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Informal Physical Reasoning Processes [article]

Kurt Ammon
<span title="2016-08-15">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The existence principle and productive functions imply the existence of physical systems whose reasoning processes cannot be modeled by Turing machines. These systems are called creative.  ...  We introduce an existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the application of any Turing-computable function to  ...  The first part of Gödel's (first incompleteness) theorem states that the undecidable formula A p (p) is unprovable in the formal system if the system is consistent (see Kleene [1952, p. 207, Theorem 28  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1608.04672v1">arXiv:1608.04672v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/voldkrys7bbbphfgmrwvr7uvje">fatcat:voldkrys7bbbphfgmrwvr7uvje</a> </span>
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Gödel's incompleteness theorem and the Anti-Mechanist Argument: revisited [article]

Yong Cheng
<span title="2020-08-05">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In Section 3, we give some additional explanations to Section 4-6 in Krajewski's [30] and classify some misunderstandings of Gödel's incompleteness theorem related to Anti-Mechanist Arguments.  ...  In Section 4 and 5, we give a more detailed discussion of Gödel's Disjunctive Thesis, Gödel's Undemonstrability of Consistency Thesis and the definability of natural numbers as in Section 7-8 in Krajewski's  ...  The following theorem is a modern reformulation of Gödel's first incompleteness theorem. Theorem 3.1 (Gödel's first incompleteness theorem (G1)).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1902.05902v2">arXiv:1902.05902v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tkpvmdpmyrhyjjwha6jvseoeta">fatcat:tkpvmdpmyrhyjjwha6jvseoeta</a> </span>
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Implications of Gödel's Incompleteness Theorem on A.I. vs. Mind

Fatih Gelgi
<span title="2007-09-05">2007</span> <i title="NeuroQuantology Journal"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vfjjshkdo5eynaqtxbx53v4cy4" style="color: black;">NeuroQuantology</a> </i> &nbsp;
The mathematician was Kurt Gödel, and the result proved in his paper became known as the Gödel Incompleteness Theorem, or more simply GödelÊs Theorem·although it was by no means the only major theorem  ...  He is also known as one of the inventors of the theory of recursive functions which formed part of the foundation for computers.  ...  E-mail: Fatih.Gelgi@asu.edu What is Gödel's Incompleteness Theor What is Gödel's Incompleteness Theor What is Gödel's Incompleteness Theor What is Gödel's Incompleteness Theorem? em? em?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14704/nq.2004.2.3.46">doi:10.14704/nq.2004.2.3.46</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vjrdjrqswjbpvj7qtsizxy6at4">fatcat:vjrdjrqswjbpvj7qtsizxy6at4</a> </span>
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Godel on the mathematician's mind and Turing Machine

Inês Hipólito
<span title="2014-06-01">2014</span> <i title="University of Economics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nr66ozjrszedhiopm46dstkjea" style="color: black;">E-LOGOS</a> </i> &nbsp;
However, (4) Gödel's first incompleteness theorem shows that F' could not see that the Gödel sentence is true.  ...  Godel's first incompleteness theorem (1931) proves that any consistent formal system in which a "moderate amount of number theory" can be proven will be incomplete, that is, there will be at least one  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.18267/j.e-logos.408">doi:10.18267/j.e-logos.408</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z4p3ftaoevhyvedvs6cxgz66yq">fatcat:z4p3ftaoevhyvedvs6cxgz66yq</a> </span>
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Gödel for Goldilocks: A Rigorous, Streamlined Proof of (a variant of) Gödel's First Incompleteness Theorem [article]

Dan Gusfield
<span title="2014-11-19">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Such an exposition is possible, because many people have realized that variants of G\"odel's first incompleteness theorem can be rigorously proved by a simpler middle approach, avoiding philosophical discussions  ...  In this exposition we give a short, self-contained Goldilocks exposition of G\"odel's first theorem, aimed at a broad, undergraduate audience.  ...  Back to Gödel How do we connect all this to Gödel's first incompleteness theorem?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1409.5944v3">arXiv:1409.5944v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e72bdbsxkrbntorno6vqk77yvq">fatcat:e72bdbsxkrbntorno6vqk77yvq</a> </span>
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A Theorem about Computationalism and "Absolute" Truth

Arthur Charlesworth
<span title="2016-01-23">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rplregtfrvbuppbjjbn5xwapiy" style="color: black;">Minds and Machines</a> </i> &nbsp;
We investigate how that weakness and some of the other problematic aspects of Gödel's argument can be eliminated or reduced.  ...  This article focuses on issues related to improving an argument about minds and machines given by Kurt Gödel in 1951, in a prominent lecture.  ...  (number codes of) theorems can be checked by a Turing machine, if S is consistent, then S is incomplete.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11023-015-9382-x">doi:10.1007/s11023-015-9382-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p5iqpuwp5faohkv5kxvzsq6l5q">fatcat:p5iqpuwp5faohkv5kxvzsq6l5q</a> </span>
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Gödel's incompleteness theorems, free will and mathematical thought [chapter]

SOLOMON FEFERMAN
<span title="2011-12-01">2011</span> <i title="British Academy"> Free Will and Modern Science </i> &nbsp;
Following an explanation of the incompleteness theorems and their relationship with Turing machines, we here concentrate on the arguments of Gödel (with some caveats) and Lucas among others for such claims  ...  Some have claimed that Gödel's incompleteness theorems on the formal axiomatic model of mathematical thought can be used to demonstrate that mind is not mechanical, in opposition to a Formalist-Mechanist  ...  This is what is called Gödel's second incompleteness theorem or his theorem on the unprovability of consistency.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5871/bacad/9780197264898.003.0007">doi:10.5871/bacad/9780197264898.003.0007</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6tz5s6axenadpj6ndb3ra53j6i">fatcat:6tz5s6axenadpj6ndb3ra53j6i</a> </span>
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A Simple Character String Proof of the "True but Unprovable'' Version of Gödel's First Incompleteness Theorem

Antti Valmari
<span title="2014-05-21">2014</span> <i title="Open Publishing Association"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qzezkwolyvf33pglm2jldjklca" style="color: black;">Electronic Proceedings in Theoretical Computer Science</a> </i> &nbsp;
A rather easy yet rigorous proof of a version of G\"odel's first incompleteness theorem is presented.  ...  Proof systems are expressed via Turing machines that halt if and only if their input string is a theorem.  ...  Incompleteness Theorem AFL '14 2014-05-29 12/14 Today Gödel's and Rosser's formulations are somewhat forgotten, because AV Character String proof of First Incompleteness Theorem AFL '14 2014  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4204/eptcs.151.25">doi:10.4204/eptcs.151.25</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h3srdyqcnnfvxj4rhvml74dgmu">fatcat:h3srdyqcnnfvxj4rhvml74dgmu</a> </span>
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Kurt Gödel and Computability Theory [chapter]

Richard Zach
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In particular, Gödel's 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church  ...  Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s.  ...  Church, then, turned out to be right: Gödel's second incompleteness theorem does not apply to his system-because the theorem only applies to consistent formal systems.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11780342_59">doi:10.1007/11780342_59</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mi2qey4monflhagqyijp3wqjfi">fatcat:mi2qey4monflhagqyijp3wqjfi</a> </span>
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On the depth of Gödel's incompleteness theorem [article]

Yong Cheng
<span title="2021-03-21">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We take for granted the widespread judgment by mathematical logicians that Gödel's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in.  ...  We focus on the methodological study of the depth of Gödel's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity.  ...  Acknowledgements I would like to thank the referees for providing detailed and helpful comments for improvements.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2008.13142v2">arXiv:2008.13142v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fcwyssrobfe5tath5lxzmb2b4e">fatcat:fcwyssrobfe5tath5lxzmb2b4e</a> </span>
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Dimensional Complexity and Algorithmic Efficiency [article]

Alexander Ngu
<span title="2022-01-17">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Intelligence is defined informally, formally, and computationally.  ...  We introduce the concept of Dimensional complexity in algorithmic efficiency and deduce that an optimally efficient algorithm has zero Time complexity, zero Space complexity, and an infinite Dimensional  ...  In the paper, Turing used computation to con rm the same relationship demonstrated by Gödel's incompleteness theorems.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.05050v2">arXiv:2201.05050v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bdjizntv7vgmdnyyajjogwydy4">fatcat:bdjizntv7vgmdnyyajjogwydy4</a> </span>
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A Note on Interaction and Incompleteness

D. Bojadziev
<span title="2003-09-01">2003</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mwzmwwmscfe4vnzrjlt5itd2ju" style="color: black;">Logic Journal of the IGPL</a> </i> &nbsp;
The notion of interaction and interaction machines, developed by Peter Wegner, includes the comparison between incompleteness of interaction machines and Gödel incompleteness.  ...  However, this comparison is not adequate, because it combines different notions and different sources of incompleteness.  ...  theorem can be proved from Gödel's (first) incompleteness theorem, by way of two other theorems).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/jigpal/11.5.513">doi:10.1093/jigpal/11.5.513</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4f4mrktdi5gojp3bthlktjfifa">fatcat:4f4mrktdi5gojp3bthlktjfifa</a> </span>
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Use and Misuse of G^|^ouml;del's Theorem

Shingo FUJITA
<span title="">2003</span> <i title="The Japan Association for Philosophy of Science"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/k7izqsbokbe2dof3is2sa2wapy" style="color: black;">Annals of the Japan Association for Philosophy of Science</a> </i> &nbsp;
therefore consistent, contrary to the second incompleteness theorem.  ...  As is well known, Godel's theorem states that if the formal system is consistent, then there always exist such and such a formula...... etc.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4288/jafpos1956.12.1">doi:10.4288/jafpos1956.12.1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xh35aamakrawzfnhgq3y5cnrim">fatcat:xh35aamakrawzfnhgq3y5cnrim</a> </span>
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Hilbert's Incompleteness, Chaitin's Ω number and Quantum Physics [article]

Tien D Kieu
<span title="2001-11-21">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of the computation of the Chaitin's Ω number, which is a representation of the Gödel's Incompletness theorem.  ...  quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth problem and  ...  Acknowledgements I am indebted to Alan Head for discussions, comments and suggestions; and would like to thank Cristian Calude for informing me the recent works on the Ω and quantum Ω numbers.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/quant-ph/0111062v2">arXiv:quant-ph/0111062v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mlafzsswubb4lhzjglwlqv4v3m">fatcat:mlafzsswubb4lhzjglwlqv4v3m</a> </span>
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Do Gödel's Incompleteness Theorems Set Absolute Limits on the Ability of the Brain to Express and Communicate Mental Concepts Verifiably?

Bhupinder Singh Anand
<span title="2007-09-05">2007</span> <i title="NeuroQuantology Journal"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vfjjshkdo5eynaqtxbx53v4cy4" style="color: black;">NeuroQuantology</a> </i> &nbsp;
Classical interpretations of Gödel's formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical  ...  We consider a constructive interpretation of classical, Tarskian, truth, and of Gödel's reasoning, under which any formal system of Peano Arithmetic -classically accepted as the foundation of all our mathematical  ...  Gödel's Theorem VI (the famous First Incompleteness Theorem of his famous 1931 paper (Gödel, 1931a) ) holds vacuously.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14704/nq.2004.2.2.39">doi:10.14704/nq.2004.2.2.39</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z6r3fnl45ba3fpxk4oj5xl26rm">fatcat:z6r3fnl45ba3fpxk4oj5xl26rm</a> </span>
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