Парадоксът на Скулем и квантовата информация. Относителност на пълнота по Гьодел
Skolem’s Paradox and Quantum Information. Relativity of Completeness according to Gödel

Summary/Abstract: In 1992, Thoralf Skolem introduced the term of «relativity» as to infinity or set theory. He demonstrated by Zermelo’s axiomatics of set theory (incl. the axiom of choice) that there exist unintended interpretations of any infinite set. The very notion of set was also «relative». We can apply his argumentation to Gödel’s incompleteness theorems as well as to his completeness theorem (1930). Then both the incompleteness of Peano arithmetic and the completeness of first-order logic turn out to be also «relative» in Skolem’s sense. Skolem’s «relativity» argumentation of that kind can be applied to a very wide range of problems and can be spoken of the relativity of discreteness and continuity, of finiteness and infinity, of Cantor’s kinds of infinities, etc. The relativity of Skolemian type helps us for generalizing Einstein’s principle of relativity from the invariance of the physical laws toward diffeomorphisms to their invariance toward any morphisms (including and especially the discrete ones). Such a kind of generalization from diffeomorphisms (when the notion of velocity always makes sense) to any kind of morphism (when ‘velocity’ may or may not make sense) is an extension of the general Skolemian type of relativity between discreteness and continuity or between finiteness and infinity. Particularly, Lorentz invariance gains constrained validity, because the very notion of velocity is limited to diffeomorphisms. In the case of entanglement, the physical interaction is discrete. ‘Velocity’ and consequently ‘Lorentz invariance’ do not make sense. That is the simplest explanation of the argument EPR, which turns into a paradox only if the universal validity of ‘velocity’ and ‘Lorentz invariance’ is implicitly accepted. Correspondingly, a more general class of topologies is to be considered, including discrete or inseparable kinds.

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