Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak (in that they are consistent relative to the Erdös cardinal κ(ω)) or inconsistent. The philosophical significance of these results is discussed.

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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 “science-popularizing ” character in the sense that first we recall a space-time diagram from Hawking–Ellis [HE73] as “God-given 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: time-travel. This applies to the first few pages.

The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum Hypothesis hold?”), combinatorics(“DoesSuslin’sHypotheseshold?”), andgrouptheory (“Is there a Whitehead group?”). These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason,

The independence results in arithmetic and set theory led to a proliferation of mathematical systems. One very general way to investigate the space of possible mathematical systems is under the relation of interpretability. Under this relation the space of possible mathematical systems forms an intricate hierarchy of increasingly strong systems. Large cardinal axioms provide a canonical means of climbing this hierarchy and they play a central role in comparing systems from conceptually distinct domains. This article is an introduction to independence, interpretability, large cardinals and their interrelations. Section 1 surveys the classic independence results in arithmetic and set theory. Section 2 introduces the interpretability hierarchy and describes some of its basic features. Section 3 introduces the notion of a large cardinal axiom and discusses some of the central examples. Section 4 brings together the previous themes by discussing the manner in which large cardinal axioms provide a canonical means for climbing the hierarchy of interpretability and serve as an intermediary in the comparison

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