Solution of a problem of Ulam on countable sequences of sets
Some 2-point sets
Chad, Knight & Suabedissen [Fund. Math. 203 (2009)] recently proved, assuming CH, that there is a 2-point set included in the union of countably many concentric circles. This result is obtained here without any additional set-theoretic hypotheses.
Some additive properties of sets of real numbers
Some additive properties of special sets of reals
Some algebraic properties of weakly compact and compact cardinals
Some applications of game determinacy
Some applications of Sargsyan's equiconsistency method
We apply techniques due to Sargsyan to reduce the consistency strength of the assumptions used to establish an indestructibility theorem for supercompactness. We then show how these and additional techniques due to Sargsyan may be employed to establish an equiconsistency for a related indestructibility theorem for strongness.
Some applications of strong Lusin sets
Some automorphisms of natural numbers in the alternative set theory
Some Baire category type theorems for
Some cardinal characteristics of ordered sets
For ordered (= partially ordered) sets we introduce certain cardinal characteristics of them (some of those are known). We show that these characteristics—with one exception—coincide.
Some classes of idempotent functions and their compositions
Some combinatorial principles defined in terms of elementary submodels
We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese-Nation property of 𝒫(ω) studied in [6] already follow from SEP. We show that it is consistent that SEP holds while 𝒫(ω) fails to have the (ℵ₁,ℵ ₀)-ideal property introduced in [2]. This answers a question...
Some combinatorial problems, connected with product-isomorphisms of binary relations
Some combinatorial properties of ultrafilters
Some combinatorics involving ultrafilters
Some combinatorics involving ξ-large sets
We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε₀ and the notion of largeness is in the sense of Hardy hierarchy.
Some comments on infinite Boolean functions
We introduce infinite Boolean functions and investigate some of their properties.
Some comments on the paper by Artigue, Isambert, Perrin and Zalc
Some complexity results in topology and analysis
If X is a compact metric space of dimension n, then K(X), the n- dimensional kernel of X, is the union of all n-dimensional Cantor manifolds in X. Aleksandrov raised the problem of what the descriptive complexity of K(X) could be. A straightforward analysis shows that if X is an n-dimensional complete separable metric space, then K(X) is a or PCA set. We show (a) there is an n-dimensional continuum X in for which K(X) is a complete set. In particular, ; K(X) is coanalytic but is not an analytic...