Hilbert Mathematics Versus Gödel Mathematics III- Hilbert Mathematics by Itself, and Gödel Mathematics Versus the Physical World within It: both as Its Particular Cases

Penchev, Vasil Dinev

doi:10.2139/ssrn.4426040

Cited by 2 publications

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References 121 publications

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“…If one manages to demonstrate the "homeomorphism of a bit of information" in the above sense, this would imply Poincare's conjecture, and the homeomorphism at issue can be also embedded in the relation of Hilbert mathematics to Gödel mathematics: e.g., as follows. One can oppose the former to the later distinguishing them as the two formal and logical alternatives of a single axiom (similar to the Fifth postulate in relation to the pair of Euclidean and non-Euclidean geometry) though being a metamathematical one: whether the Gödel incompleteness statement is an axiom or a theorem (a problem discussed in the first part of the paper: Penchev 2023 May 3;2023 January 3;2022 October 21).…”

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Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics

Пенчев

2023

SSRN Journal

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“…In more detail in the previous parts: Penchev 2023 May 3;2023 January 3;2022 October 21. Electronic copy available at: https://ssrn.com/abstract=4503027…”

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Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics

Пенчев

2023

SSRN Journal

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Hilbert mathematics versus Gödel mathematics. IV. The new approach of Hilbert mathematics easily resolving the most difficult problems of Gödel mathematics

Penchev¹

2023

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The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions.

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Hilbert Mathematics Versus Gödel Mathematics III- Hilbert Mathematics by Itself, and Gödel Mathematics Versus the Physical World within It: both as Its Particular Cases

Cited by 2 publications

References 121 publications

Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics

Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics

Hilbert mathematics versus Gödel mathematics. IV. The new approach of Hilbert mathematics easily resolving the most difficult problems of Gödel mathematics

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