Current Research

Investigation of Large Cardinal Axioms

Since P. J. Cohen introduced the forcing method in the 1960's, many of the classical unsolved problems in set theory have been shown to be independent from ZFC (Axioms of Zermelo and Fraenkel, the standard set of axioms for set theory). This means that these problems are unsolvable under ZFC. For example, the continuum hypothesis (the claim that the set of all real numbers has the least uncountable cardinality) was shown to be neither provable nor refutable from ZFC. The question of whether all of the projective sets (the sets of reals that can be obtained from Borel sets by using projections and complements) satisfy Lebesgue measurability and the Baire property also turned out to be independent from ZFC.
Anticipating the rise of independent propositions, K. Godel suggested expanding the set of axioms by adding new axioms that state the existence of transcendentally large cardinal numbers. These axioms are known as large cardinal axioms.
This direction of investigation in set theory is called Godel's program. In the study of large cardinal axioms the following are the two main themes : (1) Investigations of the influence of large cardinal axioms on numerous propositions in mathematics : Many set theorists have been investigating the relationship between large cardinal axioms and the Lebesgue measurability and the Baire property of projective sets of real numbers. D. A. Martin, J. Steel, and H. Woodin have made remarkable advances in this field. They showed that by assuming sufficiently strong large cardinal axioms, all of the projective sets satisfy Lebesgue measurability and the Baire property. They also showed which large cardinals are necessary and sufficient to prove Lebesgue measurability and the Baire property for each level of the hierarchy of projective sets. There are some other problems in analysis that are related to large cardinal axioms. In addition, there are propositions from infinitary combinatorics, such as Rado's conjecture, which turn out to be closely related to large cardinal axioms.
(2) Investigations of the strength of large cardinal axioms : By the strength of a large cardinal axiom, we usually mean the consistency strength of that large cardinal axiom together with ZFC.
If from ZFC together with large cardinal A one can prove the consistency of ZFC with large cardinal B, then we say that large cardinal A is consistency-wise stronger than large cardinal B. In modern set theory, we use the forcing method and the theory of inner models to investigate the consistency strength of various large cardinal axioms.
Unfortunately the continuum problem, arguably the most famous problem in set theory, turned out to be unsolvable even after adding large cardinal axioms to ZFC. The continuum problem is the problem of finding the cardinality of the set of all real numbers. A. Levy and R. Solovay proved that by applying forcing techniques one can manipulate the cardinality of the set of all real numbers at will without affecting the existence of large cardinals. This result showed that large cardinal axioms alone cannot solve the continuum problem. Nonetheless, as we stated above, the use of large cardinal axioms has been very fruitful in investigating sets of real numbers. Furthermore, we now know that the existence of ideals with large cardinal properties can influence the continuum problem. Many large cardinals can be defined using prime (maximal) ideals. We have been studying various large cardinal properties of ideals, and these properties can be thought of as generalizations of those that define large cardinals. We are especially interested in the precipitousness and the saturation of the non-stationary ideal, the smallest normal ideal.
The method of generic ultrapowers has been useful in the study of large cardinal axioms. The method of generic ultrapowers combines the forcing method and the techniques from large cardinal theory. We have been investigating ideals with large cardinal properties using generic ultrapowers.
H. Woodin proved that the saturation of the non-stationary ideal over ω₁ together with a measurable cardinal implies the failure of the continuum hypothesis. This was a quite unexpected result to many set theorists. Inspired by Woodin's result, we are investigating the relationship between ideals with large cardinal properties and the generalized continuum hypothesis.