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Godel's incompleteness theorems

1993 ChoiceReviews  
The work of Boole has resulted in an important concept of Boolean algebra, which is discussed in great length in a series of historical projects [3, 2, 1], also available on our webpage.  ...  1 We recall that a theory is consistent if it does not prove contradiction. 2 More details on the work of Boole, Frege, and Russell and Whitehead can be found on our webpage http://www.cs.nmsu.edu/historical-projects  ...  . 3 An important event of those years of Gödel's life was attending Hilbert's lecture in Bologna on completeness and consistency of mathematical systems.  ... 
doi:10.5860/choice.31-0356 fatcat:s5exiizvoza5dd7gxu3j64moe4

Computer Science and Metaphysics: A Cross-Fertilization

Daniel Kirchner, Christoph Benzmüller, Edward N. Zalta
2019 Open Philosophy  
The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.  ...  , to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science.  ...  Inconsistency and other results about Gödel's argument In the course of experiments,30 the theorem prover Leo-II detected that the unedited version of Gödel's formulation of the argument31 was inconsistent  ... 
doi:10.1515/opphil-2019-0015 fatcat:wljwgnhj2zffxakqpghk4xdxzm

Formally speaking

John L. Casti
2001 Nature  
This fact constitutes an algorithmic complexity version of the unsolvability of the halting problem, a computer science version of Gödel's incompleteness theorem.  ...  In other words, Gödel's incompleteness theorem shows that there is an irreducible semantic component to mathematics.  ... 
doi:10.1038/35079200 pmid:11385544 fatcat:c4nuvospavh2jncg3enhjptfem

Computer Science and Metaphysics: A Cross-Fertilization [article]

Daniel Kirchner, Christoph Benzmüller, Edward N. Zalta
2019 arXiv   pre-print
The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.  ...  , to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science.  ...  Inconsistency and Other Results about Gödel's Argument In the course of experiments, 30 the theorem prover Leo-II detected that the unedited version of Gödel's formulation of the argument 31 was inconsistent  ... 
arXiv:1905.00787v3 fatcat:slgk2xqljva7jdgapn6nzkaile

My route to arithmetization

SOLOMON FEFERMAN
2008 Theoria  
I wish to thank Anita Burdman Feferman and Karl-Georg Niebergall for their helpful comments on a draft of this paper.  ...  The request was to review a paper by Hao Wang for The Journal of Symbolic Logic on an arithmetized version of Gödel's completeness theorem. 2 It was my work on this review that led me straight into questions  ...  involving Con T such as Gödel's second incompleteness theorem and the Hilbert-Bernays-Wang arithmetized completeness theorem?  ... 
doi:10.1111/j.1755-2567.1997.tb00746.x fatcat:iy2ps3rdxncyxkj6uv273c52ce

The Gödel Editorial Project: A Synopsis

Solomon Feferman
2005 Bulletin of Symbolic Logic  
initiatives following Gödel's death in 1978 to its completion last year.  ...  The final two volumes, numbers IV and V, of the Oxford University Press edition of the Collected Works of Kurt Gödel [3]–[7] appeared in 2003, thus completing a project that started over twenty years earlier  ...  We decided to include the Vienna dissertation along with its 1930 published version because the former begins with a quite interesting discussion of the significance of the completeness theorem and the  ... 
doi:10.2178/bsl/1120231626 fatcat:mumuiuxwyze5vakpshjsg677tq

Some facts about Kurt Gödel

Hao Wang
1981 Journal of Symbolic Logic (JSL)  
At about this time he read the first edition of Hilbert-Ackermann (1928) in which the completeness of the (restricted) predicate calculus was formulated and posed as an open problem.  ...  He enjoyed much the lectures by Furtwangler on number theory and developed an interest in this subject which was, for example, relevant to his application of the Chinese remainder theorem in expressing  ...  Shortly afterwards Godel received a letter from von Neumann suggesting the theorem on consistency proofs as a consequence of Godel's original result.  ... 
doi:10.2307/2273764 fatcat:r5ion5m3prbxfgbt3y2eeevpkm

Incompleteness: The proof and paradox of kurt gödel

Rebecca Goldstein
2006 The Mathematical intelligencer  
For his part, Gödel's three fundamental results were the completeness theorem for the first-order logic of predicates (in his PhD thesis of 1929); the incompleteness theorems a year later; and his proof  ...  In addition, her view is that Gödel's theorems were designed to refute the formalist program of David Hilbert, according to which mathematics is nothing but an arbitrary human creation, 'a game played  ... 
doi:10.1007/bf02984710 fatcat:wjnvxqszx5dvtesspsnxxyp4re

Formal Methods and Science in Philosophy: Introduction to the Special Issue

Patrick Blackburn, Srećko Kovač, Kordula Świętorzecka
2020 Bulletin of the Section of Logic  
The special issue editors would like the thank the participants and referees of the Dubrovnik 2019 meeting for providing inspiration, the authors and referees of this special issue for all their hard work  ...  The author raises objections to both approaches, and then suggests an interpretation of "absolute definiteness" as semantic completeness -an approach, he argues, that does not suffice to explain Husserl's  ...  institutions coordinating the event were the Institute of Philosophy (Zagreb) and the Cardinal Stefan Wyszyński University (Warsaw).  ... 
doi:10.18778/0138-0680.2020.06 fatcat:5d4lk7tnuja3pbw4hzpfbojnmm

Introducing Kurt Gödel [review of Gödel, Collected Works, Vol. 1: Publications, 1929-1936]

Billy Joe Lucas
1989 Russell: the Journal of Bertrand Russell Studies  
what his philosophical goals were, sort out some of the tangled history of the origins of the completeness, compactness, and (downward) Lowenheim-Skolem theorems (all of which are proven either in Godel's  ...  The latter consists of notes taken in 1934, by Kleene and J. Barkley Rosser, of Godel's lectures at the Institute for Advanced Study, together with Godel's later postscript and corrections.  ... 
doi:10.15173/russell.v9i1.1746 fatcat:mohukf7krncbpjp5fepksksoh4

Kurt Gödel's reception of Charles Hartshorne's ontological proof [article]

Annika Kanckos, Tim Lethen
2021 arXiv   pre-print
To complete the picture, strong connections between Anselm of Canterbury's and G\"odel's conceptions of God and his positive properties are pointed out.  ...  In 1962 Charles Hartshorne published a modal logic proof formalizing Anselm of Canterbury's ontological argument for the necessary existence of God.  ...  Acknowledgements The research for this article is a part of the Godeliana project led by Jan von Plato, to whom we remain grateful for his support.  ... 
arXiv:2111.12562v1 fatcat:bslpqock4ban3hnqsfa2sk5mdm

Incompleteness, constructivism and truth

Fabrice Pataut
2014 Logic and Logical Philosophy  
The philosophical lesson to be drawn from Gödel's proof is that our capacities for justification in favour of minimal truth exceed what is strictly speaking formally provable in P by means of an algorithm  ...  Although Gödel proved the first incompleteness theorem by intuitionistically respectable means, Gödel's formula, true although undecidable, seems to offer a counter-example to the general constructivist  ...  I have benefited from criticisms and comments by Wilfrid Hodges and Jan Woleński when I gave a shortened version of the paper at the Second German-Polish Workshop on Logic and Logical Philosophy in Żagań  ... 
doi:10.12775/llp.1998.004 fatcat:c4d2cltrlfg6ff3briw4epspmq

A Fortuitous Year with Leon Henkin [chapter]

Solomon Feferman
2014 Studies in Universal Logic  
This is a personal reminiscence about the work I did under the direction of Leon Henkin during the last year of my graduate studies, work that proved to be fortuitous in the absence of Alfred Tarski, my  ...  I wish to thank Anita Burdman Feferman and Albert Visser for their helpful comments on a draft of this article. Stanford University Email: feferman@stanford.edu  ...  That is an algebraic version of the completeness theorem for first order logic.  ... 
doi:10.1007/978-3-319-09719-0_5 fatcat:o5qcgdwrvbbdxbnfxuesy7ivh4

Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program*

Solomon Feferman
2008 Dialectica  
This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, based on the  ...  The subtext is the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career.  ...  Part of Bernays' perplexities with Gödel's incompleteness theorems had to do with the roughly concurrent work of Hilbert (1931 and 1931a) in which a kind of finitary version of the infinitary ω-rule  ... 
doi:10.1111/j.1746-8361.2008.01136.x fatcat:az22pqays5ftjl3fovgktdowvq

Gödel, Nagel, Minds, and Machines

Solomon Feferman
2009 Journal of Philosophy  
But the story is of interest because of what was basically at issue, namely their provocative related but contrasting views on the possible significance of Gödel's theorems for minds vs. machines in the  ...  What led to the conflict were some unprecedented demands that Gödel made over the use of his material and his involvement in the contents of the book⎯demands that resulted in an explosive reaction on Nagel's  ...  As an aside, it is questionable whether Gödel indeed recognized the "justice of Gödel's concerns, Part I: the formulation of the incompleteness theorems.  ... 
doi:10.5840/jphil2009106417 fatcat:buir5daepvg7hp6ur5vn5laxyq
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