Godel's Theorem Research Articles

The Reason for the Incompleteness of Some Theorems in Modern Mathematics: Non Numbers Enter Mathematical Analysis or the Concept Definition Is Incomplete

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.

Artificial Intelligence that Exists in the Human Mind

This article asserts that some general AIs are incomprehensible to humans and that computers are innately incapable of performing certain human cognitive activities, including 1) Misconceptions of Godel's theorem and related statistics and mathematical results are at the root of objections to using AI; 2) Reductionism and objections to monistic reductionism, as well as a minimal appreciation of responses to different levels of discussion; 3) Concerns raised by a thorough understanding of computers or computing Developing a model in which "cognitive" functions can be created independently and referred to as "wetware" is pragmatic, according to a psychologist. The cognitive and essential assumptions for individuals involved in the usefulness of AI are pragmatic. From this point of view, AI is heavily invested in the future of cognitive psychology, if only because of their similar fate.

Artificial Intelligence that Exists in the Human Mind

This article asserts that some general AIs are incomprehensible to humans and that computers are innately incapable of performing certain human cognitive activities, including 1) Misconceptions of Godel's theorem and related statistics and mathematical results are at the root of objections to using AI; 2) Reductionism and objections to monistic reductionism, as well as a minimal appreciation of responses to different levels of discussion; 3) Concerns raised by a thorough understanding of computers or computing Developing a model in which "cognitive" functions can be created independently and referred to as "wetware" is pragmatic, according to a psychologist. The cognitive and essential assumptions for individuals involved in the usefulness of AI are pragmatic. From this point of view, AI is heavily invested in the future of cognitive psychology, if only because of their similar fate.

A Note on Godel's Theorem

A Note on Godel's Theorem

A Note on Godel's Theorem

A Note on Godel's Theorem

Истинность неразрешимых предложений в свете Первой теоремы Гёделя о неполноте

Истинность неразрешимых предложений в свете Первой теоремы Гёделя о неполноте

A Note on Godel's Theorem

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.

Gödel's and Other Paradoxes

Francesco Berto has recently written “The Godel Paradox and Wittgenstein's Reasons,” about a paradox first formulated by Graham Priest in 1971. The major reason for disagreeing with Berto's conclusions concerns his elucidation of Wittgenstein's understanding of Godel's theorems. Seemingly, Wittgenstein was some kind of proto-paraconsistentist. Priest himself has also, though in a different way, tried to tar Wittgenstein with the same brush. But the resolution of other paradoxes is intimately linked with the resolution of the Godel Paradox, and with understanding Wittgenstein's views of Godel's theorems. So this paper discusses some other paradoxes before looking at Wittgenstein's relevant views.

Consciousness: State reductions, mental disorders and dimensions

Quantum mechanics and godel's theorem are purely work and concept of physics. When applied to see how ways in which mind works is fascinating, at the same time at a very hypothetical level. Orch or model of consciousness explores neurological basis of consciousness and insight in psychiatric patients as well as in normal patients is also explained. Though many of these are proven concepts in physics and mathematics, there application in understanding of psychiatric phenomenon is still at hypothetical level.

An introduction to Godel's theorems

An introduction to Godel's theorems

Kurt Godel and the foundations of mathematics: horizons of truth

Part I. Historical Context - Godel's Contributions and Accomplishments: 1. The impact of Godel's incompleteness theorems on mathematics Angus Macintyre 2. Logical hygiene, foundations, and abstractions: diversity among aspects and options Georg Kreisel 3. The reception of Godel's 1931 incompletability theorems by mathematicians, and some logicians, to the early 1960s Ivor Grattan-Guinness 4. 'Dozent Godel will not lecture' Karl Sigmund 5. Godel's thesis: an appreciation Juliette C. Kennedy 6. Lieber Herr Bernays!, Lieber Herr Godel! Godel on finitism, constructivity, and Hilbert's program Solomon Feferman 7. Computation and intractability: echoes of Kurt Godel Christos H. Papadimitriou 8. From the entscheidungsproblem to the personal computer - and beyond B. Jack Copeland 9. Godel, Einstein, Mach, Gamow, and Lanczos: Godel's remarkable excursion into cosmology Wolfgang Rindler 10. Physical unknowables Karl Svozil Part II. A Wider Vision - the Interdisciplinary, Philosophical, and Theological Implications of Godel's Work: 11. Godel and physics John D. Barrow 12. Godel, Thomas Aquinas, and the unknowability of God Denys A. Turner 13. Godel's mathematics of philosophy Piergiorgio Odifreddi 14. Godel's ontological proof and its variants Petr Hajek 15. The Godel theorem and human nature Hilary Putnam 16. Godel, the mind, and the laws of physics Roger Penrose Part III. New Frontiers - Beyond Godel's Work in Mathematics and Symbolic Logic: 17. Godel's functional interpretation and its use in current mathematics Ulrich Kohlenbach 18. My forty years on his shoulders Harvey M. Friedman 19. My interaction with Kurt Godel: the man and his work Paul J. Cohen 20. The transfinite universe W. Hugh Woodin 21. The Godel phenomena in mathematics: a modern view Avi Wigderson.

Turing’s man: a dialogue

We can never emulate their meekness, soft servants of durable material: they live without pretension in complicated relays and electrical circuits. Speed, docility are their strength. One asks: “What is 2 × 2?”—“Are you a machine?” They answer or refuse to answer, depending on what you demand. There are, however, other machines as well, more abstract automatons, bolder and more inaccessible, which eat their tape in mathematical formulae. They imitate in language. In infinite loops, farther and farther back in their retreat towards more subtle algorithms, more recursive functions. They are logical and describe themselves. As when a man with a hand-mirror pressed against his nose in front of a mirror sees in infinite rows the same image multiplied in a shrinking, darkening corridor of glass. It is a Godel theorem as good as any. He sees infinity, but what he does not see is his face. (From Goran Printz-Pahlson´s poem “The Turing Machine” published in Sag minns du skeppet Refanut? Samlade dikter 1950–1983 (1984) Bonniers, Stockholm).

An Intuitionistic Completeness Theorem for Classical Predicate Logic

This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Godel's completeness theorem for classical predicate logic.

Chailakhyan’s views of the origin of consciousness and mentality

Chailakhyan's views of sources of the origin of consciousness and mentality have been analyzed. It is supposed that the information interaction alone is not sufficient for the occurrence of consciousness. Something else is necessary that distinguishes a live system from lifeless and can be transmitted to the progeny and evolve. It is assumed that this is an "information-physical interaction", namely, a power profile of conformational reorganizations of proteins during their interaction. L.M. Chailakhyan proves the point of view that this profile during the interaction of proteins cannot be calculated and therefore cannot be cognized. This conclusion is supported by the Godel theorem of incompleteness.

Consilience and the Naturalistic Foundations of Evolutionary Economics

Edward O. Wilson has defined consilience as a strong reductionist program in science. This paper proposes an ontological approach to consilience, which allows for an emergentist interpretation. The methodological basis is naturalism, which is defined through some fundamental notions such as the unity of the world and the impossibility of the detached observer. Naturalism includes, for example, evolutionary epistemology. It does not imply a reductionist approach to human mind and society, but to the contrary the convergence of Naturwissenschaften und Geisteswissenschaften. A major reference point is the naturalization of the Godel theorem, presaged by Hayek in his Sensory Order. Its implications match recent insights of the brain sciences, such as modularization and complex interaction between reason and intuition, which are being received by neuroeconomics research. The Godel theorem is also related to fundamental theorems about randomness, such as Chaitin’s theorems. The world is random in the sense of being singular, which, however, we cannot perceive adequately because our knowledge is propositional, hence relies on provability in a general sense. This diagnosis matches with certain economic approaches to entrepreneurship and uncertainty, such as Shackle’s. The relation between regularity and singularity evolves during the expansion of the universe, according to the theories of Chaisson and Brooks/Wiley, and can be measured as increasing information content that emerges from the growing differential between possible entropy and realized entropy. The difference reflects the growing number of constraints on possible states of the world. These processes are triggered by a continuous flow of energy, in particular through dissipative systems, which minimize entropy internally and maximize externally. From this, the ontological principle of bimodality is deduced, i.e. every state of the world is matterenergy and knowledge. Knowledge is interpreted as non-referential, i.e. as the outcome of a universal process of variation, selection and retention, which also applies for the human mind, including referential (propositional) knowledge only as a special case. This allows to interpret technology and institutions as carriers of knowledge.

Naturalism: Audacious Thoughts on the Consilience of Physics, Biology and Economics

Edward O. Wilson has defined consilience as a strong reductionist program in science. This paper proposes an ontological approach to consilience, which allows for an emergentist interpretation. The methodological basis is naturalism, which is defined through some fundamental notions such as the unity of the world and the impossibility of the detached observer. Naturalism includes, for example, evolutionary epistemology. It does not imply a reductionist approach to human mind and society, but to the contrary the convergence of Naturwissenschaften und Geisteswissenschaften. A major reference point is the naturalization of the Godel theorem, presaged by Hayek in his Sensory Order. Its implications match recent insights of the brain sciences, such as modularization and complex interaction between reason and intuition, which are being received by neuroeconomics research. The Godel theorem is also related to fundamental theorems about randomness, such as Chaitin's number. The world is random in the sense of being singular, which, however, we cannot perceive adequately because our knowledge is propositional, hence relies on provability in a general sense. This diagnosis matches with certain economic approaches to entrepreneurship and uncertainty, such as Shackle's. The relation between regularity and singularity evolves during the expansion of the universe, according to the theories of Chaisson and Brooks/Wiley, and can be measured as increasing information content that emerges from the growing differential between possible entropy and realized entropy. The difference reflects the growing number of constraints on possible states of the world. These processes are triggered by a continuous flow of energy, in particular through dissipative systems, which minimize entropy internally and maximize externally. From this, the ontological principle of bimodality is deduced, i.e. every state of the world is matterenergy and knowledge. Knowledge is interpreted as non-referential, i.e. as a result of a universal principle of variation, selection and retention, which also applies for the human mind, thus including referential (propositional) knowledge as a special case. Finally, the paper sketches the consequences of ontological consilience for economics, which include the recognition that growth is growth of knowledge driven by energy throughput, that institutions as a fundamental expression of order are singular and hence, irreducible phenomena, and that the human actor is the interface of complex evolutionary processes on both the biological and the social level.

Truth and Provability

The views of Redhead ([2004]) are defended against the argument by Panu Raatikainen ([2005]). The importance of informal rigour is canvassed, and the argument for the a priori nature of induction is explained. The significance of Gödel's theorem is again rehearsed.

Godel's theorem: an incomplete guide to its use and abuse

Godel's theorem: an incomplete guide to its use and abuse

On the Philosophical Relevance of Godel's Incompleteness Theorems

Godel began his 1951 Gibbs Lecture by stating: Research in the founda tions 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. (Godel 1951) Godel is referring here especially to his own incompleteness theorems (Godel 1931). Godel's first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arith metic, there exists a sentence GF of the language of the system which is true but unprovable in that system. Godel's second incompleteness theo rem states that no consistent formal system can prove its own consisten cy.1 These results are unquestionably among the most philosophically important logico-mathematical discoveries ever made. However, there is also ample misunderstanding and confusion surrounding them. The aim of this paper is to review and evaluate various philosophical interpreta tions of Godel's theorems and their consequences, as well as to clarify some confusions.

Formal Axiomatic Theories Based on a Three-Valued Logic

Formal axiomatic theories based on the three-valued logic of Lukasiewicz are considered. Main notions related to these theories, in particular, those of Luk-model, Luk-consistent theory, and Luk-complete theory are introduced. Logical calculuses that describe such theories are defined; counterparts of the classical compactness and completeness theorems are proved. Theories of arithmetic based on Lukasiewicz’s logic and on its constructive (intuitionistic) variant are investigated; the theorem on effective Luk-incompleteness is proved for a large class of arithmetic systems. This theorem is a three-valued counterpart of the famous Godel theorem on incompleteness of formal theories. Three-valued counterparts of Presburger’s arithmetic system are defined and proved to be Luk-complete but incomplete in the classical sense. Bibliography: 29 titles.