Meaning of continuum hypothesis | Babel Free

Definitions

1. The hypothesis which states that any infinite subset of ℝ must have the cardinality of either the set of natural numbers or of ℝ itself.
2. The assumption, for the purposes of mathematical modelling, that the material being studied is a continuous mass rather than being composed of discrete particles.

Equivalents

Examples

“In other words, it is consistent with the axioms^([of set theory]) to suppose that the continuum hypothesis is false. (It had been known for some time — see below — that it is consistent with the axioms of set theory to suppose that CH is true.)”

“2013, Kenneth Reinhard, Introduction, Susan Spitzer (translator), Alain Badiou, The Incident at Antioch: A Tragedy in Three Acts / L’Incident d’Antioche: Tragédie en trois actes, Columbia University Press, page xxxiii, After all, when Paul Cohen sent his proof of the undecidability of the continuum hypothesis (which Badiou considers an “event”) to Kurt Gödel in 1963, Gödel wrote back that “reading your proof had a similarly pleasant effect on me as seeing a good play."”

“In the end, it turned out that Cantor was unable to prove his continuum hypothesis precisely because it is unprovable.”

“Under the continuum hypothesis, the material being studied, whether it be solid liquid or gas, is assumed to be continuous.”

“Because matter is composed of molecules, the continuum hypothesis implies that a very small volume will contain a large number of molecules.”

“A measure for when the continuum hypothesis can be considered appropriate is the Knudsen¹ number, Kn, which is defined as the ratio of a characteristic molecular length scale, e.g., the mean free path of molecules, and a characteristic length scale of macroscopic flow properties, e.g. a length scale of the smallest scales of fluid motion.”

CEFR level