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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.
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. 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
"... We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. ..."
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We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles.
REALISM AND THE INCOMPLETENESS THEOREMS IN KURT GÖDEL’S PHILOSOPHY OF MATHEMATICS Honors Thesis
"... Kurt Gödel, an important figure in both disciplines. While possessing previous knowledge of Gödel’s famous incompleteness theorems, I first became aware of his philosophy, incidentally, through attending a seminar on the life and work of Alan Turing. Thus having some acquaintance with Gödel’s result ..."
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Kurt Gödel, an important figure in both disciplines. While possessing previous knowledge of Gödel’s famous incompleteness theorems, I first became aware of his philosophy, incidentally, through attending a seminar on the life and work of Alan Turing. Thus having some acquaintance with Gödel’s results in the foundations of mathematics and his philosophical views, I began to inquire into the connection, if any, between the two, with a specific focus on his realism or Platonism, the view that mathematical objects have an objective existence. The present study is the culmination (at least in part) of that inquiry. Rather than viewing Gödel’s realism from outside of mathematics, from a purely philosophical perspective, this paper draws attention toward the role that intramathematical considerations, i.e., mathematical results and methodology, play in the development of Gödel’s position. Of particular interest is the connection between Gödel’s incompleteness theorems and realism. I argue that Gödel’s intramathematical considerations both motivate and strengthen his argument for realism. To that end, I first provide an overview of the incompleteness theorems themselves in order to provide a foundation from which to survey their implications for realism. Second, I