Assuming PFA, every uncountable subset E of the plane meets some C1 arc in an uncountable set. This is not provable from MA(א1), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C2 arcs, and the counter-example is a perfect set.

This problem belongs to an ever-increasing list of problems known to be unsolvable from the (usual) axioms of set theory. However, some of these problems have now been solved. But what does this actually mean? Could the Continuum Hypothesis be similarly solved? These questions are the subject of this article, and the discussion will involve ingredients from many of the current areas of set theo...

A uniform Roe corona is the quotient of algebra a metric space by ideal compact operators. Among other results, we show that it consistent with ZFC isomorphism between coronas implies coarse equivalence underlying spaces, for class uniformly locally finite spaces which coarsely embed into Hilbert space. Moreover, property A, equivalent to bijective some their cofinite subsets. We also find such...

This essay was inspired by conversations with mathematicians who maintain in all seriousness that there is something canonical about foundations of mathematics in ZFC (Zermelo-Fraenkel set theory with the axiom of choice). (These took place on the FOM (Foundations of Mathematics) mailing list [7].) One of the parties to this conversation ventured to define mathematics in terms of ZFC . The best...

Map theory, or MT for short, has been designed as an \integrated" foundation for mathematics, logic and computer science. By this we mean that most primitives and tools are designed from the beginning to bear the three intended meanings: logical, computational, and set-theoretic. MT was originally introduced in [18]. It is based on -calculus instead of logic and sets, and it ful lls Church's or...

We present a structure (in a finite relational type) with a unique undefinable element. Furthermore, the unique undefinable element is undefinable even if we use the quantifier "there exist infinitely many" and quantitative variants. We show that the structures can be taken to be a graph. We also prove that "there is a structure (in a finite relational type) with a unique second order undefinab...

This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Gödel’s incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradicto...