:Boris Tsirelson/Sandbox1: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Boris Tsirelson
No edit summary
No edit summary
 
(554 intermediate revisions by one other user not shown)
Line 1: Line 1:
==Decidable or undecidable==
{{AccountNotLive}}
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 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 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).
 
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).

Retrieved from "https://citizendium.org/wiki/index.php?title=User:Boris_Tsirelson/Sandbox1&oldid=904361"