TL;DR
Under the PFA assumption, uncountable sets in the plane intersect uncountably many points of some smooth arc, but this is not provable in ZFC for less smooth arcs, with counterexamples existing for C^2 arcs.
Contribution
The paper establishes conditions under which uncountable sets in the plane intersect C^1 arcs uncountably, and shows the limitations of ZFC in this context for C^2 arcs.
Findings
Uncountable sets meet some C^1 arc in an uncountable set under PFA.
The result holds for analytic sets in ZFC.
Counterexamples exist for C^2 arcs, disproving the general statement in ZFC.
Abstract
Assuming PFA, every uncountable subset E of the plane meets some C^1 arc in an uncountable set. This is not provable from MA(aleph_1), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C^2 arcs, and the counter-example is a perfect set.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Computational Geometry and Mesh Generation · Advanced Topology and Set Theory