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Beyond Turing Machines
"... In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are pr ..."
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Cited by 40 (6 self)
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In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are presented. Other
Carnap on the foundations of logic and mathematics
, 2009
"... Throughout most of his philosophical career Carnap upheld and defended three distinctive philosophical positions: (1) The thesis that the truths of logic and mathematics are analytic and hence without content and purely formal. (2) The thesis that radical pluralism holds in pure mathematics in that ..."
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Cited by 2 (2 self)
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Throughout most of his philosophical career Carnap upheld and defended three distinctive philosophical positions: (1) The thesis that the truths of logic and mathematics are analytic and hence without content and purely formal. (2) The thesis that radical pluralism holds in pure mathematics in that any consistent system of postulates is equally legitimate and that there is no question of justification in mathematics but only the question of which system is most expedient for the purposes of empirical science. (3) A minimalist conception of philosophy in which most traditional questions are rejected as pseudoquestions and the task of philosophy is identified with the metatheoretic study of the sciences. In this paper, I will undertake a detailed analysis of Carnap’s defense of the first and second thesis. This will involve an examination of his most technical work The Logical Syntax of Language (1934), along with the monograph “Foundations of Logic and Mathematics ” (1939). These are the main works in which Carnap defends his views concerning the nature of truth and radical
Gödeltype Spacetimes: History and New Developments Visualizing ideas about Gödeltype rotating universes
"... Abstract. This paper consists mostly of pictures visualizing ideas leading to Gödel’s rotating cosmological model. The pictures are constructed according to concrete metric tensor fields. The main aim is to visualize ideas. Some kinds of physical theories describe what our universe looks like. Other ..."
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Abstract. This paper consists mostly of pictures visualizing ideas leading to Gödel’s rotating cosmological model. The pictures are constructed according to concrete metric tensor fields. The main aim is to visualize ideas. Some kinds of physical theories describe what our universe looks like. Other kinds of physical theories describe instead what the universe could be like independently of the properties of the actual universe. This second kind aims for the “basic laws of physics ” in some sense which we will not make precise here (but cf. e.g. Malament [Mal84, pp.98–99]). The present paper belongs to the second kind. Moreover, it is even more abstract than this, namely it aims for visualizing or grasping some mathematical or logical aspects of what the universe could be like. The first few pages of this material are of a “sciencepopularizing ” character in the sense that first we recall a spacetime diagram from Hawking–Ellis [HE73] as “Godgiven truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive in an easily understandable visual manner an exciting, exotic consequence of that diagram: timetravel. This applies to the first few pages.
Description of the free motion with momentums in Gödel’s universe ∗
, 2008
"... We study the geodesic motion in Gödel’s universe, using conserved quantities. We give a necessary and sufficient condition for curves to be geodesic curves in terms of conserved quantities, which can be computed from the initial values of the curve. We check our result with numerical simulations too ..."
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We study the geodesic motion in Gödel’s universe, using conserved quantities. We give a necessary and sufficient condition for curves to be geodesic curves in terms of conserved quantities, which can be computed from the initial values of the curve. We check our result with numerical simulations too. 1
—Carnap, The Logical Syntax of Language
"... “... before us lies the boundless ocean of unlimited possibilities.” ..."
The Scope of Gödel’s First Incompleteness Theorem
"... Abstract. Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of ..."
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Abstract. Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of
EMIL POST
, 1897
"... Augustów, a town at that time within the Russian empire, but after 1918 in the province of Bialystok in Eastern Poland. In 1897 his father Arnold emigrated to join his brother in America. Seven years later, in May 1904, with the success of the family clothing and fur business in New York, his wife ..."
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Augustów, a town at that time within the Russian empire, but after 1918 in the province of Bialystok in Eastern Poland. In 1897 his father Arnold emigrated to join his brother in America. Seven years later, in May 1904, with the success of the family clothing and fur business in New York, his wife Pearl, together with Emil and his sisters Anna and Ethel, joined him. The family lived in a comfortable home in Harlem. Figure 1. Emil Post, June 1924 As a child, Post was particularly interested in astronomy, but an accident at the age of twelve foreclosed that choice of career. As he reached for a lost ball under a parked car, a second car crashed into it, and he lost his left arm below the shoulder. As a high school senior, Post wrote to several observatories inquiring whether his handicap would prevent him pursuing a career as an astronomer. The responses he received, though not uniformly negative, were sufficient to discourage him from following his childhood ambition; instead, he turned towards mathematics.