arXiv subjects
Publications and source records attributed to Siiri Kivimäki.
Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $μ$, we prove the consistency of the existence of a wide $μ^+$-Aronszajn tree, i.e. a tree of height and cardinality $μ^+$ with no branches of length $μ^+$, into which every wide $μ^+$-Aronszajn tree can be embedded.
We show that the existence of a universal countably chromatic graph of size $\aleph_1$ together with the failure of continuum hypothesis is consistent. The proof is a forcing iteration of strongly proper ccc posets. The construction works for any uncountable successor cardinal $κ^+$, where $κ$ is regular.
A wide Aronszajn tree is a tree of size $\aleph_1$ with no uncountable branches. Assuming the consistency of the existence of a weakly compact cardinal, we show the consistency of the existence of a wide Aronszajn tree that is \textit{universal} in the sense that it contains an isomorphic copy of every wide Aronszajn tree.
We introduce a new kind of infinitary logic that we call Boolean expansion of ${\mathcal L}_{κκ}$. This logic involves a new kind of variable, that we call generalised Boolean variable. These variables range over the powerset of a cardinal number in a way reminiscent of random variables. From this Boolean expansion, we extract a traditional infinitary logic, called Cartagena logic. We prove several model-theoretic properties of Cartagena logic, and give multiple examples of its expressive power. The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary ${\mathcal L}^1_κ$. The latter is not known to have a generative syntax, while Cartagena logic does have a very clear one.