Dictionary

Hausdorff gap en · NOUN

Etymology

Named after German mathematician Felix Hausdorff (1868–1942), who published proof of the first example in 1909.

Meanings

1. A pair of collections of integer sequences such that there is no integer sequence lying between the two.
   • Let #92;langle#92;langlea#95;#92;epsilon#58;#92;epsilonlt;#92;omega#95;1#92;rangle#92;langleb#95;#92;epsilon#58;#92;epsilonlt;#92;omega#95;1#92;rangle#92;rangle be a Hausdorff gap. 1997, Winfried Just, Martin Weese, Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 118:
   • Of course, Hausdorff did not talk of models of set theory or prove absoluteness results. He gave a direct construction in ZFC of what is now called a Hausdorff gap and proved properties 1 through 4 for his construction. The notion of a Hausdorff gap and the proof that Hausdorff gaps are indestructible under cardinal preserving extensions (property 5 above) are due to Kunen (unpublished). 1995, Robert M. Solovay, “*1970 [Godel's ontological proof]”, in Kurt Gödel, edited by Solomon Feferman, John W. Dawson Jr., Warren Goldfarb, Charles Parsons, and Robert M. Solovay, Kurt Gödel: Collected Works: Volume III, Oxford University Press, page 419:
   • 2013, Ilijas Farah, Eric Wofsey, 3: Set theory and operator algebras, James Cummings, Ernest Schimmerling (editors), Appalachian Set Theory: 2006-2012, London Mathematical Society, Cambridge University Press, page 101, This family is one of the instances of incompactness of ω₁ that are provable in ZFC, along with Hausdorff gaps, special Aronszajn trees, or nontrivial coherent families of partial functions.
   • The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.^([Wikipedia])

Forms

Relateds

Rothberger gap

wikipedia: Felix Hausdorff