official statement of my PhD project

Many combinatorial principles attain their set theoretic strength only in the presence of the axiom of choice. Without it, it is possible that small cardinals like ω₁ can have large cardinal properties like being measurable or satisfying strong partition properties. The intended research area is to determine the consistency strengths of various infinitary combinatorial properties with respect to the Zermelo-Fraenkel axioms ZF, i.e., without the axiom of choice. This is a wide field, since the classical questions about generalised Chang's conjectures, ℵ_ω being Jonsson or Rowbottom, mutual stationarity, pcf-theory and others can be examined from this perspective.

Peter Koepke is working in this area together with Arthur Apter, CUNY. The PhD project will be embedded into this collaboration. The work will combine forcing techniques and inner model arguments. We indicate this in the case of a simple example: To prove the conjecture that Chang's conjecture (ω₃, ω₂) --> (ω₂, ω₁) with ZF is equiconsistent with the existence of an ω₂-Erdos cardinal, one uses forcing techniques of Apter and inner model techniques of Koepke. An ω₂-Erdos cardinal is collapsed by a symmetric subcollapse of a Levy collapse to ω₃. Conversely, one applies the Chang property, moves to a submodel with the axiom of choice which contains the Chang substructure, and applies known core model techniques. The PhD project will involve studying several such properties of increasing degrees of complexity.