The papers below are organized by subject matter and annotated with comments. If you prefer a simple chronological listing go here.
A.) Paradoxical decompositions and the Banach-Tarski Paradox with nice pieces.
B.) A descriptive view of ergodic theory
C.) Amenable group actions, the Rusiewicz problem, the Hahn-Banach theorem and Lebesgue measurability.
D.) Foundations of mathematics.
E.) Consistency results of foundational interest.
F.) Model Theory: (Non-regular ultrafilters, Chang's Conjecture, Lowenheim-Skolem Theorems).
G.) Saturated Ideals.
H.) Natural structure in set theory.
I.) Applications of set theory to Abelian groups and modules.
J.) Aronzahn Trees
L.) Pot Pouri
In these two papers we show that there are paradoxical decompositions of various spaces such as the n-sphere (n>1) where the pieces involved are nice in the topological sense that they have the property of Baire. This solves Marczewski's Problem (1930).
The main result also gives paradoxical decompositions without the Axiom of Choice of open dense subsets of S^n.
1.) Banach-Tarski Decompositions Using Sets with the Property of Baire, (Joint with R. Dougherty.) Journal of the American Mathematical Society, Vol.7, No 1(1994) pp.75-124 (PDF)
2.) Banach-Tarski Decompositions Using sets with the Property of Baire, (Joint with R. Dougherty.) Proceedings of the National Academy of Sciences,USA 89(1992) pp.10726-10728 (PDF)
The first paper in this collection is a discussion of a general program for using descriptive set theory as a tool for proving both classification and anti-classification results for ergodic measure preserving transformations. This is done two ways: by considering the logical complexity of various classes of transformations as subsets of the space of all measure preserving transformations, and by setting the isomorphism problem of ergodic measure preserving transformations in descriptive set theoretic language. It includes a discussion of how to put well known results in ergodic set theory, such as the Ornstein result about Bernoulli transformations and the Ornstein-Shields results on K-automorphisms in the context of descriptive set theory. In particular it shows how the Hamos-Von Neumann theorem gives a precise calculation of the complexity of isomorphism for discrete spectrum transformations. Surprisingly it turns out that the complexity is higher than the known complexity of isomorphism for K-automorphism, thus reopening the possibility of a classification of the K-automorphisms of the same complexity as the discrete specturm transformations. The other papers in this collection are examples of partial attempts at carrying this project out.
The first paper discusses the (one dimensional) problem of the complexity of the measure distal flows, while the other papers discuss the (two dimensional) problem of the complexity of the isomorphism relation between ergodic measure preserving transformations. The last paper gives the analogous results in the topological category.
1.) A Descriptive View of Ergodic Theory, in Descriptive Set Theory and Dynamical Systems, Cambridge University Press, 2000 pp. 87-173. (ps-file, PDF)
2.) The Complexity of the Measure Distal Flows, (joint with F. Beleznay), Journal of Ergodic Theory and Dynamical Systems, Vol. 16, 1996, pp 929-962. (ps-file, PDF)
3.) An anti-classification theorem for ergodic measure preserving transformations, (joint with B. Weiss), pp.1-20, To Appear. (ps-file, PDF)
4.) A Classification theorem for compact extensions and measure distal transformation of fixed height. (To appear.)
5.) The Collection of Distal Flows is not Borel, (joint with F. Beleznay), American Journal of Mathematics, Vol. 117(1), 1995 pp. 203-239 (ps-file, PDF)
The first three papers are devoted to the question of when the algebraic structure of an amenable group determines the collection of invariant means under actions of the group. Whenever the action of an amenable group is analytic (or, in particular, Borel) there are always many invariant means. The question of the existence of an action of a locally finite group on the integers admitting a unique invariant mean is shown to be independent of Z.F.C. and thus not resolvable by the usual axioms for mathematics (even with the Axiom of Choice).
The last paper shows that the Hahn-Banach theorem implies that there is a non-Lebesgue measureable subset of S^2. The proof does not use any Axiom of Choice. (The upshot is that the use of the Hahn-Banach theorem in analysis requires acceptance of non-Lebesgue measurable subsets of the real line.) The techniques of the proof come from amenable group theory.
1.) Amenable Groups and Invariant Means, Journal of Functional Analysis, Vol 126(1), (1994), pp 7-25 (ps-file, PDF)
2.) Amenable group actions on the integers; an independence result, Research announcement, Bulletin of the American Mathematical Society, October, 1989 (ps-file, PDF) pp 237-240
3.) Locally finite groups of permutations on N acting on l^{infinity}, in A tribute to Paul Erdos, Baker et. al , Cambridge University Press, 1990 pp 195-199 (ps-file, PDF)
4.) Some other problems in Set Theory, in Set Theory of the Continuum, MSRI research publications No. 26, Springer-Verlag, 1992 pp. 35-38 (ps-file, PDF)
5.) The Hahn-Banach Theorem implies the Existence of a non-Lebesgue Measurable set (without choice). (Joint with F. Wehrung) Fundamenta Mathematica 138 (1991) pp. 13-19 (PDF)
These papers discuss extensions of the Axioms of ZFC. The axioms discussed (and proposed in a halfhearted way) settle essentially all of the classical problems of set theory, such as the Continuum Hypothesis, Suslin's Hypothesis and Kurepa's Hypothesis. The axioms are generalizations of large cardinal axioms and their descriptive set theoretic consequences include the consequences of large cardinals such as the Axiom of Determinacy. The last paper discusses an "oddball" axiom that is a counterexample to many heuristics about large cardinal axioms, and shows that if 0# exists, then a version of the axiom holds over L.
(Note that C 5 above, has some implications about the difficulty of doing mathematics without the axiom of choice, which could be construed as a "foundational" result.)
1.) Generic Large Cardinals, New Axioms for Mathematics? Proceedings of the International Congress of Mathematics, Vol. II, (Berlin, 1998). pp 11-21 (ps-file, PDF)
2.) Potent Axioms, Transactions of the American Mathematical Society, 294 (1986) pp.1-28. (PDF)
3.) 0# and Some Forcing Principles, (Joint with M. Magidor and S.Shelah.) Journal of Symbolic Logic, 51 (March 1986) pp. 39-47. (PDF)
The first paper shows that it is consistent that the Generalized Continuum Hypothesis fails at every cardinal. The second paper introduces a maximally strong forcing axiom "Martin's Maximum" and shows that is consistent, settles the values of the continuum, the singular cardinals hypothesis, implies that the non-stationary ideal on the first uncountable cardinal is saturated, and (using results of Woodin) that supercompact cardinals imply that every set in L(R) is Lebesgue Measurable.
The last paper is a failed attempt to get an upper bound on "Theta" of L(R), the definable value of the continuum. Assuming appropriate large cardinals, it contains many results about when a new equivalence class can be added to a homogeneously Suslin equivalence relation by forcing, and deduces as a consequence that many generic embeddings and ideals imply that the definable continuum is small. Along the way, many examples are given showing that "standard" proper and semi-proper forcing results are special to omega_1, and can't be generalized. (e.g. that there are always non-reflecting stationary sets, Chang's Conjecture fails in collapse models, etc.)
1.) The G.C.H. Can Fail Everywhere, (Joint with H. Woodin) Annals of Mathematics133 (1991), 1-35 (PDF)
2.) Martin's Maximum, Saturated Ideals and Non-regular Ultrafilters, Part I. (Joint with M. Magidor and S. Shelah.) Annals of Mathematics. 127 (1988) 1-47. (PDF)
3.) Large Cardinals and Definable Counterexamples to the Continuum Hypothesis, (joint with M. Magidor), Annals of Pure and Applied Logic, Vol 76, (1995) pp 47-97
(ps-file, PDF)
This series of papers has results about "old fashioned model theory." The first (my Ph.d. thesis) shows that a gap one Chang's Conjecture is consistent. (The "three cardinal transfer theorem".) The second paper shows that it is constent to have highly non-regular ultrafilters on successor cardinals, and even ultrafilters with small ultrapowers. (See also "an aleph_1 dense ideal on omega_2" for stronger result.)
The last paper is an improvement on the classical Lowenheim-Skolem theory. It is provable in ZFC and has interesting combinatorial Consequences:
1.) Large Cardinals and Strong Model theoretic Transfer Properties, Transactions of the American Mathematical Society 294 (1982) pp. 427-463. (PDF)
2.) Martin's Maximum, Saturated Ideals and Non-regular Ultrafilters, Part II, (Joint with M. Magidor and S. Shelah.) Annals of Mathematics. 127 (1988) 521-545 (PDF)
3.) A new Lowenheim-Skolem Theorem. (Joint with S. Todorcevic) pp.1-35 To Appear. (ps-file, PDF)
These papers are consistency results for the existence of various kinds of saturated ideals. Saturated ideals provide generic elementary embeddings of the kinds auggested as axioms in section D. The strength of a generic embedding is determined by the nature of the forcing needed to define the embedding, the image of the ordinals and the closure of the generic ultrapower. The easiest parameter to describe is the saturation (Chain Condition) of the forcing.
The first paper shows that it is consistent to have a kappa^+ saturated ideal on every regular cardinal kappa. This forcing combines techniques of Kunen with Prikry forcing. In particular it shows the consistency of saturated ideals on the successor of singular cardinals.
The second paper shows the consistency of a uniform, countably complete aleph_1-dense ideal on aleph_2. The main result in the paper is a general combinatorial technique for creating ideals with a given Boolean algebra as a quotient algebra. A corollary of the main theorem is the consistency of an ultrafilter on omega_2 such that the ultraproduct of omega has cardinality omega_1.
1.) More Saturated Ideals, Cabal Seminar 79-81, Springer-Verlag (1983) pp. 1-27. (ps-file, PDF)
2.) An aleph_1-dense ideal on omega_2, Israel Journal of Math, Vol. 108 (1998), pp.
253-290. (ps-file, PDF)
These papers discuss the informal notion of canonical structure. Canonical structure is structure that is not arbitrarily determined by non-constructive existence assumptions. For example, structure that requires the axiom of choice to prove its existence may still be independent of any choices made in proving it exists. Cardinals of uncountable cofinality fall into this category. Other examples might include fine structure models of large cardinals.
The notion of canonical structure is different from the notion of absoluteness. We illustrate this with an example. Assuming the Axiom of Choice, the collection of real numbers has some well-ordered cardinality and this cardinality is independent of the choices made to show it exists. Similarly, one needs the Axiom of Choice to prove that the least regular uncountable ordinal (aleph_1) exists. Both of these objects are ``canonical" in our sense, but it is independent of ZFC whether they are in fact identical. We would like to say that these distinct examples of structure that may or may not determine identical objects.
Since the development of PCF theory, there have been a surprising number of new examples of canonicial structure: stationary sets of ordinals (good ordinals, approachable ordinals), the good points on scales, stationary collections of structures (the tight structures) and many others. This series of papers explores canonical structre in terms of standard set theoretic notions such as square and stationary set reflection. They include a large number of ZFC results as well as forcing constructions.
The first paper gives a square principal so weak that it is consistent with the existence of supercompact (or larger) cardinals, yet is strong enough that one is able to use it to do ``most" inductive constructions used in applications of square. Its strength is explored, and it is placed in the hierarchy of square properties. It is shown to be equivalent to various diverse principles, including one about the structure of Abelian groups.
The second paper, appearing in the innagural issue of the Journal of Mathematical Logic, explores the relationship between square properties and Shelah's PCF theory. It is shown that the hierarchy of square principles parallels a hierarchy of properties of scales. It places these hierarchies among such statements as ``there exists a special Aronzahn Tree on kappa^+." It shows that it is consistent to have ``every stationary subset of aleph_{omega+1} reflects", together with the two cardinal transfer principle: "(omega_1,omega) transfers to (aleph_{omega+1},aleph_omega)." In particular, the reflection principle is consistent with square_{aleph_omega,omega}. This result is optimal in this direction.
The third paper introduces a notion of "stationary set" for singular cardinals, mutual stationarity. This notion is applied to show that to show the theorem that the non-stationary ideal is not saturated on P_kappa(lambda) if either kappa or lambda differs from omega_1. The latter case is known to be consistent.
The fourth and fifth papers further investigate mutual stationarity, tight stationarity and square principles. A "square compactness" result is shown: some square principles propogate through singular cardinals. A covering theorem is proved that shows that below the first cardinal fixed point the set theoretic universe is essentially determined by its cardinal structure and cofinal sets in products of of regular cardinals. Some corollaries are deduced about precipitous ideals. Tightness is characterized. In the fifth paper, a couple of examples are given of mutually stationary but not tight sequences are constructed, a model with omega_1 stationary sets that don't reflect simultaneously but any countable collection of statinary sets reflects to an internally approachable model is constructed.
The sixth paper gives a model where square holds below aleph_omega, but not at aleph_omega. This shows that the square compactness proved in number 4 is, in some sense, a tight result. There is an argument that a principle of Dzamonja and Magidor is inconsistent above a supercompact cardinal.
The seventh paper explores a class of canonical filters, the club guessing filter and shows that it is consistent that the club guessing filters are saturated. A tool is a partial ordering that forces a sequences such that every closed unbounded set is tail-guessed on a closed unbounded set.
The eighth paper discusses stationary and mutually stationary sets, reflection and Chang's Conjecture. It explains the relation between Chang's Conjecture and square bracket partition relations, and shows a new about squares and Chang's Conjecture.
1.) A Very Weak Square Principle, (joint with M. Magidor), Journal of Symbolic Logic, Vol. 62(1), 1997, pp.175-196. (ps-file, PDF)
2.) Squares, Scales and Stationary Set Reflection, Joint with J. Cummings and M.
Magidor. J. Math. Log. 1 (2001), no. 1, 35--98.(ps-file, PDF)
3.) Mutually Stationary sets and the Saturation of the Non-stationary Ideal on $P_kappa(lambda)$, Joint with M. Magidor. Acta Math. 186 (2001), no. 2, 271--300. (ps-file, PDF)
4.) Canonical Structure in Universe of Set Theory, part I. Joint with M. Magidor and J. Cummings. pp1-30, to appear. (ps-file, PDF)
5.) Canonical Structure in the Universe of Set theory part II. Joint with M. Magidor and J. Cummings. pp1-27, to appear. (ps-file, PDF)
6.) The non-compactness of square. Joint with M. Magidor and J. Cummings. pp 1-10. To appear in the Journal of Symbolic Logic.(ps-file, PDF)
7.) The Club Guessing Ideal (Commentary on a theorem of Gitik and Shelah). Joint with P. Komjath. pp. 1-56. To appear.(ps-file, PDF)
8.) Stationary sets, Chang's Conjecture and partition theory, in Set Theory (The Hajnal Conference) DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 58, Amer. Math. Soc. , Providence, RI. 2002 pp. 73-94 (ps-file, PDF)
The first paper is a discussion of "Helsinki Logic" and the classificiation of Abelian groups of size omega_1. It is shown that under PFA, there is a good classification in the logic of omega_1 seperable groups, while under diamond one can construct nice groups that cannot be distinguished.
The second paper shows that under CH the collection of spectra of bilinear forms is maximal and that this is independent of ZFC. Moreover, a model from a weakly compact cardinal that has stationary sets that cannot be the spectrum of a bilinear form.
Note that there are several other papers above that have some applications to Abelian groups (e.g. H 1, D 2, G 2.)
1.) On invariants for omega_1-separable Groups, (joint with P. Eklof and S. Shelah), Trans. Am. Math. Soc., Vol 347(11), 1995, pp 4385-4402 (ps-file, PDF)
2.) The spectrum of the Gamma-invariant of a bilinear space. (joint with J. Baumgartner and O. Spinas), Joun. of Algebra Vol 189, 1997. pp 406-418 (ps-file, PDF)
These two papers discuss the existence of Aronzahn trees. The first paper shows that it is consistent to have no Aronzahn tress on any aleph_n (n>1). (Abraham had earlier showed it consistent to have no Aronzahn trees on consecutive cardinals, such as aleph_2 and aleph_3.) It also shows that it is consistent that there is no Aronzahn tree on aleph_{omega+2}.
The second paper shows that if there are two consecutive cardinals don't have Aronzahn trees then there are inner models with Wooden cardinals. This result was shown independently by Foreman/Magidor (together) and by Schindler.
1.) The Tree Property, (joint with J. Cummings) Advances in Mathematics, Vol. 133 (1998), pp 1-32. (ps-file, PDF)
2.) The consistency strength of successive cardinals with the tree property, Joint with M. Magidor, H. Schindler. 18 pages, to appear in the Journal of Symbolic Logic. (ps-file, PDF)
These papers address combinatorial problems in the style of Erdos, Rado and Hajnal. The first settles a problem of Erdos and Hajnal about graph reflection: it is shown to be consistent that every graph of size and chromatic number aleph_2 contains a graph of size and chromatic number aleph_1. (In fact a much more general fact about varieties is shown.)
The second paper addresses classical problems is partition theory: if the pairs of a successor cardinal are partitioned into two pieces, how large a homogeneous set exists? The answer (measured in "order type") depends surprising on issues of uniform indiscernibility above a measurable cardinal.
The third paper exposits a portion of partition theory and the results in the second paper for a general mathematical audience.
1.) Some downwards transfer properties for aleph_2, (Joint with R. Laver.) Advances in Mathematics. Vol. 67, No. 2 (1988) 230-238.(PDF)
2.) A partition relation for successors of large cardinals, (Joint with A. Hajnal) to appear in Mathematische Annalen. pp 1-51. (ps-file, PDF)
3.) A partition theorem for successor cardinals. in Paul Erdos and his mathematics, Volume II, Bolyai Society, Mathematical Studies, 2002, Budapest. pp. 311-328 (ps-file, PDF)
1.) Games Played on Boolean Algebras, Journal of Symbolic Logic, September 1983,(48) pp. 714-724. (PDF)
2.) A Dilworth Theorem for lambda-Suslin quasiorderings, Proc. International Congress of Logic, Philosphy and Methodology of Science, VIII, North-Holland, 1989. pp 223-244
3.)Taxonometric partitions in Methods and Applications of Mathematical Logic. Contemporary Mathematics, AMS Vol. 69. (Joint with J. Malitz and A. Ehrenfeucht.) pp. 27-34(PDF)
4.) Mathematical methods of taxonomy. (Joint with J. Malitz)