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Fermat’s last theorem proved in Hilbert arithmetic. II. Its proof in Hilbert arithmetic by the Kochen-Specker theorem with or without induction

Vasil Dinev Penchev

Published Jan 1, 2022• 9 citations
Publisher
DOI
10.2139/ssrn.4103565

Related papers

Fermat’s last theorem proved in Hilbert arithmetic. II. Its proof in Hilbert arithmetic by the Kochen-Specker theorem with or without induction

Vasil Dinev Penchev

The paper is a continuation of Part I. The case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense, in which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual . A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction.

•May 10, 2022•1 citation

Fermat’s Last Theorem Proved in Hilbert Arithmetic. II. Its Proof in Hilbert Arithmetic by the Kochen-Specker Theorem With or Without Induction

Vasil Dinev Penchev

The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.

•May 9, 2022•0 citations

Fermat’s Last Theorem Proved in Hilbert Arithmetic. Iii. the Quantum-Information Unification of Fermat’s Last Theorem and Gleason’s Theorem

Vasil Dinev Penchev

•Jan 1, 2022•11 citations

Fermat’s last theorem proved by the Kochen – Specker theorem and induction. The quantum-information unification of Fermat’s last theorem and Gleason’s theorem

Vasil Dinev Penchev

In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127 ), I have demonstrated an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction if the case for “n = 3” is granted as proved only arithmetically (a fact a long time ago). Now the same case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1968): the eventual validity of Fermat’s equation would correspond to an admissible disjunctive division of a qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen –Specker theorem. The proof involves set theory, but only within the case “n=3” and implicitly, within any next levels of “n”. Thus the application of set theory and arithmetic can remain disjunctively divided as well: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided. However, the mathematical structure relevant to the proof cannot be restricted to Peano arithmetic but to a generalized structure called “Hilbert arithmetic” and discussed in detail in another previous paper (https://dx.doi.org/10.2139/ssrn.3656179 ). Furthermore, Fermat’s last theorem in virtue of that kind of proof can be considered as another expression of Gleason’s theorem in quantum mechanics (or vice versa): the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information: the availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.

•Feb 15, 2021•0 citations

Kochen-Specker Theorem: Two Geometric Proofs

Cristian S. Calude, Peter H. Hertling, Karl Svozil

We present two geometric proofs of the Kochen-Specker theorem. A quite similar argument has been used by Cooke, Keane, and Moran, as well as by Kalmbach in her book to derive the Gleason theorem.

arXiv:1402.5195•Feb 21, 2014•6 citations

Gleason, Kochen-Specker, and a competition that never was

Ingemar Bengtsson

I review the two theorems referred to in the title, and then suggest that it would be interesting to know how much of Hilbert space one can use without forcing the proof of these theorems. It would also be interesting to know what parts of Hilbert space that are essential for the proofs. I go on to discuss cubes, graphs, and pentagrams.

arXiv:1210.0436•AIP conference proceedings•Jan 1, 2012•6 citations
The paper is a continuation of Part I. The case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense, in which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual . A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction.
The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.
In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127 ), I have demonstrated an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction if the case for “n = 3” is granted as proved only arithmetically (a fact a long time ago). Now the same case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1968): the eventual validity of Fermat’s equation would correspond to an admissible disjunctive division of a qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen –Specker theorem. The proof involves set theory, but only within the case “n=3” and implicitly, within any next levels of “n”. Thus the application of set theory and arithmetic can remain disjunctively divided as well: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided. However, the mathematical structure relevant to the proof cannot be restricted to Peano arithmetic but to a generalized structure called “Hilbert arithmetic” and discussed in detail in another previous paper (https://dx.doi.org/10.2139/ssrn.3656179 ). Furthermore, Fermat’s last theorem in virtue of that kind of proof can be considered as another expression of Gleason’s theorem in quantum mechanics (or vice versa): the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information: the availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.