Начало > Публикации > Самото число. Квантовият компютър, или защо двете теореми на Курт Гьодел за непълнотата са от фундаментален физически и философски интерес

Самото число. Квантовият компютър, или защо двете теореми на Курт Гьодел за непълнотата са от фундаментален физически и философски интерес

Васил Пенчев 
Ключови думи
квантов компютър, изкуствен интелект, непълнота, разпознаване на образ
„Is incompleteness a serious problem?“ – Is not physics mathematics? – Does the first incompleteness theorem fulfill its own conditions? – The axiom of reducibility – The axiom of extensionality – Skolemian relativity – Time – Noncomputability in Turing and incompleteness in Gödel – „Gödelization“ in Penrose – Mind and physics – Quantum computer in Deutsch – Deutsch’s thought experiment of “many-worlds vs. Copenhagen interpretation of quantum mechanics – The thermodynamic value of choice – „Wigner’s friend“ – Message and mess – Albert’s quantum-mechanical automation – Identity – A difference of principle between an internal and an external observer – „Image recognition“ formalized as a recursive function – Two time scales and the relative formulation of quantum mechanics – History as mathematics – Human intellect modeled by quantum computer – Quantum computer as a set of two Turing machines – Accelerated Turing machine in Copeland – Quantum computer as “Schrödinger’s cat” or “Wigner’s friend” – The equivalence of description by the whole and by the parts – An observer in Albert and in Einstein – Subjectivity and intersubjectivity formalized – Marta, „Albert’s friend” – Quantum mechanics as the “new” physics for mind in Penrose – Freewill theorems – Computation in Copeland and the o-machine in Turing – Internal and external computability – Achilles, the Tortoise, the Arrow, and the Liar: in a single „super story“! – The ability of quantum computer for universal image recognition – Quantum computer vs. Turing machines: supporters – Randomness and repetition – The formalization of designation (sign) – Random knowledge – „Incredulous Wigner’s friend“ and „Wigner’s incredulous friends” – The axiom of reducibility anf of extensionality interpreted semiotically – The first or the second edition of Principia and the axiom of reducibility −Self-referentiality – Our conceptual background – The set, whose set of all subsets is countable – The axiom of foundation (of regularity) and the axiom of choice – The approach of Gentzen and Tarski’s conception of truth – Again about 𝜳-function as a number in a generalized counting system – The theorem of Martin Löb about the proposition stating its proper provability – Ramsey’s redundant concept of truth – Subjective and objective probability – An interpretation in quantum terms – The common base for the Liar and Arrow paradox – An approach to the completeness of Peano Arithmetic – Syntax and semantics – The Gentzen theorem − The principle of transfinite induction – The strategy for the dual foundation of completeness − The idea for dual consistency – Transfinite induction until 𝜺𝟎 – Тhe three levels of mathematics in Gentzen – The question about mathematics and physics beyond completeness – Finitism, constructivism, and ”actualism” (formalism) – From the viewpoint of “dualistic Phytagoreanism ” – “Successor” function and “wholeness” function – Transfinite and complete induction – Transfinite calculation and transfinite algorithm – Quantum computer by two Turing machines – Superfinite induction – The duality of the finite and the infinite – The Gentzen arithmetic – About the unprovability of transfinite induction until 𝜺𝟎 – „The possibility of 2 reconciling the different points of view” – Mathematics and the physical reality in Gentzen – An reflection on the reference point to the Gödel theorems – Gödel’s sketch of the first theorem of incompleteness – After involving an antinomic statement in a proof – The worth – A theory with a contradiction and a theory with an undecidable statement – If the first theorem is applied to itself – A statement, whose validity implies its undecidability – The idea of non-Gödelian, or Hilbertian mathematics – 𝝎-consistency – The meta-mathematical exclusion of self-applying the first theorem of incompleteness – About the status of the second incompleteness theorem – That mathematics cannot found itself is undecidable, too – About the Gödel number of the first incompleteness theorem – An problem about the „prime symbols” – Hilbert or Gödel mathematics – Which is the mathematics of the real world? – The viewpoint of “dualistic Pythagoreanism”

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