Let , that is, the set of subsets of of size . Then given some cardinals , , and
states that for any set of size and any function , there is some and some such that and for any , .
As an example, the pigeonhole principle is the statement that if is finite and then:
That is, if you try to partition into fewer than pieces then one piece has more than one element.
Observe that if
then the same statement holds if:
is made larger (since the restriction of to a set of size can be considered)
is made smaller (since a subset of the homogeneous set will suffice)
is made smaller (since any partition into fewer than pieces can be expanded by adding empty sets to the partition)
is made smaller (since a partition of where can be extended to a partition of by where is the smallest elements of )
is used to state that the corresponding relation is false.
Jech, T. Set Theory, Springer-Verlag, 2003
Just, W. and Weese, M. Topics in Discovering Modern Set Theory, II, American Mathematical Society, 1996
|Date of creation||2013-03-22 17:48:54|
|Last modified on||2013-03-22 17:48:54|
|Last modified by||Henry (455)|