Abstract not found

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.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 six pages of this material are of a “science-popularizing ” character in the sense that first we recall a space-time diagram from Hawking-Ellis [18] as “God-given 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: time-travel. 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 self-contained and explain and visualize most of what we say. In more detail, this work consists of Sections 1-8. 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 space-times. The space-time built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about non-existence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces co-rotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödel-type universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) re-coordinatizes the refined GU in order that the so-called 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 space-time 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 Hawking-Ellis [18] in Figure 1 or any of the statements made in the first six pages.

3.97> v m (ph) = 1, p.51. Newbasax def = (Basaxnf Ax6; Ax3;AxE g)[f Ax6 00 ; Ax6 01 ; Ax3 0 ; AxE 0 g = f Ax1;Ax2;Ax3 0 ; Ax4;Ax5;Ax6 00 ; Ax6 01 ; AxE 0 g (cf. p.191), where: Ax6 00 (8m; k 2 Obs) wm [tr m (k)] Rng(w k ), p.190. Intuitively, observer k sees all those events which are seen by another observer m on k's life-line. Ax6 01 (8m; k

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.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 six pages of this material are of a “science-popularizing ” character in the sense that first we recall a space-time diagram from Hawking-Ellis [18] as “God-given 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: time-travel. 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 self-contained and explain and visualize most of what we say. In more detail, this work consists of Sections 1-8. 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 space-times. The space-time built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about non-existence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces co-rotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödel-type universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) re-coordinatizes the refined GU in order that the so-called 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 space-time 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 Hawking-Ellis [18] in Figure 1 or any of the statements made in the first six pages.

Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order 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 two-volume 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 first-order 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 re-evaluation 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 higher-order reflection principles, and produced a stream of

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.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 six pages of this material are of a “science-popularizing ” character in the sense that first we recall a space-time diagram from Hawking-Ellis [18] as “God-given 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: time-travel. 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 self-contained and explain and visualize most of what we say. In more detail, this work consists of Sections 1-8. 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 space-times. The space-time built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about non-existence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces co-rotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödel-type universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) re-coordinatizes the refined GU in order that the so-called 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 space-time metric in Gödel’s papers. Section 9 (p.70) contains technical data about how we constructed the figures illustrating Gödel’s universe.