Distance Graphs Whose Chromatic Number is Affected by the Underlying Set Theory

摘要

Much of mathematics, including Graph Theory, is built on the foundation of ZFC set theory. However there are other foundations and the choice of ZFC might be just a historical coincidence. In particular, ZFS foundation in many respects is "better" or as "good" as ZFC foundation: it allows a complete Lebesgue measure theory and hence analysis and physics might be built on ZFS foundation; but on the other hand it eliminates anomalies such as Banach-Tarsky paradox. The Shelah-Soifer class 5 of graphs illustrates how different mathematics could be if it were built on a different foundation. This provokes numerous questions, in particular: 1. What if we try to continue the Solovay's project and build analysis on top of ZFS? 2. How different this ZFS-analysis would be? 3. Would the physics from the point of .ZFS-analysis give better predictions?