Infinitary combinatorics without the axiom of choice
Final Report Abstract
The Axiomof Choice has been at the center of foundational attention since the beginning of the 20th century. It is trivial (in standard logic) that one can choose an element out of one non-empty set. By induction on the natural number n one can moreover choose n elements out of any n non-empty sets. The Axiom of Choice (AC) is the infinitary version of that principle: for any sequence (Ai) of non-empty sets there is a sequence (ai) of elements such that always a ∈ Ai . AC is a strong principle which is equivalent to the well-known Zorn's Lemma. It implies that every vector space has a basis, but that also implies the existence of a hardly imaginable basis of the real numbers R as a Q-vectorspace. An even more exotic consequence is the Banach-Tarski paradox which states that a solid ball can be partioned into finitely many subsets which can be rearranged toyield two copies of the original ball. The paradoxical nature of the axiomof choice motivates the study of mathematical universes, i.e., models of set theory, in which the axiom is false. Paul J. Cohen was the first to construct such models, using his method of (symmetric) forcing. A generic filter is adjoined to a given model of set theory and then one restricts to the submodel of sets that are definable from parameters which are invariant under ``many`` automorphisms in the automorphism group of the forcing partial order. There are similarities of this construction to the construction of fixedfields withgivenGalois groups inalgebra. In the project Infinitary Combinatorics Without the Axiom of Choice such methods are applied to the combinatorial study of uncountable sets. Large cardinals in the ground model are collapsed by certain forcing constructions, and then one restricts to submodels of ``symmetric`` sets. This yields models with combinatorial principles which are very strong in the context of the axiom of choice but which can now be attained and studied from weaker large cardinal assumptions. The principles studied concern model-theoretic transfer principles, almost Ramsey cardinals, mutual stationarity, small singular and measurable cardinals, and the Singular Cardinals Hypothesis. In many cases the necessity of large cardinals could also be established by using core model theory. Also other models of set theory without the axiom of choice like models of determinacy were employed. A very surprising result was that (a version of) the Singular Cardinals Hypothesis couldbe violatedina choiceless model without assuming any largecardinals. This leads to the conjecture that in some models of set theory without the Axiom of Choice the (appropriately defined) exponential function 2k on transfinite cardinals can take rather arbitrary values, which would shed new light on the independence of cardinal arithmetic from the axiomatic assumptions. Parts of this conjecture haverecently beenestablishedin an ongoingPhDproject. The project Infinitary Combinatorics Without the Axiom of Choice has demonstrated that set theory without the axiomof choice provides an axiomatic setting which allows for interesting new constructions and analyses. This is also reflected by a number of projects and papers by other set theorists in recent years.
Publications
- Symmetric Models, Singular Cardinal Patterns, and Indiscernibles, PhD dissertation, Rheinische Friedrich-Wilhelms-Universitat Bonn, 2011
IoannaMatilde Dimitriou
- Making All Cardinals Almost Ramsey, Archive for Mathematical Logic 47(7-8), 2008, 769-783
Arthur Apter and Peter Koepke
- The Consistency Strength of Choiceless Failures of SCH, The Journal of Symbolic Logic 75(3), 2010, 1066-1080
Arthur Apter and Peter Koepke
- Eventually different functions and inaccessible cardinals, Journal of the Mathematical Society of Japan 63 (2011), 137-151
Jorg Brendle and Benedikt Lowe
- Global Square and Mutual Stationarity at the ℵn, Annals of Pure and Applied Logic 162 (2011), 787-806
Peter Koepke and Philip D. Welch
