Meaning of Hausdorff gap | Babel Free

Definitions

A pair of collections of integer sequences such that there is no integer sequence lying between the two.

Examples

“The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.^([Wikipedia])”

“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proof that Hausdorff gaps are indestructible under cardinal preserving extensions (property 5 above) are due to Kunen (unpublished).”

“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.”

“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.”

CEFR level