continuum hypothesis

continuum hypothesis

1. (set theory) 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.
   • 2002, Steven G. Krantz, Handbook of Logic and Proof Techniques for Computer Science‎, Springer (Birkhäuser), page 65:

     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."

   • 2015, R. Bruce Elder, DADA, Surrealism, and the Cinematic Effect, Wilfrid Laurier University Press, page 582:

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

2. (physics, kinematics, continuum mechanics) The assumption, for the purposes of mathematical modelling, that the material being studied is a continuous mass rather than being composed of discrete particles.
   • 2002, C. S. Jog, Foundations and Applications of Mechanics, Volume 1: Continuum Mechanics, CRC Press, page 75:

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

   • 2006, Z. U. A. Warsi, Fluid Dynamics: Theoretical and Computational Approaches, 3rd edition, Taylor & Francis (CRC Press), page 1:

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

   • 2012, Clemens Chan-Braun, Turbulent Open Channel Flow, Sediment Erosion and Sediment Transport, KIT Scientific Publishing, page 7:

     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.

• (set theory): CH (abbreviation)
• (physics): continuum assumption

• generalized continuum hypothesis

• (set theory):
  • Cardinality on Wikipedia.Wikipedia
  • Aleph number on Wikipedia.Wikipedia
  • Beth number on Wikipedia.Wikipedia
  • Ω-logic on Wikipedia.Wikipedia
• (physics):
  • Continuum mechanics on Wikipedia.Wikipedia
  • Knudsen number on Wikipedia.Wikipedia