Dougherty, R., Critical points in an algebra of elementary embeddings, Annals of Pure and Applied Logic 65 211-241.Given two elementary embeddings from the collection of sets of rank less than λ to itself, one can combine them to obtain another such embedding in two ways: by composition, and by applying one to the other. Hence, a single such nontrivial embedding j generates an algebra of embeddings via these two operations, which satisfies certain laws . Laver has shown, among other things, (...) that this algebra is free on one generator with respect to these laws.The set of critical points of members of this algebra is the subject of this paper. This set contains the critical point κ0 of j, as well as all of the other ordinals κn in the critical sequence of j ). But the set includes many other ordinals as well. The main result of this paper is that the number of critical points below κn grows so quickly with n that it dominates any primitive recursive function. In fact, it grows faster than the Ackermann function, and even faster than a slow iterate of the Ackermann function. Further results show that, even just below κ4, one can find so many critical points that the number is only expressible using fast-growing hierarchies of iterated functions. (shrink)

A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A, B, C such that A is congruent to B, B is congruent to C, and A is congruent to B ∪ C; this result was the precursor of the Banach–Tarski paradox. Later, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences. The pieces involved were nonmeasurable. (...) In the present paper, we consider the problem of which systems of congruences can be satisfied using open subsets of the sphere ; of course, these open sets cannot form a partition of the sphere, but they can be required to cover "most of" the sphere in the sense that their union is dense. Various versions of the problem arise, depending on whether one uses all isometries of the sphere or restricts oneself to a free group of rotations, or whether one omits the requirement that the open sets have dense union, and so on. While some cases of these problems are solved by simple geometrical dissections, others involve complicated iterative constructions and/or results from the theory of free groups. Many interesting questions remain open. (shrink)

Hausdorff's paradoxical decomposition of a sphere with countably many points removed actually produced a partition of this set into three pieces A,B,C such that A is congruent to B, B is congruent to C, and A is congruent to B ∪ C. While refining the Banach–Tarski paradox, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences: the only nontrivial restriction is that the system should not require (...) any set to be congruent to its complement. Later, J. F. Adams showed that this restriction can be removed if one allows arbitrary isometries of the sphere to witness the congruences. The purpose of this paper is to characterize those systems of congruences which can be satisfied by partitions of the sphere or related spaces into sets with the property of Baire. A paper of Dougherty and Foreman gives a proof that the Banach–Tarski paradox can be achieved using such sets, and gives versions of this result using open sets and related results about partitions of spaces into congruent sets. The same method is used here; it turns out that only one additional restriction on a system of congruences is needed to make it solvable using subsets of the sphere with the property of Baire with free rotations witnessing the congruences. Actually, the result applies to any complete metric space acted on in a sufficiently free way by a free group of homeomorphisms. We also characterize the systems solvable on the sphere using sets with the property of Baire but allowing all isometries. (shrink)

Results of Sierpiski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is ‘narrow’ in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set ω × ω is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. (...) Such partitions or coverings can exist only when the sets forming the product are of limited size.This paper considers such coverings for products of infinitely many sets . In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering , then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on.One result given here is a relative consistency proof for a hypothesis used by S. Mrówka to produce a counterexample in the dimension theory of metric spaces. (shrink)

This paper consists of three parts supplementing the papers of K. Hauser 2002 and D. Mumford 2000: There exist regular open sets of points in with paradoxical properties, which are constructed without using the axiom of choice or the continuum hypothesis. There exist canonical universes of sets in which one can define essentially all objects of mathematical analysis and in which all our intuitions about probabilities are true. Models satisfying the full axiom of choice cannot satisfy all those intuitions and (...) they violate them in a very similar way as those models which satisfy also the continuum hypothesis. (shrink)