Search results for 'Goedel s incompleteness result' (try it on Scholar)

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  1. H. Gaifman (2000). What Godel's Incompleteness Result Does and Does Not Show. Journal of Philosophy 97 (8):462-471.
    In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. (...)
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  2.  36
    Haim Gaifman (2000). What Gödel's Incompleteness Result Does and Does Not Show. Journal of Philosophy 97 (8):462 - 470.
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  3. Geoffrey Hellman (1981). How to Goedel a Frege-Russell: Goedel's Incompleteness Theorem. Noûs 15:451-68.
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  4. Haim Gaifman (2000). What Godel's Incompleteness Result Does and Does Not Show. Journal of Philosophy 97 (8):462.
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  5. Michael Detlefsen (1976). The Importance of Goedel's Second Incompleteness Theorem for the Foundations of Mathematics. Dissertation, The Johns Hopkins University
     
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  6. Nicolás F. Lori & Alex H. Blin (2010). Application of Quantum Darwinism to Cosmic Inflation: An Example of the Limits Imposed in Aristotelian Logic by Information-Based Approach to Gödel's Incompleteness. [REVIEW] Foundations of Science 15 (2):199-211.
    Gödel’s incompleteness applies to any system with recursively enumerable axioms and rules of inference. Chaitin’s approach to Gödel’s incompleteness relates the incompleteness to the amount of information contained in the axioms. Zurek’s quantum Darwinism attempts the physical description of the universe using information as one of its major components. The capacity of quantum Darwinism to describe quantum measurement in great detail without requiring ad-hoc non-unitary evolution makes it a good candidate for describing the transition from quantum to (...)
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  7.  63
    C. Chihara (1972). On Alleged Refutations of Mechanism Using Godel's Incompleteness Results. Journal of Philosophy 69 (September):507-26.
  8.  51
    Charles Sayward (2001). On Some Much Maligned Remarks of Wittgenstein on Gödel. Philosophical Investigations 24 (3):262–270.
    In "Remarks on the Foundations of Mathematics" Wittgenstein discusses an argument that goes from Gödel’s incompleteness result to the conclusion that some truths of mathematics are unprovable. Wittgenstein takes issue with this argument. Wittgenstein’s remarks in this connection have received very negative reaction from some very prominent people, for example, Gödel and Dummett. The paper is a defense of what Wittgenstein has to say about the argument in question.
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  9.  8
    Dan E. Willard (2002). How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem Almost to Robinson's Arithmetic Q. Journal of Symbolic Logic 67 (1):465-496.
    Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π 1 sentence, valid in the standard model of the Natural Numbers and denoted as V, such that if α is any finite consistent extension of Q (...)
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  10.  7
    Timothy McCarthy (2016). Gödel's Third Incompleteness Theorem. Dialectica 70 (1):87-112.
    In a note appended to the translation of “On consistency and completeness” (), Gödel reexamined the problem of the unprovability of consistency. Gödel here focuses on an alternative means of expressing the consistency of a formal system, in terms of what would now be called a ‘reflection principle’, roughly, the assertion that a formula of a certain class is provable in the system only if it is true. Gödel suggests that it is this alternative means of expressing consistency that we (...)
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  11.  42
    Roman Murawski (1997). Gödel's Incompleteness Theorems and Computer Science. Foundations of Science 2 (1):123-135.
    In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.
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  12.  67
    Aaron Sloman (1992). The Emperor's Real Mind -- Review of Roger Penrose's The Emperor's New Mind: Concerning Computers Minds and the Laws of Physics. Artificial Intelligence 56 (2-3):355-396.
    "The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...)
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  13.  17
    Richard Zach (2005). Book Review: Michael Potter. Reason's Nearest Kin. Philosophies of Arithmetic From Kant to Carnap. [REVIEW] Notre Dame Journal of Formal Logic 46 (4):503-513.
  14.  72
    Robert F. Hadley (2008). Consistency, Turing Computability and Gödel's First Incompleteness Theorem. Minds and Machines 18 (1):1-15.
    It is well understood and appreciated that Gödel’s Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a gödel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined with (...)
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  15.  13
    Makoto Kikuchi, Taishi Kurahashi & Hiroshi Sakai (2012). On Proofs of the Incompleteness Theorems Based on Berry's Paradox by Vopěnka, Chaitin, and Boolos. Mathematical Logic Quarterly 58 (4‐5):307-316.
    By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We (...)
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  16. Francesco Berto (2009). There's Something About Gödel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.
    The Gödelian symphony -- Foundations and paradoxes -- This sentence is false -- The liar and Gödel -- Language and metalanguage -- The axiomatic method or how to get the non-obvious out of the obvious -- Peano's axioms -- And the unsatisfied logicists, Frege and Russell -- Bits of set theory -- The abstraction principle -- Bytes of set theory -- Properties, relations, functions, that is, sets again -- Calculating, computing, enumerating, that is, the notion of algorithm -- Taking numbers (...)
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  17.  92
    Michael Detlefsen (1990). On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem. Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  18. Raymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.
    Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness (...)
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  19.  51
    Cezary Cieśliński & Rafal Urbaniak (2013). Gödelizing the Yablo Sequence. Journal of Philosophical Logic 42 (5):679-695.
    We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo (...)
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  20.  21
    Newton C. A. Da Costa (2012). Gödel's Incompleteness Theorems and Physics. Principia 15 (3):453-459.
    This paper is a summary of a lecture in which I presented some remarks on Gödel’s incompleteness theorems and their meaning for the foundations of physics. The entire lecture will appear elsewhere. doi: http://dx.doi.org/ 10.5007 / 1808-1711.2011v15n3p453.
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  21.  81
    Panu Raatikainen (1998). On Interpreting Chaitin's Incompleteness Theorem. Journal of Philosophical Logic 27 (6):569-586.
    The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure (...)
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  22.  36
    Stathis Livadas (2013). Are Mathematical Theories Reducible to Non-Analytic Foundations? Axiomathes 23 (1):109-135.
    In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and (...)
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  23. Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
    Gödel began his 1951 Gibbs Lecture by stating: “Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics.” (Gödel 1951) Gödel is referring here especially to his own incompleteness theorems (Gödel 1931). Gödel’s first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized (...)
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  24.  59
    Paolo Mancosu (1999). Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems. History and Philosophy of Logic 20 (1):33-45.
    What were the earliest reactions to Gödel's incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Gödel's groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.
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  25.  84
    Yi-Zhuang Chen (2004). Edgar Morin's Paradigm of Complexity and Gödel's Incompleteness Theorem. World Futures 60 (5 & 6):421 – 431.
    This article shows that in two respects, Gödel's incompleteness theorem strongly supports the arguments of Edgar Morin's complexity paradigm. First, from the viewpoint of the content of Gödel's theorem, the latter justifies the basic view of complexity paradigm according to which knowledge is a dynamic, unfinished process, and develops by way of self-criticism and self-transcendence. Second, from the viewpoint of the proof procedure of Gödel's theorem, the latter confirms the complexity paradigm's circular line of inference through which is formed (...)
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  26.  64
    Haim Gaifman, Gödel's Incompleteness Results.
    This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observations on circularity and draw brief comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motives for it are made obvious. We suggest this as a (...)
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  27.  53
    Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.
    In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis (...)
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  28.  26
    John W. Dawson (1984). The Reception of Godel's Incompleteness Theorems. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1984:253 - 271.
    According to several commentators, Kurt Godel's incompleteness discoveries were assimilated promptly and almost without objection by his contemporaries - - a circumstance remarkable enough to call for explanation. Careful examination reveals, however, that there were doubters and critics, as well as defenders and rival claimants to priority. In particular, the reactions of Carnap, Bernays, Zermelo, Post, Finsler, and Russell, among others, are considered in detail. Documentary sources include unpublished correspondence from Godel's Nachlass.
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  29.  54
    Gregory Wheeler (2012). Explaining the Limits of Olsson's Impossibility Result. Southern Journal of Philosophy 50 (1):136-150.
    In his groundbreaking book, Against Coherence (2005), Erik Olsson presents an ingenious impossibility theorem that appears to show that there is no informative relationship between probabilistic measures of coherence and higher likelihood of truth. Although Olsson's result provides an important insight into probabilistic models of epistemological coherence, the scope of his negative result is more limited than generally appreciated. The key issue is the role conditional independence conditions play within the witness testimony model Olsson uses to establish his (...)
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  30. Glen Hoffmann (2007). The Semantic Theory of Truth: Field's Incompleteness Objection. Philosophia 35 (2):161-170.
    According to Field’s influential incompleteness objection, Tarski’s semantic theory of truth is unsatisfactory since the definition that forms its basis is incomplete in two distinct senses: (1) it is physicalistically inadequate, and for this reason, (2) it is conceptually deficient. In this paper, I defend the semantic theory of truth against the incompleteness objection by conceding (1) but rejecting (2). After arguing that Davidson and McDowell’s reply to the incompleteness objection fails to pass muster, I argue that, (...)
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  31.  45
    Panu Raatikainen, Gödel's Incompleteness Theorems. The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (Ed.).
    Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a (...)
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  32. Christopher Gauker (2001). T-Schema Deflationism Versus Gödel’s First Incompleteness Theorem. Analysis 61 (270):129–136.
    I define T-schema deflationism as the thesis that a theory of truth for our language can simply take the form of certain instances of Tarski's schema (T). I show that any effective enumeration of these instances will yield as a dividend an effective enumeration of all truths of our language. But that contradicts Gödel's First Incompleteness Theorem. So the instances of (T) constituting the T-Schema deflationist's theory of truth are not effectively enumerable, which casts doubt on the idea that (...)
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  33.  28
    Bernd Buldt (2014). The Scope of Gödel’s First Incompleteness Theorem. Logica Universalis 8 (3-4):499-552.
    Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.
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  34. William E. Seager (2003). Yesterday's Algorithm: Penrose and the Godel Argument. Croatian Journal of Philosophy 3 (9):265-273.
    Roger Penrose is justly famous for his work in physics and mathematics but he is _notorious_ for his endorsement of the Gödel argument (see his 1989, 1994, 1997). This argument, first advanced by J. R. Lucas (in 1961), attempts to show that Gödel’s (first) incompleteness theorem can be seen to reveal that the human mind transcends all algorithmic models of it1. Penrose's version of the argument has been seen to fall victim to the original objections raised against Lucas (see (...)
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  35.  23
    David Coder (1969). Godel's Theorem and Mechanism. Philosophy 44 (September):234-7.
    In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is (...)
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  36.  18
    Bernd Buldt, Gödel’s First Incompleteness Theorem.
    Slides for the second tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.
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  37.  17
    Bernd Buldt, Gödel’s Second Incompleteness Theorem.
    Slides for the third tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.
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  38.  35
    Nicholas Harrigan & Robert W. Spekkens (2010). Einstein, Incompleteness, and the Epistemic View of Quantum States. Foundations of Physics 40 (2):125-157.
    Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something (...)
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  39. Carsten Held (2015). Einstein’s Boxes: Incompleteness of Quantum Mechanics Without a Separation Principle. Foundations of Physics 45 (9):1002-1018.
    Einstein made several attempts to argue for the incompleteness of quantum mechanics, not all of them using a separation principle. One unpublished example, the box parable, has received increased attention in the recent literature. Though the example is tailor-made for applying a separation principle and Einstein indeed applies one, he begins his discussion without it. An analysis of this first part of the parable naturally leads to an argument for incompleteness not involving a separation principle. I discuss the (...)
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  40. Gregory J. Chaitin (1970). Computational Complexity and Godel's Incompleteness Theorem. [Rio De Janeiro,Centro Técnico Científico, Pontifícia Universidade Católica Do Rio De Janeiro.
     
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  41. Harvey Friedman, Fromal Statements of Godel's Second Incompleteness Theorem.
    Informal statements of Gödel's Second Incompleteness Theorem, referred to here as Informal Second Incompleteness, are simple and dramatic. However, current versions of Formal Second Incompleteness are complicated and awkward. We present new versions of Formal Second Incompleteness that are simple, and informally imply Informal Second Incompleteness. These results rest on the isolation of simple formal properties shared by consistency statements. Here we do not address any issues concerning proofs of Second Incompleteness.
     
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  42.  3
    Marcelo E. Coniglio & Newton M. Peron (2014). Dugundji’s Theorem Revisited. Logica Universalis 8 (3-4):407-422.
    In 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness showed that not every (...)
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  43.  53
    Robert Rynasiewicz (2001). Definition, Convention, and Simultaneity: Malament's Result and its Alleged Refutation by Sarkar and Stachel. Proceedings of the Philosophy of Science Association 2001 (3):S345-.
    The question whether distant simultaneity (relativized to an inertial frame) has a factual or a conventional status in special relativity has long been disputed and remains in contention even today. At one point it appeared that Malament (1977) had settled the issue by proving that the only non-trivial equivalence relation definable from (temporally symmetric) causal connectability is the standard simultaneity relation. Recently, though, Sarkar and Stachel (1999) claim to have identified a suspect assumption in the proof by defining a non-standard (...)
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  44.  6
    John D. Norton, Little Boxes: The Simplest Demonstration of the Failure of Einstein’s Attempt to Show the Incompleteness of Quantum Theory.
    The failure of Einstein’s co-authored “EPR” attempt to show the incompleteness of quantum theory is demonstrated directly for spatial degrees of freedom using only elementary notions. A GHZ construction is realized in the position properties of three particles whose quantum waves are distributed over three two-chambered boxes. The same system is modeled more realistically using three spatially separated, singly ionized hydrogen molecules.
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  45.  14
    Daniele Mundici (1981). An Algebraic Result About Soft Model Theoretical Equivalence Relations with an Application to H. Friedman's Fourth Problem. Journal of Symbolic Logic 46 (3):523-530.
    We prove the following algebraic characterization of elementary equivalence: $\equiv$ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if L = L ωω (Q i ) i ∈ ω 1 is an (ω (...)
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  46.  11
    Frank Poletti (2005). Translator's Introduction to the Incompleteness of Each Tradition: Toward an Ethic of Complexity. World Futures 61 (4):287 – 290.
    (2005). Translator's Introduction to The Incompleteness of Each Tradition: Toward an Ethic of Complexity. World Futures: Vol. 61, No. 4, pp. 287-290.
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  47.  3
    Dirk Ullrich (2014). A General Principle for Purely Model-Theoretical Proofs of Gödel’s Second Incompleteness Theorem. Logic and Logical Philosophy 6:173.
    By generalizing Kreisel’s proof of the Second Incompleteness Theorem of G¨odel I extract a general principle which can also be used for otherpurely model-theoretical proofs of that theorem.
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  48. Francesco Berto (2009). There's Something About Gdel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.
    Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical (...)
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  49. Francesco Berto (2011). There's Something About Gdel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.
    Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical (...)
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  50. Francesco Berto (2011). There's Something About Gdel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.
    Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical (...)
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