A Critical Re-Examination of the Fundamental Basis of Measuring Infinite Sets (original) (raw)

Galileo found the idea of larger or smaller infinities impossible to comprehend, then unintentionally made them equal. Over 200 years later, Cantor insisted that his formulation of transfinite numbers was not arbitrary, but then unintentionally inherited Galileo's methodological error in the way one-to-one correspondence was used. Genuine one-to-one correspondence produces exact agreement between set size, cardinality, natural density, and probability measures. The error is subtle, and literally not visible, enabling it to evade detection for more than 380 years. It takes more than a few pages to unpick the detail, since every living mathematician has had to prove that some extremely low density (and some extremely high density) sets are countably infinite as a minor part of their undergraduate studies.