Set Theory is the area of Mathematics which is focused on the mathematical study of infinity. The fundamental questions include that of Cantor’s Continuum Hypothesis. This was the first question on the now famous list of proposed by Hilbert in 1900. By the results of Godel in 1940 and Cohen in 1963, this problem has been shown to be formally unsolvable on the basis of the axioms of Set Theory. But this does not mean that the question has no answer, rather it simply shows the accepted axioms of Set Theory are incomplete. In previously funded research, a single new axiom has been discovered which resolves not only the problem of the Continuum Hypothesis but essentially all the other questions which Cohen’ method has been used in the last 50 years to show are also unsolvable. The compatibility of this new axiom with large cardinal axioms is now the central problem. The focus of this project is to bring the analysis of this new axiom to a conclusion, by either showing this new axiom cannot be refuted by large cardinal axioms, or by showing that some reasonable large cardinal axiom refutes this new axiom. In addition the project also provides research training opportunities for graduate students.
The HOD Dichotomy Theorem isolates the fundamental issue raised by the Ultimate L Conjecture. This theorem shows that assuming reasonable large cardinal axioms, HOD must be either very close to V or very far from V. The HOD Conjecture is the conjecture that the “far option†is vacuous. The only plausible approach at present to resolving the problem of the HOD Conjecture lies in the Ultimate L Conjecture. The focus of this project is in two parts. Either prove the Ultimate L Conjecture by extending inner model to the level of a supercompact cardinal, this would prove the HOD Conjecture, or refute the Ultimate L Conjecture which would strongly argue that the HOD Conjecture is false. The latter possibility is now a reasonable research target because of the increasing number of constraints which have been discovered that the inner model theory for a supercompact cardinal must satisfy. Either way, the successful conclusion of this project will resolve the problem of the compatibility of the axiom ``V=Ultimate†with large cardinal axioms. Depending on the outcome, one either verifies the HOD Conjecture or discovers why the HOD Conjecture is likely false.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
