| 研究実績の概要 |
The main goal of the project consists in developing the following forcing iteration techniques and find applications to solve open problems about combinatorics of the real line: (i) Multidimensional iterations with ultrafilter limits; (ii) Multidimensional template iterations; and (iii)
Weak creature forcing.
The year 2022 continues the main plan of 2021 to conclude part (iii). In collaboration with researchers in Vienna, we obtained great advances towards (iii) in 雑誌論文１, namely, we considerably simplified very complex creature forcing constructions, using probabilistic methods, while including "parametrized sublevels" in the forcing construction. This simplification not only allowed us to solve problems about the combinatorics of the real line (in connection with 雑誌論文３), but made creature forcing accessible for other fields. This can be used to develop weak creature forcing constructions to solve problems in Bounded Arithmetic.
Furthermore, we made new contributions to the understanding of the real line: we generalized the notion of Lebesgue measure to the context of reals modulo ideals on the natural numbers (雑誌論文２、学会発表３), and found new discoveries about the combinatorics of strong measure zero sets (学会発表１，２，４).
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