The Kolmogorov calculus as a part of minimal calculus

Plisko, V. E.

doi:10.1070/rm1988v043n06abeh001993

Cited by 5 publications

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“…369 See also Bezhanishvili et al [27], p.3, fn.2: "Kolmogorov's [162] propositional calculus is in fact equivalent to the implication-negation fragment of minimal calculus (see Plisko [222]). "…”

Section: Minimal Logic Rejects Both Lem and Ecqmentioning

confidence: 99%

Analytic Continuation of $ζ(s)$ Violates the Law of Non-Contradiction (LNC)

Sharon

2018

Preprint

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The Dirichlet series of ζ(s) was long ago proven to be divergent throughout half-plane Re(s) ≤ 1. If also Riemann's proposition is true, that there exists an "expression" of ζ(s) that is convergent at all s (except at s = 1), then ζ(s) is both divergent and convergent throughout half-plane Re(s) ≤ 1 (except at s = 1).This result violates all three of Aristotle's "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non-Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion" / Ex Contradictione Quodlibet (ECQ).In addition, the Hankel contour used in Riemann's analytic continuation of ζ(s) violates Cauchy's integral theorem, providing another proof of the invalidity of Riemann's ζ(s). Riemann's ζ(s) is one of the L-functions, which are all invalid due to analytic continuation. This result renders unsound all theorems (e.g. Modularity, Fermat's last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an L-function (e.g. Riemann's ζ(s)) is valid.We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar's 3VL, Priest's LP ).

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Section: Minimal Logic Rejects Both Lem and Ecqmentioning

confidence: 99%

Analytic Continuation of $ζ(s)$ Violates the Law of Non-Contradiction (LNC)

Sharon

2018

Preprint

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“…Derivation system. A formalization of intuitionistic logic QH in terms of a derivation system was found by Heyting [83] (1928 and 1931), extending partial formalizations by Kolmogorov [107] (1925; the →, ¬, ∃, ∀ fragment 67 with the omission of the explosion principle • × → α and with inadvertent omission of the universal specialization principle • ∀x α(x) → α(y); see [159]) and Glivenko (1928 and1929; the zero-order fragment). 68 Here is an equivalent formulation due to Spector (as in [200]; see also [199]), where the usual side conditions are eliminated as explained below.…”

Section: Intuitionistic Logic (Syntax)mentioning

confidence: 99%

“…A translation of classical predicate logic QC into intuitionistic predicate logic QH was discovered essentially by Kolmogorov (even though he only had his formalization of a fragment of QH available at the moment) [107] (see also [34], [204], [159]).…”

Section: ¬¬-Translationmentioning

confidence: 99%

Mathematical semantics of intuitionistic logic

Melikhov¹

2015

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This work is a mathematician's attempt to understand intuitionistic logic. It can be read in two ways: as a research paper interspersed with lengthy digressions into rethinking of standard material; or as an elementary (but highly unconventional) introduction to first-order intuitionistic logic. For the latter purpose, no training in formal logic is required, but a modest literacy in mathematics, such as topological spaces and posets, is assumed.The main theme of this work is the search for a formal semantics adequate to Kolmogorov's informal interpretation of intuitionistic logic (whose simplest part is more or less the same as the so-called BHK interpretation). This search goes beyond the usual model theory, based on Tarski's notion of semantic consequence, and beyond the usual formalism of first-order logic, based on schemata. Thus we study formal semantics of a simplified version of Paulson's meta-logic, used in the Isabelle prover.By interpreting the meta-logical connectives and quantifiers constructively, we get a generalized model theory, which covers, in particular, realizability-type interpretations of intuitionistic logic. On the other hand, by analyzing Kolmogorov's notion of semantic consequence (which is an alternative to Tarski's standard notion), we get an alternative model theory. By using an extension of the meta-logic, we further get a generalized alternative model theory, which suffices to formalize Kolmogorov's semantics.On the other hand, we also formulate a modification of Kolmogorov's interpretation, which is compatible with the usual, Tarski-style model theory. Namely, it can be formalized by means of sheaf-valued models, which turn out to be a special case of Palmgren's categorical models; intuitionistic logic is complete with respect to this semantics.3.2. Platonism vs. Verificationism 3.3. The issue of understanding 3.4. Solutions of problems 3.5. The Hilbert-Brouwer controversy 3.6. Problems vs. conjectures 3.7. The BHK interpretation 3.8. Understanding the connectives 3.9. Something is missing here 3.10. Classical logic revisited 3.11. Clarified BHK interpretation 3.12. The principle of decidability and its relatives 3.13. Medvedev-Skvortsov problems 3.14. Some intuitionistic validities 4. What is a logic, formally? 4.1. Introduction 4.2. Typed expressions 4.3. Languages and structures 4.4. Meta-logic 4.5. Logics 4.6. Intuitionistic logic (syntax) 4.7. Models 4.8. Classical logic (semantics) 5. Topology, translations and principles 5.1. Deriving a model from BHK 5.2. Tarski models 5.3. Topological completeness 5.4. Jankov's principle 5.5. Markov's principle 5.6. ¬¬-Translation 5.7. Provability translation 5.8. Independence of connectives and quantifiers 6. Sheaf-valued models 6.1. Sheaves 6.2. Operations on sheaves 6.3. Sheaf-valued models 7. Semantics of the meta-logic 7.1. Generalized models 7.2. Alternative semantics 7.3. Generalized alternative semantics References 1 In the introduction to her survey of logic in the USSR up to 1957, S. A. Yanovskaya emphasizes "The difference of t...

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“…Without a claim of completeness the reader is referred to e.g. [Medvedev 1955, Medvedev 1962, [Plisko 1988b, Plisko 1988a, [Uspenski 1992], [Uspenskii, V.A. and V.E.…”

Section: Introductionmentioning

confidence: 99%

Kolmogorov and Brouwer on constructive implication and the Ex Falso rule

Dalen¹

2004

Russ. Math. Surv.

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The Kolmogorov calculus as a part of minimal calculus

Cited by 5 publications

References 3 publications

Analytic Continuation of $ζ(s)$ Violates the Law of Non-Contradiction (LNC)

Analytic Continuation of $ζ(s)$ Violates the Law of Non-Contradiction (LNC)

Mathematical semantics of intuitionistic logic

Kolmogorov and Brouwer on constructive implication and the Ex Falso rule

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