The axiomatic melting point. Teaching probability theory in Prague during the 1930's.
The Banach-Tarski Paradox
The Banach-Tarski paradox for the hyperbolic plane (II)
The second author found a gap in the proof of the main theorem in [J. Mycielski, Fund. Math. 132 (1989), 143-149]. Here we fill that gap and add some remarks about the geometry of the hyperbolic plane ℍ².
The boolean prime ideal theorem holds iff maximal open filters exist
The boundedness principle in ordinal recursion
The cancellation law for pseudo-convolution
Cancellation law for pseudo-convolutions based on triangular norms is discussed. In more details, the cases of extremal t-norms and , of continuous Archimedean t-norms, and of general continuous t-norms are investigated. Several examples are included.
The cardinal equation 2m=m
The Cartesian product of sets and the Hessenberg natural product of ordinals
The characterization of -compact elements in some lattices
The combinatorics of reasonable ultrafilters
We are interested in generalizing part of the theory of ultrafilters on ω to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
The complexity of the set of squares in the homeomorphism group of the circle
The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.
The complexity of uniform distribution
The consistency of 𝔟 = κ and 𝔰 = κ⁺
Using finite support iteration of ccc partial orders we provide a model of 𝔟 = κ < 𝔰 = κ⁺ for κ an arbitrary regular, uncountable cardinal.
The consistency of some theorems concerning Lebesgue measure (Preliminary communication)
The consistency of the measurability of projective semisets
The consistency strength of the tree property at the double successor of a measurable cardina
The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the...
The covering number for category and partition relations on
We show that cov(M) is the least infinite cardinal λ such that (the set of all finite subsets of λ ) fails to satisfy a certain natural generalization of Ramsey’s Theorem.
The covering property for σ-ideals of compact, sets
The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of .
The degree sets of connected infinite graphs
The dependence of some logical axioms on disjoint transversals and linked systems