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Journal articles on the topic 'Hilbert's axioms'

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Published: 4 June 2021
Last updated: 25 July 2025

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1

Negri, Sara, and Jan von Plato. "From mathematical axioms to mathematical rules of proof: recent developments in proof analysis." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180037. http://dx.doi.org/10.1098/rsta.2018.0037.

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A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.
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Pambuccian, Victor. "Prolegomena to any theory of proof simplicity." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180035. http://dx.doi.org/10.1098/rsta.2018.0035.

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By looking at concrete examples from elementary geometry, we analyse the manner in which the simplicity of proofs could be defined. We first find that, when presented with two proofs coming from mutually incompatible sets of assumptions, the decision regarding which one is simplest can be made, if at all, only on the basis of reasoning outside of the formal aspects of the axiom systems involved. We then show that, if the axiom system is fixed, a measure of proof simplicity can be defined based on the number of uses of axioms deemed to be deep or valuable, and prove a number of new results rega
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Cohen, Paul J. "Skolem and pessimism about proof in mathematics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363, no. 1835 (2005): 2407–18. http://dx.doi.org/10.1098/rsta.2005.1661.

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Attitudes towards formalization and proof have gone through large swings during the last 150 years. We sketch the development from Frege's first formalization, to the debates over intuitionism and other schools, through Hilbert's program and the decisive blow of the Gödel Incompleteness Theorem. A critical role is played by the Skolem–Lowenheim Theorem, which showed that no first-order axiom system can characterize a unique infinite model. Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescie
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Siekmann, J., and P. Szabó. "The undecidability of the DA-unification problem." Journal of Symbolic Logic 54, no. 2 (1989): 402–14. http://dx.doi.org/10.2307/2274856.

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AbstractWe show that the DA-uniflcation problem is undecidable. That is, given two binary function symbols ⊕ and ⊗, variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following DA-axioms hold:Two terms are DA-unifiable (i.e. an equation is solvable in DA) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory DA.This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained.
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Vavilov, Nikolai. "Reshaping the metaphor of proof." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180279. http://dx.doi.org/10.1098/rsta.2018.0279.

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The simplistic view of Mathematics as a logical system of formal truths deduced from a limited set of axioms by a limited set of inference rules immediately shatters when confronted with the history of Mathematics, or current mathematical practice. To become useful, mathematical Philosophy should contemplate what Mathematics actually was, over centuries, and what it is now, rather than speculate what it should be according to the philosophical orthodoxy. The first dogma that must be completely revised is the idea of proof as a text, rather than what it is for mathematicians themselves: a proce
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CUMMINGS, JAMES, MATTHEW FOREMAN, and MENACHEM MAGIDOR. "SQUARES, SCALES AND STATIONARY REFLECTION." Journal of Mathematical Logic 01, no. 01 (2001): 35–98. http://dx.doi.org/10.1142/s021906130100003x.

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Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, a
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Rustemov, B. "AXIOMATIC THEORY OF EVERYTHIN - THE FUNDAMENTAL BASIS OF SUSTAINABLE DEVELOPMENT OF THE WORLD." Sciences of Europe, no. 153 (November 27, 2024): 84–92. https://doi.org/10.5281/zenodo.14227581.

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The article reports on the creation of an axiomatic theory claiming the status of "Theory of Everything". The scientific novelty and the main difference between this theory and its predecessors is the design of the laws of physics as causal laws. This approach made it possible to identify the metaphysical essence of Hilbert's sixth problem and solve it. Based on the accepted axioms, the basic metaphysical law was established, which made it possible to formalize a complete picture of the world in the form of an Euler-Venn diagram. This scientific picture of the world allowed us to establish the
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8

Zeman, Jan. "Hilbertova aritmetizace geometrie." FILOSOFIE DNES 10, no. 1 (2019): 45–63. http://dx.doi.org/10.26806/fd.v10i1.269.

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Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit­ obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů
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Zeman, Jan. "Hilbertova aritmetizace geometrie." FILOSOFIE DNES 10, no. 1 (2019): 45–63. http://dx.doi.org/10.26806/fd.v10i1.415.

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Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit­ obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů
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10

Richter, William, Adam Grabowski, and Jesse Alama. "Tarski Geometry Axioms." Formalized Mathematics 22, no. 2 (2014): 167–76. http://dx.doi.org/10.2478/forma-2014-0017.

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Summary This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!
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Takahashi, Tadashi, and Fumiya Iwama. "ON THE PROOF OF THE THEOREMS OF FOUNDATIONS OF GEOMETRY USING ISABELLE/HOL." Journal of Computational Innovation and Analytics (JCIA) 1, No.2 (2022): 45–69. http://dx.doi.org/10.32890/jcia2022.1.2.3.

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Isabelle/HOL is a generic proof assistant. Using Isabelle/HOL requires insight into procedures as well as into the concepts involved. In addition, how a computer manages procedures can affect mathematical concepts. Use of Isabelle/HOL can correct a current weakness in mathematical studies. The advantage of the theorem proving support system represented by Isabelle/HOL is that it mechanically guarantees the “correctness” of both human-written programs and mathematical proofs. It can allow us to clearly understand mathematical concepts and can minimize the burden of operation opportunities. Howe
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12

Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms – Part II." Formalized Mathematics 24, no. 2 (2016): 157–66. http://dx.doi.org/10.1515/forma-2016-0012.

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Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was prev
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13

Raftery, J. G. "Correspondences between gentzen and hilbert systems." Journal of Symbolic Logic 71, no. 3 (2006): 903–57. http://dx.doi.org/10.2178/jsl/1154698583.

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Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a numb
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14

McLarty, Colin. "Poincaré on the value of reasoning machines." Bulletin of the American Mathematical Society 61, no. 3 (2024): 411–22. http://dx.doi.org/10.1090/bull/1822.

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Hilbert’s Foundations of Geometry in 1899 made Poincaré think of “reasoning machines” before Hilbert did. Poincaré found the idea “deadly for teaching, and desiccating for researchers” but indispensable for telling when intuitions have been fully expressed. A machine will use stated axioms without the vague intuitions Poincaré considered vital to learning and research. Years of famously intuitive creativity, plus boundless faith in technology, as well as the impact of Hilbert, led Poincaré to see that machines could aid human intuition but not replace it, precisely because machines have no int
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15

Torres Alcaraz, Carlos. "Hilbert, Kant y el fundamento de las matemáticas." Theoría. Revista del Colegio de Filosofía, no. 8-9 (December 31, 1999): 111–29. http://dx.doi.org/10.22201/ffyl.16656415p.1999.8-9.225.

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This paper looks into Hilbert’s thought about mathematics and explores its relation which the philosophy of Kant. The focus of the research is in the role of axiomatic thinking and logical analysis in foundational studies. The paper concentrates mainly in Hilbert’s research regarding the foundations of geometry, and follows his main lines of thought up to his programme, which revolves around a consistency proof for the axioms of classical mathematics. A final analysis allows us to conclude that for him mathematics is, in a broad sense, “the science of that which is possible” in this point, Hil
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D'Ariano, Giacomo Mauro. "The solution of the sixth Hilbert problem: the ultimate Galilean revolution." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2118 (2018): 20170224. http://dx.doi.org/10.1098/rsta.2017.0224.

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I argue for a full mathematization of the physical theory, including its axioms, which must contain no physical primitives. In provocative words: ‘physics from no physics’. Although this may seem an oxymoron, it is the royal road to keep complete logical coherence, hence falsifiability of the theory. For such a purely mathematical theory the physical connotation must pertain only the interpretation of the mathematics, ranging from the axioms to the final theorems. On the contrary, the postulates of the two current major physical theories either do not have physical interpretation (as for von N
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17

Burn, R. P. "Non-Desarguesian planes and weak associativity." Mathematical Gazette 101, no. 552 (2017): 458–64. http://dx.doi.org/10.1017/mag.2017.127.

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During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such thatAxiom 1: Given two distinct points, there is a unique line containing them both.Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line conta
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18

Guo, Jiayi. "The Debates on Infinity: A Mathematical History Approach." Highlights in Science, Engineering and Technology 88 (March 29, 2024): 208–13. http://dx.doi.org/10.54097/8fz01096.

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Calculus and set theory sparked centuries of debate on infinity, which continues today. After discovering paradoxes and inconsistencies, mathematicians and philosophers questioned the underlying systems and conceptions of the infinite. Today, we easily use the infinity sign in our academic work. We often forget infinity's turbulent debut in our contemporary use of the term. David Hilbert wrote "On the Infinite" in the 1920s to persuade sceptics to embrace and use infinity. Gödel and his second incompleteness theorem defeated him very quickly. Beyond Gödel's claim of system inconsistency, Hilbe
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Fontanella, Laura. "Axioms as Definitions: Revisiting Poincaré and Hilbert." Philosophia Scientae, no. 23-1 (February 18, 2019): 167–83. http://dx.doi.org/10.4000/philosophiascientiae.1827.

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20

LANDRY, ELAINE. "THE GENETIC VERSUS THE AXIOMATIC METHOD: RESPONDING TO FEFERMAN 1977." Review of Symbolic Logic 6, no. 1 (2012): 24–51. http://dx.doi.org/10.1017/s1755020312000135.

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AbstractFeferman (1977) argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro (2005), we can be structuralists all the way down; we do not have to appeal to some
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Khvedelidze, A. "Generalizing Stratonovich–Weyl Axioms for Composite Systems." Physics of Particles and Nuclei 54, no. 6 (2023): 1025–28. http://dx.doi.org/10.1134/s1063779623060175.

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Abstract The statistical model of quantum mechanics is based on the mapping between operators on the Hilbert space and functions on the phase space. This map can be implemented by an operator that satisfies physically motivated Stratonovich–Weyl axioms. Arguments are given in favour of a certain extension of the axioms, provided that there is a priori knowledge about the composite nature of the quantum system.
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Smadja, Ivahn. "Local axioms in disguise: Hilbert on Minkowski diagrams." Synthese 186, no. 1 (2011): 315–70. http://dx.doi.org/10.1007/s11229-011-9984-7.

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23

Lehrer, Ehud, and Eran Shmaya. "A qualitative approach to quantum probability." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2072 (2006): 2331–44. http://dx.doi.org/10.1098/rspa.2006.1672.

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A likelihood order is defined over linear subspaces of a finite dimensional Hilbert space. The question arises as to when such an order can be represented by a quantum probability. We introduce a few behaviorally plausible axioms that provide the answer in two cases: pure state and uniform measure. The general problem is answered by using duality-like conditions. The general problem of characterizing the partial orders that admit a quantum representation by behaviorally justified axioms remains open.
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Lobovikov, Vladimir. "Combining Universal Epistemology with Formal Axiology in a Multimodal Formal Axiomatic Theory “Sigma + 2C”, and Philosophical Foundations of Mathematics." Respublica Literaria 4, no. 4 (2023): 88–113. http://dx.doi.org/10.47850/rl.2023.4.4.88-113.

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The paper is devoted to investigating Kant’s apriorism underlying Hilbert’s formalism in philosophical foundations of mathematics. The target is constructing a formal axiomatic theory of knowledge in which it is possible to invent formal inferences of formulae-modeling-Hilbert-formalism from the assumption of Kant apriorism concerning mathematics. The scientific novelty: a logically-formalized axiomatic system of universal philosophical epistemology called “Sigma +2C” is invented for the first time as a generalization of the already published formal epistemology system “Sigma +C”. In compariso
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Majid, S. "On the emergence of the structure of physics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2118 (2018): 20170231. http://dx.doi.org/10.1098/rsta.2017.0231.

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We consider Hilbert’s problem of the axioms of physics at a qualitative or conceptual level. This is more pressing than ever as we seek to understand how both general relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a principle of self-duality or quantum Born reciprocity as a key structure. Here, I outline some of my recent work around the idea of quantum space–time as motivated by this non-standard philosophy, including a new toy model of gravity on a space–time consisting of four points forming a square. This
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KLEV, ANSTEN. "DEDEKIND AND HILBERT ON THE FOUNDATIONS OF THE DEDUCTIVE SCIENCES." Review of Symbolic Logic 4, no. 4 (2011): 645–81. http://dx.doi.org/10.1017/s1755020311000232.

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We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas Hilbert dismisses elucidation and consequently treats the primitives as schematic.
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Smadja, Ivahn. "Erratum to: Local axioms in disguise: Hilbert on Minkowski diagrams." Synthese 186, no. 1 (2011): 441–42. http://dx.doi.org/10.1007/s11229-011-0039-x.

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28

Wawrzycki, J., and T. Wawrzycki. "Representation Structure of the \(\mathrm {SL}(2, \mathbb {C})\) Acting in the Hilbert Space of the Quantum Coulomb Field." Acta Physica Polonica B 56, no. 6 (2025): 1. https://doi.org/10.5506/aphyspolb.56.6-a3.

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We give a complete description of the representation of \(\mathrm {SL}(2,\mathbb {C})\) acting in the Hilbert space of the quantum Coulomb field and a constructive consistency proof of the axioms of the quantum theory of the Coulomb field. Abstract Published by the Jagiellonian University 2025 authors
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29

Simpson, Stephen G. "Ordinal numbers and the Hilbert basis theorem." Journal of Symbolic Logic 53, no. 3 (1988): 961–74. http://dx.doi.org/10.2307/2274585.

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In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates
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Lobovikov, Vladimir Olegovich. "A Logically Formalized Axiomatic Epistemology System Σ + C and Philosophical Grounding Mathematics as a Self-Sufficing System". Mathematics 9, № 16 (2021): 1859. http://dx.doi.org/10.3390/math9161859.

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The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized ax
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Gusin, Pawel, Daniel Burys, and Andrzej Radosz. "Measures of Distance in Quantum Mechanics." Universe 10, no. 1 (2024): 34. http://dx.doi.org/10.3390/universe10010034.

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Combining gravity with quantum theory is still a work in progress. On the one hand, classical gravity is the geometry of space-time determined by the energy–momentum tensor of matter and the resulting nonlinear equations; on the other hand, the mathematical description of a quantum system is Hilbert space with linear equations describing evolution. In this paper, various measures in Hilbert space will be presented. In general, distance measures in Hilbert space can be divided into measures determined by energy and measures determined by entropy. Entropy measures determine quasi-distance becaus
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Buss, Samuel R. "On Gödel's theorems on lengths of proofs I: Number of lines and speedup for arithmetics." Journal of Symbolic Logic 59, no. 3 (1994): 737–56. http://dx.doi.org/10.2307/2275906.

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AbstractThis paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact, unbounded) proof speedup of (i + l)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and • as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as axioms and allows all generalizations of axioms as axioms.Our f
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D'Esposito, Vittorio, Giuseppe Fabiano, Domenico Frattulillo, and Flavio Mercati. "Doubly Quantum Mechanics." Quantum 9 (April 24, 2025): 1721. https://doi.org/10.22331/q-2025-04-24-1721.

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Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin-12 measurements by promoting the group of spatial rotations SU(2) to the quantum group SUq(2) and generalizing the axioms of Quantum Theory in a c
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34

Kanamori, Akihiro. "Gödel and Set Theory." Bulletin of Symbolic Logic 13, no. 2 (2007): 153–88. http://dx.doi.org/10.2178/bsl/1185803804.

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Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axi
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OZAWA, MASANAO. "ORTHOMODULAR-VALUED MODELS FOR QUANTUM SET THEORY." Review of Symbolic Logic 10, no. 4 (2017): 782–807. http://dx.doi.org/10.1017/s1755020317000120.

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AbstractIn 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models
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Tertychny-Dauri, V. Yu. "Hilbert’s 6-th problem and principle of completeness in dynamics." Journal of Physics: Conference Series 2090, no. 1 (2021): 012106. http://dx.doi.org/10.1088/1742-6596/2090/1/012106.

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Abstract The following offers a new axiomatic basis of mechanics and physics in their most important dynamics domain, i.e. a principle (axiom) of completeness intended to generalize Newton’s second law of motion for the case of a non-stationary variable-mass point (system) that varies with time. This generalization leads to hyperdynamic dependencies describing such motion from new accurate qualitative standpoints.
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ASANO, MASAKO. "ON A CLASS OF TOPOLOGICAL QUANTUM FIELD THEORIES IN THREE DIMENSIONS." International Journal of Modern Physics A 11, no. 25 (1996): 4577–96. http://dx.doi.org/10.1142/s0217751x96002121.

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We investigate the Chung–Fukuma–Shapere theory, or Kuperberg theory, of three-dimensional lattice topological field theory. We construct a functor which satisfies Atiyah’s axioms of topological quantum field theory by reformulating the theory as a Turaev–Viro type state sum theory on a triangulated manifold. This corresponds to giving the Hilbert space structure to the original theory. The theory can be extended to give a topological invariant of manifolds with boundary.
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38

Krajewski, Stanisław. "Anti-foundationalist Philosophy of Mathematics and Mathematical Proofs." Studia Humana 9, no. 3-4 (2020): 154–64. http://dx.doi.org/10.2478/sh-2020-0034.

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Abstract The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert’s Thesis, accord
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39

Baldwin, John T., and Andreas Mueller. "Autonomy of Geometry." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 11 (February 5, 2020): 5–24. http://dx.doi.org/10.24917/20809751.11.1.

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In this paper we present three aspects of the autonomy of geometry. (1) An argument for the geometric as opposed to the ‘geometric algebraic’ interpretation of Euclid’s Books I and II; (2) Hilbert’s successful project to axiomatize Euclid’s geometry in a first order geometric language, notably eliminating the dependence on the Archimedean axiom; (3) the independent conception of multiplication from a geometric as opposed to an arithmetic viewpoint.
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40

Willard, Dan E. "Self-verifying axiom systems, the incompleteness theorem and related reflection principles." Journal of Symbolic Logic 66, no. 2 (2001): 536–96. http://dx.doi.org/10.2307/2695030.

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AbstractWe will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive proofs. Our axiom systems will not be strong enough
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41

Błaszczyk, Piotr, and Anna Petiurenko. "Commentary to Book I of the Elements. Hartshorne and beyond." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 13 (December 31, 2021): 43–99. http://dx.doi.org/10.24917/20809751.13.7.

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(Hartshorne, 2000) interprets Euclid’s Elements provides an interpretation of Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1-I.27, covering the so-called absolute geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Postulate 5, we present a model of nonEuclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the ordered field of hyperreal numbers R*.
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42

Ruoff, Dieter. "Zur Unabh�ngigkeit von Hilberts Axiomen des affinen Raumes." Journal of Geometry 25, no. 1 (1985): 1–18. http://dx.doi.org/10.1007/bf01222941.

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43

Morillon, Marianne. "Notions of compactness for special subsets of ℝI and some weak forms of the axiom of choice". Journal of Symbolic Logic 75, № 1 (2010): 255–68. http://dx.doi.org/10.2178/jsl/1264433919.

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AbstractWe work in set-theory without choice ZF. A set is countable if it is finite or equipotent with ℕ. Given a closed subset F of [0, 1]I which is a bounded subset of ℓ1(I) (resp. such that F ⊆ c0(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice ACℕ) implies that F is compact. This enhances previous results where ACℕ (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example I = ℝ), then, in ZF, the closed unit ball of the Hilbert space ℓ2 (I) is (Loeb-)compact in the weak topology. However, the weak comp
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44

Brezhnev, Yurii V. "The Born rule as a statistics of quantum micro-events." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2244 (2020): 20200282. http://dx.doi.org/10.1098/rspa.2020.0282.

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We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would b
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45

DROBYSHEVICH, SERGEY, and HEINRICH WANSING. "PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS." Review of Symbolic Logic 13, no. 4 (2019): 720–47. http://dx.doi.org/10.1017/s1755020319000261.

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AbstractWe present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a
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46

Ciuciura, Janusz. "Gently Paraconsistent Calculi." Axioms 9, no. 4 (2020): 142. http://dx.doi.org/10.3390/axioms9040142.

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In this paper, we consider some paraconsistent calculi in a Hilbert-style formulation with the rule of detachment as the sole rule of interference. Each calculus will be expected to contain all axiom schemas of the positive fragment of classical propositional calculus and respect the principle of gentle explosion.
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47

Jorge, Juan Pablo, and Hernán Luis Vázquez. "Retornando al Hotel de Hilbert." Revista de Educación Matemática 36, no. 2 (2021): 67–87. http://dx.doi.org/10.33044/revem.32687.

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Se construyen particiones particulares del conjunto de los naturales a través de procesos recursivos generando, de esta manera, numerables ejemplos de conjuntos numerables y disjuntos cuya unión es un conjunto también numerable. El proceso es constructivo por lo cual no se hace uso del axioma de elección. Se presenta un programa que genera una de estas particiones especiales y se muestra cómo generar infinitas de las mismas. Esta línea de razonamiento puede tener múltiples aplicaciones en la teoría de conjuntos y de modelos. Probamos que la cantidad de formas de realizar estas particiones de l
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ANGLBERGER, ALBERT J. J., NOBERT GRATZL, and OLIVIER ROY. "OBLIGATION, FREE CHOICE, AND THE LOGIC OF WEAKEST PERMISSIONS." Review of Symbolic Logic 8, no. 4 (2015): 807–27. http://dx.doi.org/10.1017/s1755020315000209.

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AbstractWe introduce a new understanding of deontic modals that we callobligations as weakest permissions. We argue for its philosophical plausibility, study its expressive power in neighborhood models, provide a complete Hilbert-style axiom system for it and show that it can be extended and applied to practical norms in decision and game theory.
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49

Ozawa, Masanao. "Transfer principle in quantum set theory." Journal of Symbolic Logic 72, no. 2 (2007): 625–48. http://dx.doi.org/10.2178/jsl/1185803627.

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AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum
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50

Da Silva, Eleonoura Enoque. "Abordagem categorial para a linguagem da teoria quântica." Revista Ágora Filosófica 1, no. 1 (2016): 233–44. http://dx.doi.org/10.25247/p1982-999x.2015.v1n1.p233-244.

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O propósito do nosso trabalho é investigar a construção da “categoria dos espaços Hilbert”, com o objetivo de superar certos limites contidos nas linguagens da teoria quântica. A importância de se estudar a teoria quântica por uma via categorial é a possibilidade da construção de uma categoria que já incorpora na sua própria linguagem alguns dos axiomas, fenômenos e princípios básicos da mecânica quântica.
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