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Journal articles on the topic 'Godel's Incompleteness theorem'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Katrechko, Sergey. "Goedel's Incomplete Theorem: Does it Say Something about the Incompleteness of Math." Respublica Literaria 4, no. 4 (2023): 82–87. http://dx.doi.org/10.47850/rl.2023.4.4.82-87.

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The report will propose some methodological approach (“method of eliminating monsters”) going back to Lakatos (“Proof and refutation”) to the analysis of Godel’s theorem, which shows that despite its mathematical correctness, Godel’s theorem is not applicable to mathematics (recursive arithmetic), since the Gödel expression is not a formula, but a formulaoid (Yesenin-Volpin), or a mathematical “monster” that must be eliminated from the field (language) of mathematics. This makes it possible to “remove” Godel's thesis about the incompleteness of mathematics, although it does not cancel the task

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2

Jech, Thomas. "On Godel's Second Incompleteness Theorem." Proceedings of the American Mathematical Society 121, no. 1 (1994): 311. http://dx.doi.org/10.2307/2160398.

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3

Moore, A. W. "What Does Godel's Second Incompleteness Theorem Show?" Noûs 22, no. 4 (1988): 573. http://dx.doi.org/10.2307/2215458.

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4

Gauker, C. "T-schema deflationism versus Godel's first incompleteness theorem." Analysis 61, no. 2 (2001): 129–36. http://dx.doi.org/10.1093/analys/61.2.129.

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5

Gauker, Christopher. "T-schema deflationism versus Godel's first incompleteness theorem." Analysis 61, no. 270 (2001): 129–36. http://dx.doi.org/10.1111/1467-8284.00282.

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6

Xie, Ling. "The Reason for the Incompleteness of Some Theorems in Modern Mathematics: Non Numbers Enter Mathematical Analysis or the Concept Definition Is Incomplete." European Journal of Theoretical and Applied Sciences 1, no. 3 (2023): 171–80. http://dx.doi.org/10.59324/ejtas.2023.1(3).19.

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The concept of non-number and number is found, and the concept of non-number and number is defined by symbols. The basic theory of number theory is solved; it is proved that {Godel's Incomplete Theorem} are incomplete. and it is also proved that the first number of natural number must be 1, not 0.

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7

Tselishchev, Vitaly V. "The truth of undecidable sentences in the perspective of Godel's first incompleteness theorem." Vestnik Tomskogo gosudarstvennogo universiteta, no. 421 (August 1, 2017): 53–58. http://dx.doi.org/10.17223/15617793/421/7.

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8

Lv, Sicheng. "Formalism in the History of Philosophical Mathematics: Beyond Logicism and Intuitionism." Highlights in Science, Engineering and Technology 88 (March 29, 2024): 522–30. http://dx.doi.org/10.54097/ywkbwm53.

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An examination of the similarities and differences between these three schools may shed light on the richness and variety of mathematical philosophies, as well as the significance of such philosophies to the comprehension and growth of mathematics. Formalism, logicism, and intuitionism are the names of these schools. The purpose of this study is to investigate the three primary schools of thought within the philosophy of mathematics, examine the many ways in which these schools interpret and approach mathematics, and investigate the areas in which these schools share and diverge in their persp

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9

Franzén, Torkel. "Transfinite Progressions: A Second Look at Completeness." Bulletin of Symbolic Logic 10, no. 3 (2004): 367–89. http://dx.doi.org/10.2178/bsl/1102022662.

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§1. Iterated Gödelian extensions of theories. The idea of iterating ad infinitum the operation of extending a theory T by adding as a new axiom a Gödel sentence for T, or equivalently a formalization of “T is consistent”, thus obtaining an infinite sequence of theories, arose naturally when Godel's incompleteness theorem first appeared, and occurs today to many non-specialists when they ponder the theorem. In the logical literature this idea has been thoroughly explored through two main approaches. One is that initiated by Turing in his “ordinal logics” (see Gandy and Yates [2001]) and taken v

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10

Enderton, H. B. "Vladimir A. Uspensky. Godel's incompleteness theorem. A reprint of LV 889 with minor corrections. Theoretical computer science, vol. 130 (1994), pp. 239–319." Journal of Symbolic Logic 60, no. 4 (1995): 1320. http://dx.doi.org/10.1017/s0022481200018132.

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11

Tselishchev, Vitaly V. "Intensionality of the Godel’s Second Incompleteness Theorem." Vestnik Tomskogo gosudarstvennogo universiteta. Filosofiya. Sotsiologiya. Politologiya, no. 40 (December 1, 2017): 98–111. http://dx.doi.org/10.17223/1998863x/40/10.

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12

Bessonov, Alexandr V. "Godel's incompleteness theorems do not disrupt Hilbert’s program." Vestnik Tomskogo gosudarstvennogo universiteta. Filosofiya. Sotsiologiya. Politologiya, no. 40 (December 1, 2017): 311–18. http://dx.doi.org/10.17223/1998863x/40/29.

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13

Raattkainen, Panu. "On the Philosophical Relevance of Godel's Incompleteness Theorems." Revue internationale de philosophie 234, no. 4 (2005): 513–34. http://dx.doi.org/10.3917/rip.234.0513.

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14

Longo, Giuseppe. "Reflections on Concrete Incompleteness." Philosophia Mathematica 19, no. 3 (2011): 255–80. https://doi.org/10.1093/philmat/nkr016.

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How do we prove true, but unprovable propositions?  Godel produced a statement whose undecidability derives from its "ad hoc" construction.  Concrete or mathematical incompleteness results, instead, are interesting unprovable statements of Formal Arithmetic.  We point out where exactly lays the unprovability along the ordinary mathematical proofs of two (very) interesting formally unprovable propositions, Kruskal-Friedman theorem on trees and Girard's Normalization Theorem in Type Theory.  Their validity is based on robust cogniti

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15

MANCOSU, PAOLO. "Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems." History and Philosophy of Logic 20, no. 1 (1999): 33–45. http://dx.doi.org/10.1080/014453499298174.

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16

Ling, Xie. "The Reason for the Incompleteness of Some Theorems in Modern Mathematics: Non Numbers Enter Mathematical Analysis or the Concept Definition Is Incomplete." European Journal of Theoretical and Applied Sciences 1, no. 3 (2023): 171–80. https://doi.org/10.59324/ejtas.2023.1(3).19.

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Abstract:

The concept of non-number and number is found, and the concept of non-number and number is defined by symbols. The basic theory of number theory is solved; it is proved that {Godel's Incomplete Theorem} are incomplete.  and it is also proved that the first number of natural number must be 1, not 0.

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17

Dokulil, Miloš. "Kurt Gödel’s Religious Worldview." Journal of Interdisciplinary Studies 32, no. 1 (2020): 95–118. http://dx.doi.org/10.5840/jis2020321/26.

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Kurt Gödel is well-known as a first-class logician-mathematician, but less well for his proof of God. Godel's Incompleteness Theorems proved that all formal axiomatic systems have inherent limitations. He created also “Gödel numbering,” a special code for writing mathematical formulae. His proof of God was presented logically on the basis of modal axioms. Gödel was sure of God’s personal influence and believed in eternal life of the human soul. He was more than only a “Baptized Lutheran” whose belief was “theistic.” Yet Gödel’s individual assurance of God’s “personal existence“ cannot be viabl

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18

Paola, Robert A. Di, and Franco Montagna. "Some properties of the syntactic p-recursion categories generated by consistent, recursively enumerable extensions of Peano arithmetic." Journal of Symbolic Logic 56, no. 2 (1991): 643–60. http://dx.doi.org/10.2307/2274707.

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The dominical categories were introduced by Di Paola and Heller, as a first step toward a category-theoretic treatment of the generalized first Godel incompleteness theorem [1]. In his Ph.D. dissertation [7], Rosolini subsequently defined the closely related p-categories, which should prove pertinent to category-theoretic representations of incompleteness for intuitionistic systems. The precise relationship between these two concepts is as follows: every dominical category is a pointed p-category, but there are p-categories, indeed pointed p-isotypes (all pairs of objects being isomorphic) wit

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19

Gupal, N. A. "Methods of Numeration of Discrete Sequences." Cybernetics and Computer Technologies, no. 2 (June 30, 2021): 63–67. http://dx.doi.org/10.34229/2707-451x.21.2.6.

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Introduction. Numeration, or code, discrete sequences act fundamental part in the theory of recognition and estimation. By the code get codes or indexes of the programs and calculated functions. It is set that the universal programs are that programs which will realize all other programs. This one of basic results in the theory of estimation. On the basis of numeration of discrete sequences of Godel proved a famous theorem about incompleteness of arithmetic. Purpose of the article. To develop synonymous numerations by the natural numbers of eventual discrete sequences programs and calculable f

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20

FUCHINO, Sakaé. "[Challenged by [challenged by [challenged by the Incompleteness Theorems]]] - a review of K. Godel, "Uber fromal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I", translated and commented by Susumu Hayashi and Mariko Yasugi (2006), / Kazuyuki Tanaka, "Challenged by Godel", (2012)." Journal of the Japan Association for Philosophy of Science 41, no. 1 (2013): 63–80. http://dx.doi.org/10.4288/kisoron.41.1_63.

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21

"GODEL'S INCOMPLETENESS THEOREM AND COMPLETENESS AXIOM." Философия науки, no. 2 (2019). http://dx.doi.org/10.15372/ps20190202.

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22

Yuhua, Fu, and Fu Anjie. "A Revision to Godel's Incompleteness Theorem by Neutrosophy." October 15, 2007. https://doi.org/10.5281/zenodo.809318.

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According to Smarandache’s neutrosophy, the Godel’s incompleteness theorem contains the truth, the falsehood, and the indeterminacy of a statement under consideration. It is shown in this paper that the proof of G¨odel’s incompleteness theorem is faulty, because all possible situations are not considered (such as the situation where from some axioms wrong results can be deducted, for example, from the axiom of choice the paradox of the doubling ball theorem can be deducted; and many kinds of indeterminate situations, for example, a proposition can be proved in 9999 cases, and only in 1 case it

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23

J., Ulisses Ferreira. "A Note on Godel's Theorem." April 27, 2022. https://doi.org/10.5281/zenodo.6496957.

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This short and informal article shows that, although Godel's theorem is valid using classical logic, there exists some four-valued logical system that is able to prove that arithmetic is both sound and complete. This article also describes a four-valued Prolog in some informal, brief and intuitive manner.

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24

"TWO FALSE DOGMAS RELATED WITH GODEL'S SECOND INCOMPLETENESS THEOREM. II." Философия науки, no. 2 (2016). http://dx.doi.org/10.15372/ps20160204.

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25

"Godel's incompleteness theorems." Choice Reviews Online 31, no. 01 (1993): 31–0356. http://dx.doi.org/10.5860/choice.31-0356.

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26

Christianto, V., and F. Smarandache. "A short remark on Gödel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective." International Journal of Neutrosophic Science, 2019, 21–26. http://dx.doi.org/10.54216/ijns.000102.

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It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as re

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27

Fu, Yuhua. "A Revision to Godel’s Incompleteness Theorem by Neutrosophy." September 5, 2008. https://doi.org/10.5281/zenodo.9572.

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According to Smarandache’s neutrosophy, the G¨odel’s incompleteness theorem contains the truth, the falsehood, and the indeterminacy of a statement under consideration. It is shown in this paper that the proof of G¨odel’s incompleteness theorem is faulty, because all possible situations are not considered (such as the situation where from some axioms wrong results can be deducted, for example, from the axiom of choice the paradox of the doubling ball theorem can be deducted; and many kinds of indeterminate situations, for example, a propositio

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28

Christianto, Victor, and Florentin Smarandache. "A short remark on Godel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective." International Journal of Neutrosophic Science 0 / 2019 (December 1, 2019). https://doi.org/10.5281/zenodo.4018910.

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It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential par

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29

Dong, Jiani. "Set Theory Revisited: Related Theorems and Realms and Their Development." Arts, Culture and Language 1, no. 7 (2024). http://dx.doi.org/10.61173/v0qqpd87.

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The concept of set dates back to the beginning of counting, and logical concepts regarding classes have existed since the tree of Porphyry, which was created in the third century CE. In light of this, it is difficult to determine where the idea of a set came from in the first place. However, sets are not collections in that this word is commonly understood, nor are they classes in the sense that logicians understood them before the middle of the 19th century. This study further presents a comprehensive trail on the development of set theory by comparing and organizing the disciplines, theories

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30

"There's something about Godel: the complete guide to the incompleteness theorem." Choice Reviews Online 47, no. 12 (2010): 47–6913. http://dx.doi.org/10.5860/choice.47-6913.

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31

Brierly, Joseph E. "Relativity Tied to Repulsion Gravity." Journal of Physics & Optics Sciences, June 30, 2021, 1–3. http://dx.doi.org/10.47363/jpsos/2021(3)137.

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This article refutes the Time Dilation Equation and Length Contraction that are derived in the Special Theory of Relativity. The conclusion reached in this article is that Time Dilation and Length Contraction cannot be characterized by simple equations due to repulsion gravity. The conclusion follows from gravity being a natural force of repulsion rather than the assumption that gravity is an attraction force. That gravity is a repulsion force follows from the Sir Arthur Eddington experiment designed to prove that gravity affects light. Few looked at that experiment as anything other than prov

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