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Visualizing some ideas about Gödeltype rotating universes
, 2008
"... 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 ..."
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Cited by 2 (1 self)
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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 [25, pp.9899]). 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 six pages of this material are of a “sciencepopularizing ” character in the sense that first we recall a spacetime diagram from HawkingEllis [18] as “Godgiven truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive carefully in an easily understandable visual manner an exciting, exotic consequence of that diagram: timetravel. This applies to the first six pages. The rest of this work is of a more ambitious character. The reader does not have to believe anything 1. We do our best to make the paper selfcontained and explain and visualize most of what we say. In more detail, this work consists of Sections 18. Section 1 (p.2) is the just mentioned “popular ” part. Section 2 (p.8) lays the foundation for discussing rotating universes. E.g. it shows how to visualize such spacetimes. The spacetime built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about nonexistence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces corotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödeltype universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) recoordinatizes the refined GU in order that the socalled gyroscopes do not rotate in this coordinatization. Section 8 (p.67) gives connections with the literature. E.g. it presents detailed computational comparison with the spacetime metric in Gödel’s papers. Section 9 (p.70) contains technical data about how we constructed the figures illustrating Gödel’s universe. 1 Not even the diagram recalled from HawkingEllis [18] in Figure 1 or any of the statements made in the first six pages.
Visualizing some ideas about Gödeltype rotating universes.
, 2008
"... 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 ..."
Abstract
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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 [25, pp.9899]). 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 six pages of this material are of a “sciencepopularizing ” character in the sense that first we recall a spacetime diagram from HawkingEllis [18] as “Godgiven truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive carefully in an easily understandable visual manner an exciting, exotic consequence of that diagram: timetravel. This applies to the first six pages. The rest of this work is of a more ambitious character. The reader does not have to believe anything 1. We do our best to make the paper selfcontained and explain and visualize most of what we say. In more detail, this work consists of Sections 18. Section 1 (p.2) is the just mentioned “popular ” part. Section 2 (p.8) lays the foundation for discussing rotating universes. E.g. it shows how to visualize such spacetimes. The spacetime built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about nonexistence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces corotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödeltype universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) recoordinatizes the refined GU in order that the socalled gyroscopes do not rotate in this coordinatization. Section 8 (p.67) gives connections with the literature. E.g. it presents detailed computational comparison with the spacetime metric in Gödel’s papers. Section 9 (p.70) contains technical data about how we constructed the figures illustrating Gödel’s universe. 1 Not even the diagram recalled from HawkingEllis [18] in Figure 1 or any of the statements made in the first six pages.
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
Visualizing some ideas about Gödeltype rotating universes.
, 2008
"... 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 ..."
Abstract
 Add to MetaCart
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 [25, pp.9899]). 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 six pages of this material are of a “sciencepopularizing ” character in the sense that first we recall a spacetime diagram from HawkingEllis [18] as “Godgiven truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive carefully in an easily understandable visual manner an exciting, exotic consequence of that diagram: timetravel. This applies to the first six pages. The rest of this work is of a more ambitious character. The reader does not have to believe anything 1. We do our best to make the paper selfcontained and explain and visualize most of what we say. In more detail, this work consists of Sections 18. Section 1 (p.2) is the just mentioned “popular ” part. Section 2 (p.8) lays the foundation for discussing rotating universes. E.g. it shows how to visualize such spacetimes. The spacetime built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about nonexistence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces corotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödeltype universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) recoordinatizes the refined GU in order that the socalled gyroscopes do not rotate in this coordinatization. Section 8 (p.67) gives connections with the literature. E.g. it presents detailed computational comparison with the spacetime metric in Gödel’s papers. Section 9 (p.70) contains technical data about how we constructed the figures illustrating Gödel’s universe.