limit cardinal

A limit cardinal is a cardinal κ such that λ+<κ for every cardinal λ<κ. Here λ+ denotes the cardinal successor of λ. If 2λ<κ for every cardinal λ<κ, then κ is called a strong limit cardinal.

Every strong limit cardinal is a limit cardinal, because λ+2λ holds for every cardinal λ. Under GCH, every limit cardinal is a strong limit cardinal because in this case λ+=2λ for every infiniteMathworldPlanetmath cardinal λ.

The three smallest limit cardinals are 0, 0 and ω. Note that some authors do not count 0, or sometimes even 0, as a limit cardinal. An infinite cardinal α is a limit cardinal if and only if α is either 0 or a limit ordinalMathworldPlanetmath.

Title limit cardinal
Canonical name LimitCardinal
Date of creation 2013-03-22 14:04:40
Last modified on 2013-03-22 14:04:40
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Definition
Classification msc 03E10
Related topic SuccessorCardinal
Defines strong limit cardinal