Monads of indiscernibles
More on distributive ideals
More on set-theoretic characteristics of summability of sequences by regular (Toeplitz) matrices
More remarks on the intersection ideal
We prove in ZFC that every additive set is additive, thus we solve Problem 20 from paper [Weiss T., A note on the intersection ideal , Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445] in the negative.
More set-theory around the weak Freese–Nation property
We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for , we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals,...
Multiple gaps
We study a higher-dimensional version of the standard notion of a gap formed by a finite sequence of ideals of the quotient algebra 𝓟(ω)/fin. We examine different types of such objects found in 𝓟(ω)/fin both from the combinatorial and the descriptive set-theoretic side.
Mycielski ideals generated by uncountable systems
Negative universality results for graphs
It is shown that in many forcing models there is no universal graph at the successors of regular cardinals. The proof, which is similar to the well-known proof for Cohen forcing, is extended to show that it is consistent to have no universal graph at the successor of a singular cardinal, and in particular at . Previously, little was known about universality at the successors of singulars. Analogous results show it is consistent not just that there is no single graph which embeds the rest, but that...
Non-dominating ultrafilters
Non-isomorphic Hyper-real Fields from Non-isomorphic Ultrapowers.
Nonreflecting stationary subsets of
We explore the possibility of forcing nonreflecting stationary sets of . We also present a generalization of Kanamori’s weakly normal filters, which induces stationary reflection.
Normal restrictions of the noncofinal ideal on
We discuss the problem of whether there exists a restriction of the noncofinal ideal on that is normal.
Norms on possibilities. II: More ccc ideals on .
Note on a theorem of J. Baumgartner
OCA and towers in
We shall show that Open Coloring Axiom has different influence on the algebra than on . The tool used to accomplish this is forcing with a Suslin tree.
On a certain prewellordering
On a classification of denumerable order types and an application to the partition calculus
On a paper by Karel Prikry concerning Ulam's problem on families of measures
On a problem of Steve Kalikow
The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.
On a sharp estimation in the theory of binary relations on a finite set