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Dan E Willard - One of the best experts on this subject based on the ideXlab platform.
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Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles
Journal of Symbolic Logic, 2020Co-Authors: Dan E WillardAbstract:We 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 to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the “ Tangibility Reflection Principle ”. We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further.
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An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency
Journal of Symbolic Logic, 2020Co-Authors: Dan E WillardAbstract:AbstractThis article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae.Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions.
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how the law of excluded middle pertains to the second Incompleteness Theorem and its boundary case exceptions
arXiv: Logic, 2020Co-Authors: Dan E WillardAbstract:Our earlier publications showed semantic tableau admits partial exceptions to the Second Incompleteness Theorem where a formalism recognizes its self consistency and views multiplication as a 3-way relation (rather than as a total function). We now show these boundary-case evasions will collapse if the Law of the Excluded Middle is treated by tableau as a schema of logical axioms (instead of as derived Theorems).
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on the tender line separating generalizations and boundary case exceptions for the second Incompleteness Theorem under semantic tableaux deduction
Foundations of Computer Science, 2020Co-Authors: Dan E WillardAbstract:Our previous research has studied the semantic tableaux deductive methodology, of Fitting and Smullyan, and observed that it permits boundary-case exceptions to the Second Incompleteness Theorem, when multiplication is viewed as a 3-way relation (rather than as a total function). It is known that tableaux methodologies do prove a schema of Theorems, verifying all instances of the Law of the Excluded Middle. But yet we show that if one promotes this schema of Theorems into formalized logical axioms, then the meaning of the pronoun “I” in our self-referencing engine changes, and our partial evasions of the Second Incompleteness Theorem come to a complete halt.
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LFCS - On the Tender Line Separating Generalizations and Boundary-Case Exceptions for the Second Incompleteness Theorem Under Semantic Tableaux Deduction
Logical Foundations of Computer Science, 2019Co-Authors: Dan E WillardAbstract:Our previous research has studied the semantic tableaux deductive methodology, of Fitting and Smullyan, and observed that it permits boundary-case exceptions to the Second Incompleteness Theorem, when multiplication is viewed as a 3-way relation (rather than as a total function). It is known that tableaux methodologies do prove a schema of Theorems, verifying all instances of the Law of the Excluded Middle. But yet we show that if one promotes this schema of Theorems into formalized logical axioms, then the meaning of the pronoun “I” in our self-referencing engine changes, and our partial evasions of the Second Incompleteness Theorem come to a complete halt.
Albert Visser - One of the best experts on this subject based on the ideXlab platform.
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from tarski to godel or how to derive the second Incompleteness Theorem from the undefinability of truth without self reference
Journal of Logic and Computation, 2019Co-Authors: Albert VisserAbstract:In this paper, we provide a fairly general self-reference-free proof of the Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of Truth.
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From Tarski to Gödel—or how to derive the second Incompleteness Theorem from the undefinability of truth without self-reference
Journal of Logic and Computation, 2019Co-Authors: Albert VisserAbstract:In this paper, we provide a fairly general self-reference-free proof of the Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of Truth.
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from tarski to g odel or how to derive the second Incompleteness Theorem from the undefinability of truth without self reference
arXiv: Logic, 2018Co-Authors: Albert VisserAbstract:In this paper, we provide a fairly general self-reference-free proof of the Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of Truth.
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another look at the second Incompleteness Theorem
Logic group preprint series, 2017Co-Authors: Albert VisserAbstract:In this paper we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.
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the second Incompleteness Theorem and bounded interpretations
Studia Logica, 2012Co-Authors: Albert VisserAbstract:In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has `consistency power' over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n, all translations of V-sentences are U-provably equivalent to sentences of complexity less than n. We call a sequential sentence with consistency power over T a pro-consistency statement for T. We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A. We show that, if A is $${{\sf S}^{1}_{2}}$$ , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA. The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing.
Victor Christianto - One of the best experts on this subject based on the ideXlab platform.
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ON GÖDEL'S Incompleteness Theorem(S), ARTIFICIAL INTELLIGENCE/LIFE, AND HUMAN MIND
viXra, 2015Co-Authors: Victor Christianto, Florentin SmarandacheAbstract:In the present paper we have discussed concerning Godel’s Incompleteness Theorem(s) and plausible implications to artificial intelligence/life and human mind. Perhaps we should agree with Sullins III, that the value of this finding is not to discourage certain types of research in AL, but rather to help move us in a direction where we can more clearly define the results of that research. Godel’s Incompleteness Theorems have their own limitations, but so do Artificial Life (AL)/AI systems. Based on our experiences so far, human mind has incredible abilities to interact with other part of human body including heart, which makes it so difficult to simulate in AI/AL. However, it remains an open question to predict whether the future of AI including robotics science can bring this gap closer or not. In this regard, fuzzy logic and its generalization –neutrosophic logicoffer a way to improve significantly AI/AL research.[15]
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on godel s Incompleteness Theorem s artificial intelligence life and human mind
viXra, 2015Co-Authors: Victor Christianto, Florentin SmarandacheAbstract:In the present paper we have discussed concerning Godel’s Incompleteness Theorem(s) and plausible implications to artificial intelligence/life and human mind. Perhaps we should agree with Sullins III, that the value of this finding is not to discourage certain types of research in AL, but rather to help move us in a direction where we can more clearly define the results of that research. Godel’s Incompleteness Theorems have their own limitations, but so do Artificial Life (AL)/AI systems. Based on our experiences so far, human mind has incredible abilities to interact with other part of human body including heart, which makes it so difficult to simulate in AI/AL. However, it remains an open question to predict whether the future of AI including robotics science can bring this gap closer or not. In this regard, fuzzy logic and its generalization –neutrosophic logic- offer a way to improve significantly AI/AL research.[15]
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is it possible to apply godel s Incompleteness Theorem to scientific theories
viXra, 2014Co-Authors: Victor ChristiantoAbstract:Godel's Incompleteness Theorem is normally applied to mathematics. But i just found an article by Michael Goodband who argues that GIT can also be applied to scientific theories, see http://www.mjgoodband.co.uk/papers/Godel-science-theory.pdf. My own idea can be expressed generally as follows: any theory boils down to an exposition of a statement/proposition. According to GIT, in any theory there is at least one statement which is unprovable, and therefore any theory can be considered as incomplete or has indeterminacy. One implication of this deduction is that any theory should be made falsifiable (Popper), and also perhaps we can use conditional Bayesian probability to describe acceptance of a theory.
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On Gödel's Incompleteness Theorem, Artificial Intelligence & Human Mind
Scientific GOD Journal, 2013Co-Authors: Victor ChristiantoAbstract:In this essay, I discuss Godel’s Incompleteness Theorem and plausible implications to Artificial Intelligence/Life and human mind. Perhaps we should agree with Sullins III, that the value of this finding is not to discourage certain types of research in AL, but rather to help move us in a direction where we can more clearly define the results of that research. Godel’s Incompleteness Theorem has its own limitations, but so do Artificial Life systems. Based on our experiences, human mind has incredible abilities to interact with other part of human body including heart, which makes it difficult to simulate in AI/AL systems. However, it remains an open question to predict whether the future of AI including robotics science can bring this gap closer or not.
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on godel s Incompleteness Theorem artificial intelligence human mind
Scientific GOD Journal, 2013Co-Authors: Victor ChristiantoAbstract:In this essay, I discuss Godel’s Incompleteness Theorem and plausible implications to Artificial Intelligence/Life and human mind. Perhaps we should agree with Sullins III, that the value of this finding is not to discourage certain types of research in AL, but rather to help move us in a direction where we can more clearly define the results of that research. Godel’s Incompleteness Theorem has its own limitations, but so do Artificial Life systems. Based on our experiences, human mind has incredible abilities to interact with other part of human body including heart, which makes it difficult to simulate in AI/AL systems. However, it remains an open question to predict whether the future of AI including robotics science can bring this gap closer or not.
Payam Seraji - One of the best experts on this subject based on the ideXlab platform.
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godel s second Incompleteness Theorem for σn definable theories
Logic Journal of the IGPL, 2018Co-Authors: Conden Chao, Payam SerajiAbstract:Godel's second Incompleteness Theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA isn+1-definable and T isn-sound, then T dose not prove the sentencen-Sound(T) that expresses then-soundness of T. The optimal- ity of the generalization is shown by presenting an+1-definable (indeed a complete �n+1-definable) andn−1-sound theory T such that PA ⊆ T andn−1-Sound(T) is provable in T. It is also proved that no recursively enumerable and �1-sound theory of arithmetic, even very weak theories which do not contain Robinson's Arithmetic, can prove its own �1-soundness.
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Gödel’s second Incompleteness Theorem for Σn-definable theories
Logic Journal of the IGPL, 2018Co-Authors: Conden Chao, Payam SerajiAbstract:Godel's second Incompleteness Theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA isn+1-definable and T isn-sound, then T dose not prove the sentencen-Sound(T) that expresses then-soundness of T. The optimal- ity of the generalization is shown by presenting an+1-definable (indeed a complete �n+1-definable) andn−1-sound theory T such that PA ⊆ T andn−1-Sound(T) is provable in T. It is also proved that no recursively enumerable and �1-sound theory of arithmetic, even very weak theories which do not contain Robinson's Arithmetic, can prove its own �1-soundness.
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constructibility and rosserizability of the proofs of boolos and chaitin for godel s Incompleteness Theorem
arXiv: Logic, 2016Co-Authors: Saeed Salehi, Payam SerajiAbstract:The proofs of Chaitin and Boolos for Godel's Incompleteness Theorem are studied from the perspectives of constructibility and Rosserizability. By Rosserization of a proof we mean that the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. It is known that Godel's own proof for his Incompleteness Theorem is not Rosserizable, and we show that neither are Kleene's or Boolos' proofs. However, we prove a Rosserized version of Chaitin's (Incompleteness) Theorem. The proofs of Godel, Rosser and Kleene are constructive in the sense that they explicitly construct, by algorithmic ways, the independent sentence(s) from the theory. We show that the proofs of Chaitin and Boolos are not constructive, and they prove only the mere existence of the independent sentences.
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Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories
Journal of Logic and Computation, 2016Co-Authors: Saeed Salehi, Payam SerajiAbstract:Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $\Pi_n$-sentences or equivalently the $\Sigma_n$-soundness of the theory, and the other is $n$-consistency the restriction of $\omega$-consistency to the $\Sigma_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the Incompleteness Theorem using the $\Sigma_n$-soundness assumption is constructive, it is shown that there is no constructive proof for the Incompleteness Theorem using the $n$-consistency assumption, for $n\!>\!2$.
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godel rosser s Incompleteness Theorem generalized and optimized for definable theories
Journal of Logic and Computation, 2016Co-Authors: Saeed Salehi, Payam SerajiAbstract:Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $\Pi_n$-sentences or equivalently the $\Sigma_n$-soundness of the theory, and the other is $n$-consistency the restriction of $\omega$-consistency to the $\Sigma_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the Incompleteness Theorem using the $\Sigma_n$-soundness assumption is constructive, it is shown that there is no constructive proof for the Incompleteness Theorem using the $n$-consistency assumption, for $n\!>\!2$.
Song Taiping - One of the best experts on this subject based on the ideXlab platform.
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study on the logical basis of modern physics from zeno paradox and gdel Incompleteness Theorem
Journal of Nanyang Normal University, 2008Co-Authors: Song TaipingAbstract:Starting from Zeno paradox and Gdel Incompleteness Theorem,this paper discusses the basic features of Newton's mechanics,furthermore analyses in detail the logical bases of modern physics such as theory of relativity,quantum mechanics,standard model of particle physics,grand unified theory and superstring theory,and points out the inherent dynamism and development trends of modern science.
