# Ultrafilter quantifiers and Hindman's theorem

A filter on is a consistent notion of largeness for subsets of . "Largeness" has the following properties.

• if is large and then is large
• if and are large then is large
• the empty set is not large

At most one of and is large; an ultrafilter is a filter which always has an opinion about which it is.

• either or is large

The usual notation for " is large" is , where is the ultrafilter. This casts ultrafilters as sets, rather than notions of largeness. To bring the notion of largeness to the foreground we can use ultrafilter quantifiers; we write , read "for -most , holds" (where we have also identified with the predicate "is a member of ").

• if and then
• if and then

From this point of view says that the set of elements with property is everything, and that the set of elements with property is non-empty. behaves like a mixture of and , with the considerable advantage that logical operations pass through unchanged without having to worry about De Morgan's laws.